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o
3.2.12
with xes) = o,s E [-r,O] where a,b,r are positive constants and
0
and in this sense solutions of (3.2.12) for r = 0 are nonoscillatory for t E [0,(0). We recall that by definition, the system (3.2.12) with r > 0 is said to be oscillatory, if every solution x(t) with 0 < x(O) f (a/b) is such that, x(t) - (a/b) has infinitely many discrete zeros for t E [0,(0). The equation (3.2.12) is said to be nonoscillatory, if it has at least one solution x(t) with < x(O) (a/b) such that [x(t) - (a/b)] has only a finite number of zeros on [0,(0). The proof of the next result is quite elementary.
°
t=
Theorem 3.2.3. Let a, b be positive constants and let r be a nonnegative constant. Then the system (3.2.12) is nonoscillatory on [0,(0) about the steady state (a/b) for all r ~ 0. Proof. If we let x*
= alb, it is then found that solutions of (3.2.12) satisfy
x(t) - x* = [x(O) - x*] exp [ - b
J.' xes - r)] ds
from which the conclusion is immediate.
[]
We consider next, the following system with a delay in production:
dx(t)
dt
(3x(t-r)
1 + xn(t _ r) -,x(t)
where r, (3, "I E (0,00). If we assume exists and satisfies
1 . "I 1 + (x *) n = fi
j
0.
I(t)
3.2.20
in which
Izl(t)=
Iz(s)/,
sup
t E [0,00).
sE[t-r,tJ
By a result of Halanay [1966, see also Lemma 1.4.6], there exist constants Cl > that ·3.2.21
o, 01 > 0 such If t E
h, then we have similarly,
~Iz(t I::; -1'1 z(t) 1+1' [n(l- j) -1]1> I(t), for which there exist positive numbers C2, 02 satisfying 3.2.22 From (3.2.21) and (3.2.22), we derive 3.2.23
t~O
where c = max( C1, C2), 0 proof is complete.
= mine 01 , 02).
The result follows from (3.2.23) and the []
We remark that the type of stability established in Proposition 3.2.4 is known as the "absolute" or "delay-independent" type. Such a type of stability can occur (with some exceptions) whenever there is a "dominant" negative feedback without delays. On the other hand, if there are delays in the negative feedback (destruction), the stability is usually delay dependent. This chapter will elaborate this aspect in considerable detail. Briefly let us now consider,
duet) --;u= -au(t -
it) + bu(t - 1'2);
t>O
3.2.24
181
§9.2. Delays in production
and rewrite (3.2.24) for t > 2(11
d~(t) t
+ 12) as follows:
= -au(t) +ajt
[-au(s -
II)
+ bu(s -
12)]ds
t-Tl
3.2.25
+ bu(t - 12) One derives from (3.2.25) that
d~~t)
::; -au(t) + [( a2 + al b DI + I b I] £l(t)
£l(t) =
sup
sE[ t-2( Tl +T2 ),t 1
u(s);
3.2.26
1"=11+1"2
from which one concludes
a > [(a 2 +albl)l+
Ibl] => u(t) ~ 0 as
t ~
00.
3.2.27
Note that if the delay II is large, then the condition (3.2.27) can fail. While the condition a > [( a2 + al b 1)1 + I b IJ in (3.2.27) is not necessary, for the conclusion of (3.2.27), it is possible by methods of Chapter 2 to show the loss of stability of the trivial solution for large enough delay I I in (3.2.24). Note also that if I I = 0 and a > 1b I, then the size of the delay 12 does not matter, as far as the convergence of u( t) ~ 0 is concerned. The rate at which u( t) ~ 0 as t ~ 00 is usually delay dependent. The following are some examples of systems with delays in production. It is left to the reader to show that solutions corresponding to positive initial values remain positive for all t > 0, and to examine the local asymptotic stability of the positive steady states.
[1< -
dN(t) = r N(t _ I) N(t)]. dt 1 + cN(t)
d~?) dN(t) dt
= -rN(t)
(1)
+ e--yN(t-T).
(2)
= -rN(t) + bN(t _ l)e--yN(t-r).
dN(t) = -rN(t) dt
+ aN(t)[N(t a + j3[N(t -
dN(t) _ -r N(t) dt -
1")Jm. I)Jm
a
+ a + [N (t -
I)] n
.
(3)
(4) (5)
§3.2. Delays in production
182
dx(t) dt
= ax([t _
dx(t) _ ([ t _ - -rx dt dx(t) dt
m
])
= -rx(t) + e-iX(t)
'
m)) _ bx 2 (t).
(K1 +-cx([t x([t - m])) - mJ) x(t) =
dx(t)
=
dx(t)
dx(t)
dt
= [a
= [a
sup
x( s).
a
1
+ bx(t -
+ bx([t -
(8)
(9)
mJ)]n
(1= K(s)x(t - s)ds) (a - bx(t)).
d:~t) = a dt
(7)
.
sE[t-r,t]
dt = -rx(t) + a + [x([t -
d~~t)
(6)
(10)
00
K(s)x(t - s)ds - bx 2 (t).
r)] [c - x(t)],
(11)
a,b,c,E (0,00).
m])] [c - x(t)x(t - r)],
mEN.
(12) (13)
3.3. Competition and cooperation
Let Xl (t) and x 2 (t) denote the population densities (or biomasses) of two species competing for a common pool of resources in a temporally uniform environment. For an extensive discussion of the processes of competition we refer to the article by Miller [1976]. Let bi and mi (i = 1,2) denote the respective density dependent birth and death rates so that in the absence of time delays the population densities are governed by
3.3.1
In order to make the system (3.3.1) denote a model of competition of the "interference type" (see Miller [1976], Brian [1956]), we make the following assumptions on the birth and death rates.
183
§3.3. Competition and cooperation
(i) bi, mi (i = 1,2) are continuous with continuous partial derivatives for all Xi 2: o(i = 1,2); also we assume
8b o· -8 >, i
for
Xi
Xi> 0;
i,j = 1,2;
3.3.2
3.3.3 (iii) for some
xi > 0, xi > 0 we have b1(xn - m1(x~,0) = 0 b2 ( xi) - m2(0, xi) = 0;
3.3.4
(iv) there exist positive constants b1, b2 such that
b1(bt) - m1(bl,X2) < 0
3.3.5
b2(b2) - m2(xl,b2) < 0 (v) there exist numbers
0:
> 0, 13 > 0 such that bi (0:) b2 (fJ) -
=0 m2( 0:,13) = O. m1
(0:,13)
3.3.6
The conditions on bi in (3.3.2) mean that the birth rates are positively density dependent and any crowding effects acting negatively on the birth rates are included in the death rates; the assumptions on mi indicate intraspecific and interspecific competition. The equations (3.3.3) imply that (0,0) is ~ trivial steady state of the system (3.3.1) as it is customary in all models of population ecology; (3.3.4) means that in the absence of anyone species, the other has a positive steady state. The inequalities (3.3.5) will imply, that each of the species cannot grow to unbounded levels since
dXi(t) < 0 dt - . The equations (3.3.6) will guarantee the existence of a positive steady state (0:,13) for the system (3.3.1).
§3.3. Competition and cooperation
184
A simple example of (3.3.1) - (3.3.6) is the familiar Volterra-Lotka model system described by
3.3.7
where
ri, aij
(i,j
= 1,2) are positive constants satisfying either or
all
Tl
al2
aZl
TZ
a22
- 0, will it follow that the corresponding solutions of (3.3.1) continue to remain nonnegative for all t 2: O. On the (Xl,XZ) state space of (3.3.1) we have
(0,0), (x;,O), (O,x;), (a,;3) as possible steady states and a plausible question is whether (3.3.3) will lead to the invariance of the boundaries Xl = 0 and X2 = of the state space; such an invariance will imply that if Xl (0) > 0, xz(O) > 0, then the population trajectory (Xl(t), xz(t)) of (3.3.1) cannot reach the outside of the nonnegative quadrant Xl 2: 0, X2 2: O. These questions are easily answered for (3.3.7); we will discuss these aspects for a system of the form (3.3.1) with time delays.
°
Let us first derive a set of sufficient conditions for the local asymptotic stability of the positive steady state (a, (3) of (3.3.1) and subsequently show that the same set of conditions are also sufficient to maintain such a stability even if there are time delays in production (or recruitment) and destruction by competing species. Local asymptotic stability of (a,;3) is easily examined by an analysis of the associated linear variational system in the perturbations Xl, X z where
such a linear variational system is found to be
3.3.8
185
§9.9. Competition and cooperation
where (.J;)" p.
__
[8b i OXj
_
8m i ]
evaluate d at
({.J a, fJ),
i,j = 1,2.
3.3.9
OXj
The steady state (a, (3) of (3.3.1) is locally asymptotically stable, if the trivial solution (0,0) of (3.3.8) is_asymptotically stable. We can derive the following result from elementary considerations;
"In the system (3.3.1)-(3.3.6), if the following holds 8m}
obI
am2
8m2
8bz
amI
> -ax!+ -ox} OX1
3.3.10
- >aX2 -+OX2 aX2 ' then the steady state (a, (3) of (3.3.1) is (locally) asymptotically stable." The proof of the above result is easy, if we note that the characteristic equation associated with the linear system (3.3.8) is given by
/311 -(j12 ) -0 -/321 ,\ - /322 -
det (,\ -
3.3.11
or equivalently 3.3.12 and the roots of (3.3.11) will have negative real parts implying the asymptotic stability of the trivial solution of (3.3.8). If we apply the condition (3.3.10) to the Volterra-Lotka model (3.3.7), then (3.3.10) leads to
which together with a > 0, (3
> 0 lead to
It is known, that if a steady state with positive components exists for (3.3.7), then allan -a12a21 > 0 is a sufficient condition for the global asymptotic stability of (a,{3) in (3.3.7). In fact, using a function V(Xl,X2) defined by
§9.9. Competition and cooperation
186
with suitable positive constants Cl, C2, it is possible to show that (0',13) will be globally asymptotically stable for (3.3.7) whenever alla22 - a12a21 > 0; details of this verification are left to the reader. The following is an interpretation of (3.3.10); the intraspecific negative feedback effects on the i-th species ~, (i = 1,2) dominate its own positive feedback as well as its influence on its competitor ~r;:.i i =1= j . A detailed discussion of this can be found in the article by the author (Gopalsamy (1984bJ). ]
,
We proceed to an examination of the dynamics of a system of two competing species with sufficiently strong intraspecific negative feedbacks and with delays in production (recruitment or birth rate) and destruction by competitor species. We have seen in Chapters 1 and 2 that delays in intraspecific negative feedbacks can render otherwise stable systems oscillatory. Delays in production are more general and common in most biological populations. We shall now formulate a competition model with delays in production and destruction. Let Tij (i,j = 1,2) be a set of nonnegative constants with r = max{rij li,j = 1, 2} and suppose that the two competing species display delayed reproduction and interspecific interaction while the intraspecific interactions involve no time delays. Such a competition system in a constant environment can be modelled by an autonomous delay differential system of the form
dx~y)
= bl (XI( t - TU)) - ml (Xl (t), X2(t - T'2))
dX~t(t)
= b2 (X2(t - Tn») -
3.3.13 m2
(Xl(t - T21), X2(t»)
in which the birth rates bI , b2 and the death rates ml, m2 satisfy the same conditions as in (3.3.1). Along with (3.3.13) we suppose that the initial population sizes are specified by the following:
Xi(S)
= (Pi(s) > 0,
¢iEC([-r,O],IR+),
S E [-r, 0]; ¢i~O
i
on
= 1" 2' [-r,O],
r =
max r"
1 ~i,j~2
I)
3.3.14
i=1,2.
Since (3.3.13) - (3.3.14) are not of the Kolmogorov-type, we have to verify that the solutions of (3.3.13) - (3.3.14) will remain nonnegative so long as such solutions are defined. Let {~~n>c s)} , s E (-r, 0], i = 1,2, (n = 1,2,3, ... ) be a sequence of strictly positive continuous functions such that (in a pointwise sense)
1im~;n)(s)=¢i(S) SE(-T,O] \
n--+oo
i=1,2.
3.3.15
181
§9.9. Competition and cooperation
Let {xin\t) ,x~n)(s)} be the solution of (3.3.13) corresponding to the initial condition Consider the solution
n
where 7* is the positive minimum of 7ij, i,j = 1,2. Suppose xi )(t) does not remain positive for all t E [0,7*]; then there exists a t* in (0, 7*J for which
It will follow from (3.3.13) and the positivity of the initial condition that
dx~n)(t*) dt
::::: b (x(n)(t* 1
1
7
11
») > 0
3.3.16
where we have used the properties of bI and ml; it is found that (3.3.16) contradicts the definition of t* and thus we have
Similarly,
We can repeat the above procedure for intervals of the form [7*,27*], [27*,37*] etc. n Thus, it will follow that so long as (xi )(t), x~n) (t) is defined, we have
If we consider the limit as n -+
00,
we get 3.3.17
with Xl(t) ~ 0,X2(t) ~ 0 where {Xl(t),X2(t)} is the solution of (3.3.13) - (3.3.14) and this is a consequence of the continuous dependence of solutions on ini tial conditions (Hale [1977], p.41). A second question for (3.3.13) - (3.3.14) is concerned with the existence of solutions of (3.3.13)-(3.3.14) defined for all t 2: O. Suppose now a solution of
§3.3. Competition and cooperation
188
(3.3.13) does not exist for all t 2: 0;
one of either
Xl
then there is a t1 > 0 such that for at least
or X2, we have lim Xi(t) =
i = 10r2.
00,
t-+tl-
3.3.18
To be specific let us suppose that limt_tl- Xl (t) = 00; then let t2 be the first time for which Xl(t2) = 51, t2 < t l . It will follow from (3.3.13) that
dXI(t Z) ( ) ----;u-=b 1 Xl(tz-Tll)
-ml
( Xl(t 2),X2(t z - T12) )
») - m, (XI(t2),X2(t2 - T!2»)
< h, (XI(t 2 < bl (51 )
-
ml
(5 ,xz(t 1
2 -
TIZ))
0 such that sup 8~O
{I Xl(S) - ex 1+1 X2(S) - ,BI} S b =}
I Xl(t) - ex I + I X2(t) -,8 l-t 0
as t
-t 00.
Theorem 3.3.2. Assume that (3.3.10) holds. Then (ex, (8) of (3.3.23) is (locally) asymptotically stable. Proof. The linear variational system corresponding to (ex,,B) in (3.3.23) is found to be dX 1 (t) --;u= -mllX 1 (t) + bll 0 kll(S)Xl(t - s)ds
T
dX2 (t)
--;u- =
/.00
- m12
/.00 k12(S)X2(t _ s)ds
-m22 X 2(t) -
roo m2110 k21 (S)X (t -
+ b22 /.00 k22(S)X2(t _
1
s)ds.
3.3.24
s)ds
§3.3. Competition and cooperation
191
A Lyapunov functional v for (3.3.24) is given by
vet)
= V(t,X 1 ,X2) = IX1 (t) I + IX2(t) I + bl l
/,= kll(s ){L,
[XI (u) [du } ds
+b22/,= k,,(S){L.[X,(U) [dU} ds + m21 + ml
t'
3.3.25
k21 (S){L,[ XI(u) [du} ds
'/,= k
12
(S){L,[X,(U) [du }ds.
The remaining details of proof are left to the reader as an exercise.
[]
We shall consider the dynamics of cooperation (or mutualism) between two species. Such an interaction can be modelled by a system of the form
3.3.26
where
II, h
are continuously differentiable such that
An example of such a model is
3.3.27
where ri,
Ki,
ai
E (0,00)
and
(¥i
> Ki ,i = 1,2.
3.3.28
Depending on the nature of Ki , (i = 1,2), the mutualism model (3.3.27) can be classified as facultative, obligate or a combination of both. For more details of mutualistic interactions we refer to Vandermeer and Boucher [1978], Boucher et al. [1982], Dean [1983], Wolin and Lawlor [1984] and Boucher [1985].
§J.J. Competition and cooperation
192
The system (3.3.27) has a unique positive equilibrium N*
= (Ni, N z) satis-
fying 3.3.29
It is possible to show by means of phase plane methods or by an application of Kamke's Theorem (Coppel [1965}i more details of Kamke's Theorem will appear in the next chapter where we will study the global behavior of the system (3.3.27» that solutions of (3.3.27) satisfy the following:
N2(t)
N;
-+
as
t
-+ 00.
3.3.30
We are interested in the study of the following time lagged model corresponding to (3.3.27);
dN 1 (t) _ dt
--- -
°
dNz(t) --dt
N ()
lIlt
= 12 N 2 (t )
[Kl + a1Nz(t 1 + N2(t -
[K2 + a2 N l(t 1 + NI (t -
TZ)
-
N 1 (t )]
T2)
3.3.31
Tl) - N 2 ( t )]
Td
where Tl 2:: 1 T2 2:: 0, T1 + T2 > 0. The system (3.3.31) means that the mutualistic or cooperative effects are not realized instantaneously but take place with time delays. We shall show that the steady state N* of (3.3.31) is linearly asymptotically stable irrespective of the sizes of the delays Tl and T2. We associate with (3.3.31) initial functions of the form 3.3.32 where
iEC([-Ti,O],R+) and i(O»O for
i=1,2.
Lemma 3.3.3. The initial value problem (3.3.31) - (3.3.32) has a unique solution which exists for all t 2: 0, is positive (componentwise) and is uniformly bounded for t 2: o.
Proof. The solutions of (3.3.31) and (3.3.32) exist uniquely on an interval of the form [0, T) for some T > 0 and remain positive. Let [0, T) be the maximal interval of such an existence. We have from (3.3.31),
liNi[]{i - Ni(t)] ~ Ni(t) ~ liNdai - Ni(t)] for
0~t
< T.
3.3.33
193
§3.3. Competition and cooperation
This implies that Ni(t) is nondecreasing as long as Ni(t) ~ Ki and nonincreasing as long as Ni(t) ~ ai. An implication of this is that Ni(t) is positive and bounded for all t E [0, T). Hence T = 00, and this completes the proof. [] Let us examine the linear asymptotic stability of N* of (3.3.31). We let
Ni(t) = Nt
+ Xi(t)
i
for
= 1,2
and linearise the system (3.3.31) around N* ; such a linear system is
3.3.34
The characteristic equation associated with (3.3.34) is
3.3.35
The steady state N* of (3.3.31) is said to be linearly and "absolutely" stable in the delays if the trivial solution of (3.3.34) is asymptotically stable for all 71 2:: 0) 72 ~ O. "Absolute" stability corresponds to the fact that delay induced stability switches (see section 3.7 below) cannot occur.
Theorem 3.3.4. Assume that (3.3.28) holds. Then the positive steady state N* of (3.3.27) is linearly asymptotically stable absolutely in the delays. Proof. It is sufficient to show that for all values of 71 2:: 0, istic equation does not have roots of the form
,.\ = a + i/3 We replace ,.\ in (3.3.35) by a leading to a 2 - {32
with
a
72
2:: 0, the character-
2:: O.
+ i{3 and then separate the real and imaginary parts
+ aQ + R = _SRe- ar COS{3r} 2a{3 + flQ = SRe- ar sinfl7
3.3.36
§3.3. Competition and cooperation
194
where 7"
= 7"1 + 72, Q = T"lN: +T"2N;, R = T"lr2N;N; S = (a1 - KJ)(a2 - K 2) . (1 + Ni)2(l + N;)2
Squaring both sides of (3.3.36 ) and then adding, we obtain
(a 2 + 13 2)2
+ a 2Q2 + R2(1- s2 e-2aT) + [2aQ + ri(N:)2 + T"2(N;?]f3 2 + 2a(a 2Q + aR + QR) = O.
3.3.37
We first claim 3.3.38
O<S 0 8Y2 for Y1 2: 0,
812 >0' 8Y2 - ,
Y2 2: 0
=
3.4.2
892 > 0 8Y2 -
(ii) fiCO) = OJ
91(0,yZ) == 0
12(0, Y2) == 0;
12(y,O) == 0;
(iii) there exists a number
3.4.3 92(0) = 0
Y; > 0 such that 3.4.4
(iv) there exists a pair of constants a*, /3* > 0 such that
!I(a*) - 91(a*, /3*) = 0
3.4.5
f2( a*, (3*) - 92((3*) = OJ (v) there exist positive constants
T}t, T}z such that
fI(T}t) - 91(T}t, Y2) < 0; fI(T}!, T}2) - 92(T}2) < O.
Yz
2: 0,
3.4.6
§3.4. Prey-predator systems
198
The conditions (3.4.2) mean that the system (3.4.1) is of the prey-predator typej (3.4.3) means that (0,0) is a trivial steady state of (3.4.1)j (3.4.4) implies that, in the absence of the predator, the prey species has a nontrivial (positive) steady state; (3.4.5) shows that (3.4.1) has a nontrivial steady state in the interior of the positive quadrant of the state space of (3.4.1); (3.4.6) will imply (see below) that no species can grow unbotmdedly. An example of (3.4.1) - (3.4.6) is the familiar Volterra-Lotka prey-predator system
where ri, aij (i,j state to exist)
= 1,2) are positive constants such that (fora nontrivial steady
As in the case of our analysis of two species competition, we have to verify that when Yl(O) > 0, Y2(0) > 0, the corresponding solutions of (3.4.1) will be nonnegative for those t 2:: 0 for which they exist. Since the coordinate axes Yl = 0 and Y2 = 0 are invariant sets for (3.4.1) by virtue of (3.4.3), it will follow that the solutions of (3.4.1) starting from the interior of the positive quadrant cannot enter the outside of that quadrant; this is a consequence of the invariance of the coordinate axes for (3.4.1). In the following, we suppose that Yl (0) and Y2(0) satisfy 0 < Yl (0) < 1]1 and 1]2; (on the otherhand) if ei ther one of Y1 (0) 2:: 1]1 and Y2 (0) 2:: 1]2 or both hold, it will follow that the corresponding Yi(t) will decrease as t increases by virtue of (3.4.6) at least for small t > O. Suppose Y1(t) is not defined for all t 2:: OJ then for some tl < 00, t~~ _ Y1 (t) = 00 and let t2 > 0 be the first time for which Yl (t2) = 1]1; we will. have from (3.4.6) and (3.4.1) that
o < Y2 (0)
0;
3.4.7
where all the partial derivatives in (3.4.7) are evaluated at (a*, (3*). The steady state (a*, (3*) of (3.4.1) is asymptotically stable, if the trivial solution of (3.4.7) is asymptotically stable and this will be the case, if the roots of the characteristic equation associated with (3.4.7) given by
det. (A - (Ill - 911) - 121
912 ) - 0 A - (122 - 922) -
3.4.8
where
09i 9ij = -0 ;
at
(a*,{3*)
Yj
have negative real parts. If AI, A2 are the roots of (3.4.8), then
Al
+ A2 = (Ill
- 911) + (h2 - 922)
AIA2 = (ill - 911)(122 - 922) + h2I2l . The following result is an immediate consequence of (3.4.7) - (3.4.8).
3.4.9
200
§J.4. Prey-predator systems
Theorem 3.4.1. In the prey-predator system (3.4.1) - (3.4.6) assume that the
partial derivatives satisfy the following additional conditions:
evaluated at
(a*, (3*);
3.4.10
then the steady state (a*, (3*) is (locally) asymptotically stable for the preypredator system (3.4.1) The literature on prey-predator model systems involving time delays is quite extensive. Most of the models have been derived from the Kolmogorov-type systems with time delays incorporated in the average growth rates of the prey and predator species. Volterra [1931] has proposed the following system of integradifferential equations for a prey-predator model system
3.4.11
where TI, r2, 51 ,52 are positive constants and FI , F2 are nonnegative continuous delay kernels suitably defined on [0,(0). The equations (3.4.11) do not contain negative effects of predator crowding. Brelot [1931] has cosidered a modified format of (3.4.11) in the form
3.4.12
which incorporates crowding effects typified by the positive constants (AI, A2)' Under suitable conditions on the various parameters in (3.4.12), one can show that (3.4.12) has a (locally) asymptotically stable steady state. We will consider in detail a system more general than (3.4.12) in the next chapter. A number of integrodifferential equation models of the type (3.4.11) and (3.4.12) have been investigated by Cushing [1977].
§9.4. Prey-predator systems
201
Starting from Hutchinson's [1948) delay logistic equation, May [1973) has proposed the following system
[1 _NI(tK- T)]_ aNI (t)N2(t)
dNt(t) = rNI(t) dt dN2 (t) -;u- = -bN2(t)
3.4.13
+ f3N2(t)NI(t)
where r, T, k, a, (3, b are positive constants; (3.4.13) contains a single discrete delay; one can modify (3.4.13) and incorporate a continuously distributed delay in it so that 3.4.14
The first model of a prey-predator system which departs from the Kolmogorov-type formulation is due to Wangersky and Cunningham [1957] who have proposed 3.4.15 where aI,a2,b l ,q,c2,T are positive constants; (3.4.15) means that a duration of time units elapses when an individual prey is killed and the moment when the corresponding increase in the predator population is realised. Cushing [1979J has considered a model of the form
T
dNI(t) = rNI (t)[l- N 1 (t) - aN2(t)] dt K
+
dN (t) = -6N (t) 2
+ bN1 (t)
1= 0
3.4.16
(3(a)N2(t - a)e- 6a da
where r, k, a, 6, b are positive constants and (3 is related to age dependent fecundity of the predator species. With the models (3.4.11) - (3.4.16) in the background, let us consider a time delayed model in the spirit of (3.4.1);
dNI(t) = It ( N 1 (t -;u-
T11) ) - gl ( NI(t),N2(t - T12) ) 3.4.17
dN2 (t) = -;u-
fz ( N 1(t - T21),N2(t - T22) ) - g2 ( N2(t) )
§3.4. Prey-predator systems
202
where tij are nonnegative constants with 7 = 1;:a.:X: 0, 8N > 0, h(NI,N2 ) > I
2
large enough
N1
°
for
N2
>
°
and
3.4.19
> 0.
It is not difficult to show by the methods used before for the competition system
°
that whenever NI(s) > 0,N2 (s) > for s E [-7,0], solutions of (3.4.18) exist for all 7 ~ and remain nonnegative for an7 ~ 0. Also (a*, (3*) of (3.4.5) is a steady state of (3.4.18). The proof of the following is similar to that of theorem 3.3.1 and hence we omit the details of proof.
°
Theorem 3.4.2. In the prey-predator model (3.4.17), let the conditions (3.4.2)(3.4.6), (3.4.18), (3.4.19) hold for the birth and death rates. Furthermore, if the conditions (3.4.10) hold, then for all nonnegative delays tij in (3.4.17), the nontrivial steady state (a* ,(3*) of (3.4.17) is (locally) asymptotically stable. An alternative to the system (3.4.17) is an integrodifferential system of the form
3.4.20
under appropriate conditions on the nonnegative delay kernels kij (i,j Details of further analysis of (3.4.20) are left to the reader.
=
1,2).
203
§9.4. Prey-predator systems
The prey-predator model systems (A) to (D) listed below have been investigated by Nunney [19S5a, b, c]; N denotes the predator density and R denotes the resource (or prey) density:
d~it)
= N(t)F(R(t)) - N(t)M(R(t))
d~;t) ~ B(R(t)) -
d~;t) = d~;t)
(A) D(R(t)) - N(t)G(R(t)).
N(t) [F(R(t)) - M(R(t))]
d~(t)
I
.
(B)
= B(R(t - T)) - D(R(t)) - N(t)G(R(t)).
d~?) = N(t _ T)F(R(t d~;t)
}
T)) - N(t) M(R(t)) }
(C) = B(R(t)) _ D(R(t)) - N(t)G(R(t)). = N(t _ T)F(R(t - T)) - N(t)M(R(t))
}
(D)
t
d~;t) = B(R(t -
T)) - D(R(t)) - N(t)G(R(t)).
The following are examples of models of one prey and one predator systems in the absence of delays; the interested reader should formulate appropriate models with various time delays (such as discrete, continuous, piecewise constant etc.).
dH(t) = rH(l- H) - aHP dt I{
d~;t)
= -bP
I
+ j3 H P.
d~;t) =rH(l- ; ) _aP(l_e- CH ) dP(t) dt
(1)
I
(2)
= -bP + j3P (1 _ e- cH ) .
dH( t) = r H dt
(1 - H) - +HI
d~; t)
[1 - ~].
= bP
K
aP j3 H
(3)
204
§S'.4. Prey-predator systems
(4)
(5)
I
dH(t) = rH[K - H]_ aHP dt 1 + cH j3 + H
(6)
dP(t) =p[-f3 ~-bP]. dt + f3 + H dH(t) = rH[K - H]_ aHP dt 1 + cH
d~~t) dH(t) dt dP(t)
---;It
=
I (7)
p[ _a +bH - CP]_
= aH -
I
bHP - €H 2
1 + aH
bHP
2
(8)
= -cP + 1 + aH - T}P .
3.5. Delays in production and destruction One of the techniques for the analysis of local asymptotic stability of steady states in autonomous delay-differential equations is based on an examination of the roots of the characteristic equation associated with the corresponding linear variational systems. As one can see from the following, that such a method based on the characteristic equation is quite difficult and often is an analytically almost impossible task if the system has several delay parameters; a reward for such a task is, however, that one can derive necessary and sufficient conditions for local asymptotic stability. In the case of ordinary differential equations, a stability analysis based on the characteristic equation is almost trivial due to the availability of the Routh-Hurwitz criterion. We have already considered several techniques based on Lyapunov functionals and we will consider other related techniques in the next section.
§3.5. Delays in production and destruction
205
For purposes of our illustration we first consider the system
dx(t)
dt = x(t)f{x(t), yet - T)} dy(t)
at =
3.5.1
y(t)g{ x(t - r), yet)}
in which T is a nonnegative constant, f and 9 are continuously differentiable in their arguments. Suppose there exists a point (x*, y*) , x* > 0, y* > 0 such that
f(x*,y*) = 0 = g(x*,y*). The local asymptotic stability of the steady state (x*, y*) of (3.5.1) is studied by an analysis of the asymptotic behavior of the related variational system obtained from (3.5.1) by setting
x(t) = x*
+ X(t),
yet) = y*
+ Yet),
and neglecting the nonlinear tenns in the perturbations X and Y so that
dX(t) -;It = x* fxX(t) dY(t) -;It
+ x* fyY(t -
r)
3.5.2
= y*gxX(t -
T)
+ y*gyY(t)
where the partial derivatives fx,fy,gx,gy are evaluated at (x*,y*). We formulate our result as follows:
Theorem 3.5.1. If the coefficients of the system (3.5.2) are such that
fx(x*,y*) < 0, Ifx( x*, Y*)1 > /gx( x*, Y*)I,
gy(x*, y*) < 0, Igy(x*,y*)1 > Ify(x*,y*)/,
3.5.3
tben for any r ?: 0, tbe trivial steady state (0,0) of (3.5.2) is asymptotically stable. Proof. First let r = 0 in (3.5.2); one can show that (3.5.3) will imply that the steady state (0,0) of (3.5.2) is asymptotically stable. Now, assume r > 0 be fixed and for convenience let -a = fx( x*, Y*)j
-b=fy(x*,y*)
-c = gx(x*, Y*)j
-d = gy(x*, y*).
§3.5. Delays in production and destruction
206
Consider any solution of (3.5.2) in the form
[X(t)] Yet)
=
[A]
zt
B e
where A, B, z are constants (not necessarily real), satisfying the system of equations (z + ax*)A + bx*e- zr B = 0 3.5.4 cy*e- zr A + (z + y*d)B = o. A necessary and sufficient condition for the existence of nontrivial solutions of (3.5.4) is that the constant z in (3.5.4) satisfies the characteristic equation z + ax* det. [ cy*e- ZT
bx*e- ZT ] z + y*d
=0
3.5.5
or equivalently Z2
If we let Z
+ z(az* + dy*) + adx*y* -
bcx*y*e- 2zr
= O.
3.5.6
= 2Z7 in (3.5.6), we can rewrite (3.5.6) in the form (Z2 +pZ + q)e z
+r =
O.
3.5.7
with
+ Y*d)} q = adx*y*47 2
p = 27(ax*
r = -bcx*y*47
2
3.5.8
•
To investigate the nature of the real parts of the roots of (3.5.7) we use Theorem 13.7 from Bellman and Cooke [1963, pp. 443-444]. In order to apply this theorem we let 3.5.9 H(Z) = (Z2 + pZ + q)e z + r and note that a necessary and sufficient condition for all the zeros of H(Z) to have negative real parts is that 3.5.10 F(w)G'(w) > 0 at all the roots of G( w)
H(iw)
= 0 where = F(w) + iG(w)
and
wE (-00,00).
3.5.11
207
§9.5. Delays in production and destruction
From (3.5.9) and (3.5.11) we derive,
F( w) = (q - w 2 ) cos W
-
pw sin w + r
3.5.12
G(w) = (q - w )sinw + pwcosw. 2
It is known (see Bellman and Cooke [1963], p.447) t-hat all the roots of G(w) = 0 are real. Let Wj (j ~ 0,1,2, ... ) denote the zeros of G(w) with Wo = 0. For Wo, (3.5.10) demands that
F(O)G'(O) = (r + q)(p + q) >0.
3.5.13
With a simple computation, we obtain that the nonzero roots of G( w) roots of cot w = (w 2 - q)/wp
=
°are the 3.5.14
and hence for such nonzero roots of G( w) = 0, we have
F(w) =
r_(s~:) [(W' _q)' + w'p']
G'(w) = -
(S~pw) [(w' - q)' + w'(p' + p) + pq]
3.5.15
from which it will follow that the sign of F( w )G' (w) is the same as that of
L(w) = (S~pw)' [(w' _p)' + w'p'] _r (S~pw)-
3.5.16
(3.5.14) and (3.5.16) together imply that
3.5.17 Since
Irl
0. Thus by theorem 13.7 of Bellman and Cooke [1963], a necessary and sufficient condition for all the roots of (3.5.9) to have negative real parts is,
Irl < q,
p
> 0,
q ~ 0.
It is easily seen from (3.5.3) that
Ir/- q = (Ibcl- ad)4r 2 x*y* < 0,
208
§3.5. Delays in production and destruction
and therefore the trivial solution of the variational system (3.5.2) is asymptotically stable and the proof is complete. 0 For more details related to the result of Theorem 3.5.1 and an estimation of the rate of convergence of solutions o~ (3.5.2) to the trivial solution we refer to Gopalsamy [1983a] where examples can be found. For a mathematical analysis of physiological models with time delays in production and destruction, we refer to the articles of an der Heiden [1979J and an cler Heiden and Mackey [1982]. Let us consider a system somewhat more general than (3.5.1); let 1,2) be a set of nonnegative constants and consider the system dx(t) = -;u-
x(t)i ( x(t -
7'11),
yet -
7'12)
7'ij
(i,j ::::
)
3.5.18 dy(t) = y(t)g ( x(t ---;It
) 7'2J), yet - 7'22)
in which i and g satisfy the same conditions as in Theorem 3.5.1. Note that our analysis above corresponds to (3.5.18) with 7'11 = 0 and 7'22 = 0 and 7'12 = 7'21. Hence, let us suppose at least one of 7'11 , 7'22 is not zero. IT we let
x(t) :::: x*[l
+ X(t)]
yet) :::: y*[l + Yet)]
in (3.5.18), then the linear variational system in X, Y is of the form dX ( t) :::: allx * X ( t ~
-
7'11
) + a12Y * Y ( t -
7'12
)
3.5.19
dY(t) ~
::::
* ( a21x X t -
) *y( t --7'22 ) 7'21 + a22Y
where all, al2, a21, a22 denote the partial derivatives ix, i y , gx, gy respectively evaluated at (x*, y*). The following questions are of interest for (3.5.19); (i) ifthe trivial solution of (3.5.19) is asymptotically stable in the absence of delays, will it continue to be so for all delays; (ii) is there a threshold value for the delay parameters so that (3.5.19) can become unstable, if (3.5.19) is stable in the absence of delays; that is, can an estimate on the delay parameter be obtained for stability to hold; (iii) if the system (3.5.19) is unstable in the absence of delays, will it remain unstable for all delays or it will switch to stability; (iv) will the system exhibit stability switches, i.e. switch from stability to instability and back to stability and so on? In the next two sections we investigate certain aspects of the above questions for
§9.5. Delays in production and destruction
209
a general linear system with a single as well as several different delays. Usually linear analyses of models with delays in production and destruction lead to equations of the form (3.5.19) and their integrodifferential analogues; a characteristic of such systems is that they need not necessarily have terms without delays. The following are some examples of models with delays in production and destruction:
d~~t)
= rN(t _ r1)
[1 - N(t)Nl! - r z )].
dN(t) = rN(t) [K - N(t - 7"z)]. dt 1 + eN (t - r2)
d~it)
= J.=X(S)N(t-s)d+-N(t) J.= H(S)N(t-s)dsj.
dx(t) _ f3x(t - r) _ () ( _ ) ( ) 'Yx txt r. 1 + xn t - 7" dt dN1(t) dt dNz(t) dt
= N 1(t-rd []{1 +a1N z(t- r2) -Nl(t)]) 1 + N 2 (t - rz) . = Nz(t _ 7"4) [Kz + aZN1(t - 7"5) - Nz(t)]. 1+N1 (t-7"s)
d~?) = rN(t -
r)[l- Nit) - eUCt)])
duet)
aT =-au(t) + bN(t -1]).
d~~t) = x(t dy(t)
di
= yet - 7") [-Kz
d~~t) = x(t -
x(t) - ay(t - r) J }
7")[]{1 -
+ f3x(t -
r)J.
e-.(t-rl ) -
X(t)])
= yet - 7")[K2(1-
e-x(t-r)
yet)].
dxd(t) = -,x(t) + ae-/Jx(t) , t
x(t) =
d~~t)
r)[K1 (1-
-
sup
xes).
sE[t-r,t]
dx(t) J.OO -;tt=-'Yx(t)+aexp [- 0 K(s)x(t-s)dsJ.
§3.5. Delays in production and destruction
210
d~~t) = -,x(t) + ax(t _ r),e-PX(t-T). dx(t) = -,x(t) + axn([tDe-Px([tj). dt dx(t)
---;It = xCi) [a - blog[x(t)] - clog[x(t - 7)J]. It should have become clear from the foregoing, that local analyses of various models with time delays lead to investigations of linear delay differential equations. In the next section and in the remainder of this chapter, we consider the asymptotic and oscillatory behavior of linear vector - matrix systems using certain algebraic facts related to matrices and vectors.
3.6. X(t) = AX(t)
+ BX(t -
r)
Let us first consider the delay differential system
dx(t) dt
= Bx(t _
7)
3.6.1
where x(t) E IRn and B is a real constant n X n matrix. The following result shows that if the trivial solution of (3.6.1) is asymptotically stable for 7 = 0, then it will remain so for 7E [0,70); we also obtain an estimate on 70 (for more details see Goel et al. [1971]). Theorem 3.6.1. Let the eigenvalues of the matrix B be denoted by
Suppose that the trivial solution of the non delay system
dy(t) = By(t) dt
3.6.2
is asymptotically stable implying that 3.6.3
If j
= 1,2, ... , n,
3.6.4
§9.6. X(t)
= AX(t) + BX(t -
211
T)
then the trivial solution of (3.6.1) is asymptotically stable.
Proof. The characteristic equation corresponding to (3.6.1) is 3.6.5 Since
n
II (A +
det. [AI - Be- Ar ) = 0 =}
Qje-
Ar
)
= 0,
3.6.6
j=l
it will follow that the roots of the characteristic equation (3.6.5) are the roots of j = 1,2,3, ... ,n.
If we let AT =
Z
3.6.7
in (3.6.7), we can rewrite (3.6.7) in the form 3.6.8
j = 1,2,3, ... ,no
Equations of the form (3.6.8) with real Qj have been discussed in the literature on delay differential equations; since Q j can be complex, we provide a complete discussion of (3.6.8). For convenience, let us consider a fixed j and let the corresponding Q j be denoted by Q with ~e( Q) > O. We let (J,O
being real,
and introduce the substitution L
H(L)
=z -
(J>
0,
I B I < 7r /2
iO so that (3.6.8) becomes (for the fixed j)
= Le L + iOe L + (JT = O.
3.6.9
We note that ~e(L) = ~e(z) and hence ~e(L) < 0 will imply ~e(z) < 0 and conversely. In order to use Theorem 13.7 of Bellman and Cooke [1963), we proceed by letting L = iy (y real) in (3.6.9) so that where = F(y) + iG(y) F(y) = (JT - (y + B) sin y G(y) = (y + 0) cos y.
H(iy)
3.6.10 3.6.11
3.6.12
By the above Theorem of Bellman and Cooke [1963], a necessary condition for all the roots of H(L) = 0 to have negative real parts is that
212
§S.6. X(t) = AX(t) + BX(t - 7)
(i) the zeros of F(y) and G(y) are real, simple and they alternate; (ii) G'(y)F(y) - G(y)F'(y) > 0 for y E Rj 3.6.13 a set of sufficient conditions for H(L) = 0 to have roots only with negative real parts is that (a) all the zeros of G(y) are real and for each such zero, (b) the relation (3.6.13) holds.
The roots of G(y) = 0 are given by yO and Yn where
Yn
= ±( n + 1/2)7r;
n
= 0,1,2,3, ...
At the roots of G(y) = 0, we have and
3.6.14
It is readily verified that G'(Yn)F(Yn) > 0 when (7r/2) -IBI- (77 > 0; this result translated back to (3.6.8) implies that all the roots of (3.6.7) and hence of (3.6.6) will have negative real parts whenever (3.6.4) holds and this completes the proof.
,,;
We note that if ~e(a) 2:: 0 then the necessary condition G'(y)F(yO) > 0 is violated implying that the system (3.6.1) cannot switch from the instability to stability with an increase in T. We ask the reader to investigate this in detail. Let us consider the linear vector - matrix delay-differential system
dX(t) --;It
= AX(t) + BX(t -
7)
3.6.16
and examine the following: if the trivial solution of (3.6.16) is asymptotically stable when 7 = 0, for what positive values of 7 such a stability is maintained. There are several possible ways of answering the above question each leading to a different estimate of 7. The following result is due to Rozhkov and Popov [1971] (see also Tsalyuk [1973], Gosiewski and Olbrot [1980]). Theorem 3.6.2. Let A and B be real n x n constant matrices such that the trivial solution of
d~;t) = (A + B)Y(t)
3.6.17
§9.6. X(t)
= AX(t) + BX(t -
219
r)
is asymptotically stable and let M,o: be positive constants satisfying 3.6.18
If r is small and
MIIBllr(IIAIl + IIBI!) < 1, 3.6.19 a then the trivial solution of (3.6.16) is asymptotically stable. Furthermore, if X(t) denotes any solution of (3.6.16), then IIX(t)1I
~ M{
sup
SE[-T,T]
IIX(s)lI}e-/3(t-T);
3.6.20
in which f3 is the unique root of
T MIIBII( e/3 - 1) (IlAlI
f3 1- - =
0:
+ liB II e/3
T )
-----~----.::..-
3.6.21
/30:
Proof. We rewrite (3.6.16) in the form
X(t) = (A + B)X(t) - B L/(s)ds; = (A + B)X(t) - B
t;:: r
l~J AX(s) + BX(s -
r)) ds;
t;:: r
leading to
and hence
/lX(t)/I ::;
/lX/I.Me-a('-T)
+ M/lB/I[dsU>-a(.-s) (/I A /III X(u) /I 3.6.22
+IIBIIIIX(u-r)ll)du}i t?r where II X 11*
=
sup II X(t)
tE[-T,T]
II·
§3.6. X(t) = AX(t)
214
+ BX(t -
r)
Define t ~-r
3.6.23
and note that since f3 is a root of (3.6.21),
Z(t)
=
Mil X lI,e- o ('-r) + Mil B II
l' e-
O ('-'){
Lr
(II A IIIIZ(u)1I
+ IIBIIIIZ(u -
r)1I
)dU} ds 3.6.24
for t
~
r. We have from (3.6.22) - (3.6.24) that
3.6.25 where
Wet) = IIX(t)lI- Z(t).
3.6.26
From the definition of W in (3.6.26), Wet) < 0 for t E [-r, r] and Wet) is continuous for t ~ O. If IIBII 0, then we have from (3.6.25) that Wet) < 0 for t E (r, r + t) for some possibly small t> O. We shall show that Wet) < 0 for all t > rj for instance, if Wet) i- 0 for all t > r, then there exists a finite number t* such that t* = inf {t > r + tj Wet) ~ O}
t-
so that W(t*)
=0
and
Wet) < 0 for
t E [-r, t*),
t
=f r.
But in such a case we have from (3.6.25) that W(t*) < 0 and this is a contradiction. The result follows. [] The next result due to Khusainov and Yun'kova [1981] provides an alternative estimate on the delay parameter r in (3.6.16) for maintaining the asymptotic stability of the trivial solution of (3.6.17). Theorem 3.6.3. Assume that the trivial solution of (3.6.17) is asymptotically stable. Let C denote the real symmetric positive definite matrix satisfying
3.6.27
§9.6. X(t) = AX(t) where I is the n x n identity matrix. Let 70
= ( 2(IIAII + IIBII)IICBII )
+ BX(t -
70
215
7)
be the positive constant defined by
-1 (
Amin(C)/Amax(C)
)1/2
3.6.28
where Amin(C) and Amax(C) respectively denote the smallest and largest eigenvalues of C. Then the trivial solution of (3.6.16) is asymptotically stable for all 7
0, the equation veX) = a defines a closed surface in IRn which we denote by avo. We let 3.6.30 Vo = {X E IRnlv(X) :::; a}. Let us first verify the following observation; suppose that for t an a > 0, and that
X(t) E avo Then for every
X(t - 7)/1
0, there exists a
70
for
t - 27 :::;
such that for
S :::;
~ 7
there exists
t.
3.6.31
< 70 we will have IIX(t) -
7
EIIX(t)lI·
One can derive using the estimates of Lyapunov functions in Barbashin [1970] that (3.6.31) implies, 1/2
IIX(s)lI:::; ( Amax(C)/Amin(C) ) t-2r::;s9 sup
But we also have
IIX(t) - X(t - r)1I =
II
Lr (
AX(s) + BX(s -
IIX(t)lI.
r») dsll
:s r (IIAII + IIBII) t-,s;f,,,;, IIX(s )11 =
r(IIAII + IIBII) {Amax(C)/Amin(C)} 1/2 1IX (t)11
=
EIIX(t)/I
3.6.32
§3.6. X(t)
216
= AX(t) + BX(t -
r)
provided r
< ro = ,(II A I + IIBII) - I (A min(C)/Amax(C)),'2,
For arbitrary a > 0 and r > 0 we find a number 8(0', r) > 0 such that
IIX(t)11 < 8(0', r) for t E [-r,O] =* X(t) E
Vcr
t E [-r, r].
for
For instance, we have from
X(t) = X(O) +
l
(AX(S)
+ BX(s -
r)) ds;
t E [O,r]
that
If we choose 8 such that
then we have X(t) E above relation so that
Vcr
8( a r)
for t E (-r, r]. Thus, we are led to choose 8 from the
= e- IIAllr
[1 + /lBllr]-1 [0'/ Amax( C)]1/2 .
With these preparations we consider the rate of change of v along the solutions of (3.6.16).
~ V ( X(t»)
= _(XT(t) , CX(t»
«(
+ X(t -
r) - X(t)) T B T , CX(t))
+ (XT(t) , CB[X(t -
r) - X(t)]).
If X(t) E avcr and Xes) E Vcr for t - 2r ~ s ~ t, then we have that
~ v ( X(t») ~ -/lX(t)/l2 + 2€IICBIIIIX(t)1l 2
X (2e Il CBII-1).
: 0. Since S is closed, to E S and therefore there exists an i E {I, 2, ... ,n} such that
x(t) :; yet),
°:; t :; to
3.6.68
with
Xi(tO) = Yi(tO)'
3.6.69
But we have from (3.6.65) - (3.6.66),
x(to) :::; TA(to, O)e - [ ' TA(to, s) [A - AD Jx(s) ds
:::; TB(to, O)e - [ ' TB(to, s)[ E - ED Jy( s) ds :S;y(to).
3.6.70
Since at least one of the inequalities in (3.6.65) or (3.6.66) is strict, it follows that x(to) < veto) and this contradicts (3.6.69). Thus, the set S is empty and hence x(t) < yet) for all t ~ 0. The proof is complete. []
§J.6. X(t) = AX(t)
224
+ BX(t - 7")
Corollary 3.6.9. In tbe linear system of autonomous ordinary differential equa-
tions
dx(t)
-dt- + Ax(t) = let the constant matrix A
= (aij)
0 t
> O'
" be real with
x(O) aij :::;
x(O) = Xo > 0 E Rn =:::;. x(t) > 0 E Rn
= Xo 0, i
3.6.71
ERn
of j.
for
Then
t
~
O.
3.6.72
Proof. We can rewrite (3.6.71) in the fonn
x(t) = TA(t, O)xo Since Xo > 0,
x(t) > -
J.'
J.'
TA(t, s)[ A - AD Jx(s) ds.
3.6.73
TA(t, s)[ A - AD Jx(s) ds.
3.6.74
By Theorem 3.6.8 we obtain,
x(t) > yet) for t
~
3.6.75
0
where yet) is a solution of
y(t) = -
J.'
TA(t, s)[ A - AD Jy(s) ds.
3.6.76
But (3.6.71) has the unique solution yet) == 0 on [0,00) and hence the result follows [] from (3.6.75). Corollary 3.6.10. Let the constant matrices A = (aij), B = (bij) be such that (i) aii > 0, i = 1,2, ... ,n (ii) bi ; > 0, i = 1,2, ... ,n j bij :::; 0, i of j , i,j = 1,2, ... ,n (iii) 3.6.77 If
dx(t)
= 0,
t
~ 0;
x(O) = Xo
d~~t) + By(t) = 0,
t
~ OJ
y( 0) = yo > I x 0
--;It + Ax(t)
I
§S.6'. X(t) = AX(t) + BX(t - r)
225
o.
3.6.78
I xCt) I < yet) Proof.
By Corollary 3.6.9, yet)
yet)
for all
t 2:
> O. But y satisfies
= Ta(t,O)yo -
J.'
Ta(t, s) [B - BD 1yes) ds,
3.6.79
while x(t) satisfies
I x(t) I < TA(t, 0)1 Xo 1Since TB(t, O)Yo
J.'
TA(t, s)[ A - AD 1x(s) ds.
3.6.80
> TA(t, 0)/ Xo I, we have
J.' 1- J.'
yet) > TA(t, 0)1 "0 1-
~ TA(t, 0)1 xo
Ta(t, s)[ B - BD 1y(s) ds TA(t, s) [A
- AD 1y(s) ds.
An application of Theorem 3.6.8 to (3.6.80) and 3.6.81) leads to (3.6.78).
3.6.81 []
The following results are concerned with comparison and convergence characteristics of systems of delay differential equations and inequalities of the form
du.(t) n n dt < - "" ~ a"u I) ) ·(t) + "" 6 b··u I) I·(t - r:I}·(t»
_ I_
i=l
i = 1,2,"', n.
,
i=l
Proposition 3.6.11. (Tokumaru et. al. [1975]) Consider the systems
dx(t)
-;It ::; Ax(t) + Bi(t) , t > 0
3.6.82
dz(t)
--;J,t = Az(t) + Bi(t) , t > 0
3.6.83
where
z(t) z(s) 2: xes),
s E [-r, 0],
= {Zl(t) r
... , Zn(t)}T
E (0,00)
3.6.84
+ BX(t -
§3.6. X(t) = AX(t)
226
x(t) = {
sup
Xl(S),...
sE[t-r,t]
i(t) = {
Xn(S)}T
sup
3.6.85
sE[t-r,t]
Zl(S),...
sup
r)
sE[t-r,t]
Zn(S)}T.
sup sE[t-r,t]
Suppose further iij,
i,j=1,2, ... ,n
i,j=1,2, ... ,n.
bij~O,
Then
xCi)
~
z(t)
for
t
~
0.
Proof. Our strategy of proof is to show first that every solution y of
d~~t) > Ay(t) + By(t), y(s»x(s),
t > 0, yet) E R n
3.6.86
sE[-r,O]
°
satisfies yet) > x(t) for all t ~ and then apply a limiting process. Suppose there exists a positive number fJ and an integer j such that for the j -th component of xCi) and yet), Xj(fJ) = yiCfJ)· Then there exists fJo such that for some j}.
fJo = inf{fJlxj(fJ) = yj(fJ) But fJo
3.6.87
°
> since xes) < yes) for s E [-r,O]. For this fJo
and there exists a k, 1 ~ k ~ n for which Xk(fJO)
Xk(fJO) ~
~
= Yk(fJo)i hence
n
n
m=l n
m=l n
I: akmXm(fJO) + L
bkmxm(fJo)
I: akmYm(fJO) + I: bkmYm(fJO)
m=l
m=l
3.6.88 But by the definition of fJo, we have Yk(fJO) ~ Xk(fJO) which contradicts (3.6.88) and therefore we have yet) > x(t) for all t ~ 0.
§3.6. X(t)
= AX(t) + BX(t -
227
r)
To complete the proof we let €* E Rn denote a vector each of whose components is equal to an arbitrary positive number €. Let ze(t) denote the solution of
Ze(t) = AZe(t) + BiE(t) + €*
> AZc(t) + BiE(t),
t>0
3.6.89
with the initial condition
Zc(S)
= x(s) + €* >
x(s), s E [-r, 0).
By the above discussion,
Z€(t) > x(t) for t > 0. Since Z€ depends continuously on
€,
3.6.90
we can conclude
Z(t) = lim z€(t) ;::: x(t) for t E--O
~
° []
and the proof is complete.
For the convenience of the reader we recall from Chapter 1 the following result on a scalar differential inequality due to Halanay [1966]: Proposition 3.6.12. (Halanay [1966]) Let to be a real number and r be a nonnegative number. If f : [to - r, (0) I---? IR+ satisfies
dfd(t) t and if a >
~ -af(t) + (3[
(3 > 0, then there exist I
sup
f(S)];
3.6.91
SE[t-T,t}
°
°
> and ~ > such that 3.6.92
The above result of Halanay has been generalized to a class of vector-matrix systems of differential inequalities by Tokumaru et.al. [1975]. In preparation to present their result we note a few properties of M -matrices (see below for a definition) formulated for convenience in the form of the following two propositions.
228
§9.6. X(t)
= AX(t) + BX(t -
r)
Proposition 3.6.13. (Araki and Kondo [1972]) Let P = (Pij) be an n X n matrix with Pij :::; 0 for i '1= j. Then the following conditions are mutually equivalent. 1. There exists a positive vector x such that Px > O. 2. The matrix P is nonsingular and p- 1 2:: 0 (elementwise). 3. All the successive principal minors of P are positive; i.e. Pll
det [PH P21
PH
det
[
> 0, P12] P22
P12
> 0,
PIn
1
~~~ .. ~2.2...........~~~ > O. Pnl
Pn2
••.
Pnn
4. The real parts of all the eigenvalues of P are positive.
P can be put in the form P
= pI -
A where A =
(aij) , aij
2:: O. The following
facts about M matrices can also be found in Araki and Kondo [1972]. Proposition 3.6.14. Let A = [aij] be a real n x n matrix. LetB = (pI - A] where I denotes the n x n identity matrix. Then the following hold. 1. If we increase some elements of an M -matrix so that no element changes sign, then the new matrix is an M -matrix. 2. If we multiply a row or column of an M -matrix by a positive number, then the new matrix is an M -matrix. 3. The matrix pI - A is an M -matrix if and only if p > AA where AA denotes the nonnegative eigenvalue of A. 4. An M -matrix has a positive eigenvalue AA such that, if f3 is the maximum
element on the main diagonal, then f3 2:: AA and for any eigenvalue W A of A,
5. If A is an M-matrix, then A - III is an M-matrix, if and only if Il < AA' Definition. A matrix Q = [qij] with qij :::; 0, i 1= j is said to be an M-matrix if anyone of the four equivalent conditions of Proposition 3.6.13 bolds.
§S.6. XCi)
= AX(i) + BX(i -
r)
229
Theorem 3.6.15. (Tokumaru et. al. [1975)) Let A, B be real n x n matrices and let xCi) E Rn denote a solution of the system of differential inequalities
xCi)
~
-Ax(t) + Bi(t),
t>O
3.6.93
°
where i is defined by (3.6.85). If B ~ and if A - Bis an irreducible M -matrix, then there exist a number b > and a vector XO E Rn with positive components satisfying for t ~ 0. 3.6.94
°
Proof. We shall first show that the system
i(t) = -Az(t) + Bz(t), z(s)~
k,
S
t>o
E [-r, OJ,
kE
3.6.95
h~s a solution of the form
O satisfying 3.6.97
z
°
then z(t) = ke- ot is a solution of (3.6.95) with initial value z( s) = ke- 6s , s E [-r,O}. Define a map F(.) : [0,00)
t--t
Rnxn as follows:
F(O")
=A-
Be UT
•
3.6.98
Let -\(0") denote the minimum of the absolute values of the eigenvalues of F( 0"). We first verify that -\( (J") is an eigenvalue of F( 0"). We can write
F(ul=aI- [aI-(A-Be OTl ] =aI-P(O")
3.6.99
§3.6. X(t) = AX(t) + BX(t - r)
230
and observe that the matrix P( a) 2: 0 where Q is the maximum of the diagonal elements of A-B. From the properties of M- matrices (Araki and Kondo [1972]), F(a) is an M- matrix, ifand only if one of the following hold: where [F(a)]-l E Rnxn (i) [F(a)J-l ~
°
(ii) Q> p(p'(a»)
where
p(P(a») denotes the spectral radius of pea).
If al < a2, then irreducibility of F(at) will imply that of F(a2) and furthermore, p[P(al)] S p[P(a2») since peal) S P(a2). By hypothesis, F(O) is irreducible; hence F(a) is irreducible for a ~ 0. If F(a) is an M- matrix, (F(a)]-1 is an irreducible nonnegative matrix by (i) above. The well known Perron-Frobenius theorem (Gantmacher (1959]) guarantees that p([F(a)]-l) is an eigenvalue of
[F(a)]-l and the associated eigenvector k(a) is positive (componentwise). It is clear that >.(a) = IIp( [F(a)]-l) and >.(a) is an eigenvalue of F(a) with k(a) as the corresponding eigenvector of F(a). We have from >.(a) = Q - p[F(a)J that -\(a2) S -\(a1) for a1 S a2 and -\(0) > since F(O) = A - B is an Mmatrix. From the properties of M -matrices, F( a) cannot be an M - matrix for a sufficiently large a > 0. Hence A( a) S for large enough a > 0. It follows from all these facts, that A( a) > so long as F( a) is an M - matrix and A( a) continuously approaches zero. Therefore, the equation A( a) = a has a positive root ao and the corresponding eigenvector k( ao) is positive. It follows now that z(t) = k(ao)e- tTot is a solution of i(t) = -Az(t) + Bi(t). For any continuous initial value xes), s E [-r,O), xes) E IR+., one can find a f3 > such that x(o) S f3k( ao) == k and z(t) = ke- 6t , 8 = ao is a solution of i(t) = -Az(t)+Bi(t). The result follows by an application of proposition 3.6.11.
°
° °
°
n
In order to illustrate the applicability of the result of Proposition 3.6.13, we consider the linear system dx .(t) 3 -dt '- = """ ~ a"x t} } ·(t j=1
3
-
·(t»
T't}
+ """ b··x '(ft ~
I}
}
m t}.. ]).,
i = 1,2,3
3.6.100
j=l
where x(t) = {Xl(t),X2(t),X3(t)} E R 3 ;aij,bij E lR;i,j = 1,2,3;rij : [0,00) 1--+ [0, roJ; mij EN, {i,j = 1,2, 3}, (PJ denotes the greatest integer contained in pER and Xi(t) denotes the right derivative of Xi at t. Except for notational complexity and inconvenience, there is no difficulty in extending the following analysis of (3.6.100) to vector systems with any finite number of
§3.6. X(t)
= AX(t) + BX(t -
1')
231
components. We assume i,j = 1,2,3.-
For t 2: 21'0
+ 2(m + 1), we can write (3.6.100) in the form
3.6.101
3.6.102
where i,j = 1,2,3.
For any fixed t 2: to = [21'0 + 2(m + 1)], the possible sign pattern of the components XI(t), X2(t), X3(t) of the vector x(t) E R3 is as follows: we can without loss of generality assume that Xl(t) 2: 0 since otherwise, we can multiply the corresponding equation governing Xl in (3.6.100) by (-1) and restore XI(t) 2:: O. With this choice for Xl, we have the following sign pattern for x(t) for any fixed value of t:
{+,+,+}, {+,+,-}, {+,-,+}, {+,-,-} (If x(t) E IRn, then we will have 2 n- 1 possibilities of sign combinations for the components of x(t)). We write
§3.6. X(t) = AX(t) + BX(t - T)
232 where
J1
= {t ~ toIXi(t) ~ O,i =
1,2,3}
J 2 = {t ~ toIXl(t) ~ 0,X2(t) ~ 0,X3(t)
< O}
J 3 = {t ~ toIXl(t) ~ 0,X2(t) < 0,X3(t) ~ O} J 4 = {t ~ to/Xl (t) ~ 0, X2(t)
< 0, X3(t) < OJ.
For any t E J1 , we have from (3.6.102),
where
I x(t) 1= {l x l(t)I, IX2(t)l, IX3(t)I}T !AI2=IAlxIAl,
IAI=(!aijj)
IB 12 = IB I x / B I,
1B I =
I X ICt) = I Xj 1Ct)
(I bij I)
{Ixd(t), IX2!Ct), IX31(t)}T
=
sup
Ix j ( S ) I
sE[t-2(ro+m+l),t)
C1
=
al1+bll max(O, a21 + b21 ) [ max(O, a31 + b3J)
max(0,aI2+ b12) max(0,aI3+b13)] a22 + bzz maxCO, a23 + b23 ) . max(O, a32 + b32 ) a33 + b33
We rewrite (3.6.103) so that
~ Ix(t)1 :0; -
[-
Cd x I(t) - {I A I'ro +21 A II B I(ro +m+ 1)+ I B I'(m+ 1) } I x I(t)]. 3.6.104
If we assume now that
-{ C.
+ I A I'ro + 21 A II B I(ro + m+ 1) + I B I'(m + 1)}
is an M-matrix, then by Proposition 3.6.13, it will follow that there exist 81 > 0 and a positive vector kl' such that 3.6.105 Now let t E
Jz .
Define T2 as follows:
Tz =
[~ ~ ~].
°°
-1
§3.6. X(t)
= AX(t) + BX(t -
r)
233
It is easy to see that T2X(t) = I x I(t) so that
:t 1x I(t) :::; T2(A + B)T;-l [T2X(t)] +
{I A 12ro + 21 A II B 1(/6 + m +
1)+
I B 12(m + 1)}1 x I(t) :::; C21x I(t) + {I A 12/0 + 21 A II B I(ro + m + 1) + I B 12(m +
3.6.106
1)}1 x I(t)
where
C2 =
all + bl l max{O, (a2l + b21 )} [ max{O, -(a31 + b31 )}
max{O,a12 + b12 } a22 + b22 max{O, -(a32 + b32 )}
max{O, -(a13 + bI3 )}] max{O, -( a23 + b23 ) • a33 + b33 3.6.107
Again if we suppose that the matrix
is an M -matrix, then it will follow as before, that there exist 82 > 0 and a positive vector k2 such that t E J2 • 3.6.108 Suppose now t E J3 j define a matrix T3 so that
Ta =
[~ ~l ~]
One can derive again, that
d dt I x I(t) :::; T3(A
+ B)T3- 1 x I(t) + {I A 12/0 + 21 A II B 1(/0 + m + 1)+ IB 12(m + 1)}1 x I(t) 3.6.109 2r :::; C3 1x I(t) + [I A 1 o + 21 A II B I(ro + m + 1) I B 12(m + 1)]1 x I(t) 3.6.110 1
where max{O, -(aI2 + bI2 )} a22 + b22 max{O, -(a32 + b32 )}
max{0,aI3 + b13 } ] max{O, -(a23 + b23 )} a33
+ b33
•
3.6.111
§3.6. X(t)
234
= AX(t) + BX(t -
T)
If the matrix
is an M-matrix, then there will exist 83
>
°and positive vector k3 E IR~ such that 3.6.112
Finally if t E J 4 , one considers the matrix
and derives that
~ 1x I(t) ::; T4(A + B)T4- I I x I(t) + {I A 12To + 21 A II B I(To + m + 1)+ 1B 12(m + I)}I x I(t) 3.6.113 2T ::; C4 x I(t) + {I A O + 21 A II B I(To + m + 1) 3.6.114 + I B 12(m + I)}I x I(t) 1
1
where max{O, -( al3 + b13 )}] max{O, a23 + b23 )} [ max{O,a32 + b32 } a33 + b33 3.6.115 from which one can conclude that there exist a positive vector k4 E 1R3 and 84 > 0 such that C4
=
all + bl l max{O, -( a2I + b2 J)} max{O, -(a31 + b31 )}
+ bI2 )} + b22
max{O, -( al2 a22
We can summarize the above analysis in the form of the following:
Proposition 3.6.16. If the following matrices
- (C j
+ [IA 12 TO + 21 A I B 1(To + m + 1)+ IB 12(m+ 1)]) j = 1,2,3,4
3.6.116
§3.6. X(t) = AX(t)
+ BX(t -
r)
235
are M -matrices, then the trivial solution of (3.6.100) is asymptotically stable (in fact exponentially asymptotically stable). Proof follows immediately from our discussion and
I x(t) I :::; ke- 6t
where
3.6.117
As an example of a linear system of differential equations with unbounded delays, we consider a linear system of the form
x(t) = Bx(t)
x(t) E IR n ,
+ AX(At), t>O A> 0,
3.6.118
A, BE IRnxn.
The linear system (3.6.118) has been investigated in detail by Lim [1976] from where we have extracted the following result.
Theorem 3.6.17. Let
Let A
= [aij]
°
0, v > 1,
t > to.
3.6.127
§9.6. X(t) = AX(t)
+ BX(t -
We also assume mo
= max II to. Assume the contrary and let t* > to be the first point where
II XI(t*) II = y(t*). By assumption, II Xl(t) II Hence, (3.6.139) implies
< yet) for t < t* and so IIxI(t)IIV < yV(t) for t < t*. 0=
II XI(t*) 11- y(t*) < 0
and this contradiction proves the assertion that
II Xl(t) 11- yet) < 0
for
t > to·
Thus,
t 2:: to·
3.6.140
§3.6. X(t) = AX(t)
+ BX(t -
239
r)
Now let tl > to. For t E [to, tIl, we can use the following majorizing equation for
Xl(t), 3.6.141 which has a particular solution yet) = ~. For t ?:: t l , we use the majorizing equation (3.6.131) and proceed as before to ·obtain 3.6.142 Comparing (3.6.142) and (3.6.140) one obtains (3.6.130).
[]
3.7. Stability switches
A primary purpose of this section is to develop techniques to perform a comparative study of linear systems with and without delays. In particular, we are interested in finding conditions for delays not to destabilize an otherwise stable linear system. It is also of interest to examine whether delays can stabilize otherwise unstable systems. Furthermore, it is possible for a system to be stable for a small delay; if the delay is longer, the system can become unstable and for still longer delays the system can regain stability; this sequence of switching from stability to instability and back to stability can repeat if the delays progressively become longer (see Cooke and Grossman [1982]). Our analysis below will be restricted to systems with one delay; a number of generalizations are indicated in the exercises. The results of this section and the relevant exercises are selected from the works of Cai Sui Lin [1959J, Qin Yuan-Xun et 81. [1960], Chin Yuan Shun [1962J, Wang Lian [1962] and Chang Hsueh Ming [1962J. Consider the linear delay differential system
dx~?) =
t
(ajk;k(t)
+ bjkXk(t -
r») ;
j=1,2, ... ,n
3.7.1
k=l
where ajk, bjk are real constants and r ?:: O. Let the characteristic equation associated with (3.7.1) be denoted by 3.7.2 The following result provides conditions for the absence of delay-induced switch from stability to instability.
§3.7. Stability switches
240
Theorem 3.7.1. A set of necessary and sufficient conditions for the trivial tion of (3.7.1) to be asymptotically stable for all r ~ 0 is the following:
solu~
(i) the real parts of all the roots of 3.7.3 are negative. (ii) for any real y and any r
~
0, the following holds: 3.7.4
in which i = yCI. Proof. It is easy to verify the necessity of the conditions (3.7.3) and (3.7.4). For instance, if (i) does not hold then the trivial solution of (3.7.1) is not asymptotically stable for r = O. If there exist a real number y and some r ~ 0 such that
D(iy, r) = 0
3.7.5
then for such r, the characteristic equation (3.7.2) will have a pair of pure imaginary roots and hence the trivial solution of (3.7.1) is not asymptotically stable. To prove the sufficiency part of the result, we have to show that when the conditions (i) and (ii) hold, all the roots of (3.7.2) have negative real parts. We note that we can rewrite (3.7.2) in the form 3.7.6 where
AI,'"
ajk, bjk
(j, k = 1,2, ... , n) are known constants and
An are polynomials in
ajk,
bjk
and
e-
Ar
.
We assume that
3.7.7 Thus, for ~e( -\) ~ 0 and r ~ 0, the coefficients Aj (j == 1,2, ... ,n) in (3.7.6) are bounded in absolute value. Let 3.7.8
241
§3.7. Stability switches
and define
M = max For IA I 2:: M and
~e( A)
(1, (n + l)N) > O.
3.7.9
2:: 0 we will then have
I( _l)n An + A1A n- 1 + ... + Ani;::: 1),[" [1- I~II -... - I~II] ~Mn[l- (n:~)N] which implies that in the domain IAI 2:: M and no roots and this is valid for all r 2:: o.
~e(A)
> 0,
3.7.10
::::: 0, the equation (3.7.2) has
Now let us examine the region IAI < M, ~e(A) 2:: 0 and show that under (i) and (ii), this region also cannot have roots of (3.7.2). By condition (i) we know that for r = 0, the roots of (3.7.2) are all in the half-plane ~e(A) < O. Now for r t= 0, the only possibility that the roots of (3.7.2) can fall in the region ~e(A) > 0 is that for some r > 0, one or more roots of (3.7.2) lie on the imaginary axis of the complex A plane between -M and Mj but (3.7.4) will not permit any of the roots of (3.7.2) to lie on the imaginary axis of the A- plane for any r 2:: and therefore all the roots of (3.7.2) will be such that ~e(A) < when (i) and (ii) simultaneously hold. (]
°
°
The result of Theorem 3.7.1 provides conditions for the absence of a delay induced switch from stability to instability in (3.7.1). The next result gives sufficient conditions for the absence of a delay-induced switch from instability to stability in (3.7.1).
Theorem 3.7.2. Suppose we have in (3.7.2) that (-ltD(O,r)
= (-l)nD(O, O) < 0
and furthennore, that D(A,O) = 0 has an odd number of roots with positive real parts. Then (3.7.2) has at least one root with a positive real part for all r 2:: O. Proof. Define a continuous real valued function arbitrarily fixed r 2:: 0, we let
f (0:) = (-1 t
D( 0:, r);
f : [0,00)
T
2::
o.
-7
(-00,00). For an
3.7.11
§3.7. Stability switches
242
We note
f(O) = (-lr D(O, r) < 0
(by hypothesis)
and
f(a)
-+
00
as
a
-+
00
r 2 0.
for all
It will follow that there exists a real number say a* such that a* > 0 and
f(a*) = (-ltD(a*,r) = 0 for any
r 2 0
showing that the trivial solution of (3.7.1) is unstable for all r 2
o.
[J
We remark that the additional assumption in Theorem 3.7.2 regarding the existence of odd number of roots with positive real parts is necessary, although, this assumption has not been used in the proof. A counter-example showing the necessity of this assumption is formulated in Exercise 30 at the end of this chapter. The condition (ii) of Theorem 3.7.1 is not in a form convenient for applications. So, let us examine this condition further by considering two possibilities: y = 0 and y =f 0. For y = 0, the condition (3.7.4) becomes
D(O, r)
= det[ajk + bjk ] =f
°
3.7.12
and (3.7.12) is valid for all r 2 0. For y =f 0, let us suppose that r varies on the interval [0,211" Ilyll implying that IrYI will vary in [0,211"]. This will mean that iry e will vary over the unit circle. Thus, for y =f 0, we can let ry be another independent variable say (J (i.e. (J = -ry), and derive the following two conditions in the place of (ii) in Theorem 3.7.1:
( iii) {
det[ajk
+ bjk ] =f 0
H(y, (J)
= det[ajk + bjkeiu -
for nonzero real
iy8j k]
=f 0
3.7.13
y and any real (J.
Since (J and yare regarded as two real independent variables, we can write
H(y,(J) where
= F(y,(J) +iG(y,(J)
3.7.14
§3.7. Stability switches The equation H(y,a)
243
= 0 will lead to two equations
O}
F(y,a) = G(y, a) = 0
3.7.15
from which by eliminating either of y or a, we can derive an equation of the form
u (y) = 0 in terms of y
or
U ( cos 0' , sin (7)
=0
3.7.16
i- 0 or cos 0' and sin a.
Now if U(y) = 0 has no real nonzero roots y, then the second of (3.7.13) holds or if U(y) = 0 has a real nonzero root and if for such a y i- 0, the two equations in (3.7.15) have no common real root a, then also the second of (3.7.13) holds. Thus, we have to check only these facts to prove the validity of the second of (3.7.13). We note that checking these aspects will involve only algebraic equations and not the solving of transcendental equations (see the examples below). For convenience, we summarize the above discussion. Theorem 3.7.3. A necessary and sufficient condition for the trivial solution of (3.7.1) to be asymptotically stable for all 7 ~ 0 is that tbe following bold. (a) all the roots of 3.7.17
have negative real parts. ((3) the equation U(y) = 0 of (3.7.16) either has no real root or if U(y) = 0 of (3.7.16) bas real root, then for such a real root y, the two equations (3.7.15) bave no common real root a. An alternative to (f3) above is provided as follows: (fi') the equation U(cos a, sin a) = 0 of (3.7.16) mayor may not have real roots 0'; if such roots exist, then for those roots, the two equations F(y, 0') = 0, G(y,O') = 0 of (3. 7.15) have no common real nonzel'O roots y. Let us now consider a few illustrative examples; first we discuss the scalar equation
dx(t)
-;It
= ax(t)
+ bx(t - 7).
3.7.18
§3.7. Stability switches
244
Suppose a + b > 0; then the trivial solution of (3.7.18) is not stable for If A is a root of the characteristic equation of (3.7.18) for 7 > 0 satisfying a
+ be-AT -
A = 0,
7
== O.
3.7.19
then look at the roots of
D(A,7) = ae AT For real A ~ 0 and
7
+b -
Ae AT = O.
3.7.20
> 0 we find, D(O, 7) =
a
+b> 0
and lim D(A, 7) = lim (a - A)e AT ')'''''''00
+b=
-00.
A-+OO
°
It follows that there is at least one real root A = >'(7) > satisfying (3.7.20) implying that for all 7 > 0, the trivial solution of (3.7.18) remains unstable. No switch from instability to stability can take place due to increase in 7.
Let us suppose that a+b < 0 in (3.7.18) so that the trivial solution of (3.7.18) is asymptotically stable for 7 = 0. We want to find additional conditions on a and b (if any) so that the trivial solution of (3.7.18) will be asymptotically stable for all 7 > O. We have to verify the condition (ii) of Theorem 3.7.1. We note H(y,a)
implying that
= (a + bcos a) + i( -y + bsin a) = F(y,a) + iG(y,a) = 0
F( y, a) = a + b cos a G(y,a)
=0
= -y + bsin a
= 0,
and hence A necessary and sufficient condition for U (y) =
Now if U(y)
= 0 has nonzero real roots, then
°
not to have nonzero real roots is
§3.7. Stability switches
245
Thus for b =1= 0, we have from F(y, 0') = 0 = G(y, a), that
tan a =
_l!... a
For any real y, the above equation has real roots a which will simultaneously satisfy F(y, a) = 0 = G(y, a). Thus, a set of necessary and sufficient conditions for the trivial solution of (3.7.18) to be asymptotically stable for all r 2:: 0 is given by the following: a + b < 0 and b2 - a 2 ~ 0 which can also be written as a
+ b < 0,
b - a 2::
o.
In the next example, we consider the second order equation
d?x(t) -;Ji2
dx(t)
+ a-;Jt + bx(t) + cx(t -
r)
= O.
The characteristic equation associated with the above equation is
D()", r)
= )..2 + a).. + b + ce-,xT = O.
Condition (i) of Theorem 3.7.1 requires that the real parts of the roots of
D()",O) =
)..2
+ a).. + (b + c) =
0
are negative and this will be the case, if and only if
a> 0,
b + c > O.
Condition (ii) of Theorem 3.7.1 leads to an analysis of the roots of
H(y, a)
= F(y, a) + iG(y, a) = 0 = (_y2 + b + c cos a) + i( ay + c sin a) =
or equivalently
+ b + c cos a = ay +c sin a = O.
F(y, a) = _y2 G(y,a) =
0
0
§3.7. Stability switches
246 If we let A
G(y,O')
=
a2
-
=
2b and B
b2
-
c2 , then we have from F(y, 0')
o and
= 0 that
whose roots are given by
y=± [
-A ± {A2 - 4BP/2jl/2 2
Hence, a necessary and sufficient condition for the nonexistence of nonzero real roots of fey) = 0 is
A2 - 4B < 0;
either
A
or
2
-
4B
=0
and
A 2:: 0
or
A2 -4B > 0,
B > 0,
A> 0
or
A2 - 4B > 0,
B=O,
A 2:: 0.
One can further simplify the above conditions to obtain, either A 2:: 0, B 2:: 0 or A < 0, A2 - 4B < 0. If fey) = 0 has nonzero real roots, we can get from
G(y, 0')
= ay + c sin a =
°
real values of 0' which will also satisfy F(y, a) = 0. Thus, a set of necessary and sufficient conditions for the asymptotic stability of the trivial solution of the second order equation is
(i)
a> 0,
A 2:: 0
(ii) either or where A
=a
2
-
2b
and
b+c>O and
B 2:: 0
A2 -4B < 0 A < 0, 2 2 B =b - c .
Let us consider a third example provided by the following prey-predator system with mutually interfering predators;
3.7.21
§3.7. Stability switches
247
where a, b, c, K are positive constants and T 2:: 0 while 0 < m < 1; x(t) and yet) respectively denote the biomasses of the prey and predator populations. The above system has a positive steady state (prove this) E* : (x*, y*) satisfying
,(1- :*) =
ay*m
bx*(y*)m = cy*
(=?
bx*(y*)m-l =
c).
3.7.22
The linear variational system associated with E* is
dX(t) dt dY(t)
= _1x* X(t)
_ amx*(y*)m-lY(t)
K
3.7.23
--;It = b(y*)m X(t - T) + bm(y*)m-lx*Y(t - T) - cY(t) which has the characteristic equation given by
or
D()..,T)
=)..2
+ )..{c+ 1x* -
bm(y*)m-lx*e- AT }
K
+ 1x*c _Ix*bm(y*)m-lx*e- AT K
K
+ amx*(y*)m-1b(y*)me- AT
3.7.24
=0. When
T
= 0, D(A,O)
=)..2
+ )..(c+ ~x* -
bm(y*)m-l x*)
+ Ix* c - 1x*m(y*)m-l x* K
K
+ amx*(y*)m-l b(y*)m
=0.
It is easy to check, bm(y*)m-l X * = me < c < e + Ix* K =}
c + 1x* - bmx*(y*)m-l K
We have from
, * = , - a ( y *)m -x K
> O.
3.7.25
§3.7. Stability switches
248 that
Ix* c K
= b - a(y*)m]c = b - a(y*)mJbx*(y*)m-l = ,bx*(y*)m-l _ a(y*)mbx*(y*)m-l > ,b x* x*(y"*)m-l _ abx*(y*)m(y*)m-l K
and hence
Thus, by the Routh-Hurwitz criteria, all the roots of D()..,O) = 0 have negative real parts. We check whether a delay induced switching to instability can take place.' We let A = iy in D(A, 7) = 0 and derive
D(iy, r) = -y'
[c
+ iy + ~x* -
+ ~x*c = O.
bm(y*)m-'x*e- iYT 1
1
[;x*bm(y*)m-. x* - amx*(y*)m-'b(y*)m e- iYT 3.7.26
Separating the real and imaginary parts in (3.7.26), y2 _lx*c
= -ybm(y*)m-l x * sin(Y7)
K
-
[~x*bm(y*)m-.x* -
1
amx*(y*)m-'b(y*)m cos(yr)
-y( c + ~ )x* = -ybm(y*)m-· x* cos(yr) +
3.7.27
[ (;) x*bm(y*)m-· x*
- amx*(y*)m-'b(y*)m] sin(yr).
3.7.28
Square and add both sides of the above two equations;
y' + y' { c'
+ (;x*)' + 2C~X* _ 2;x* c _
+ (;x*c)' -
( bm(y*)m-. x*) '}
{;x*bm(y*)m-. x* _ amx*(y*)m-'b(y*)m}'
= O.
3.7.29
§9.7. Stability 8witche8
249
A sufficient condition for the nonexistence of delay induced instability is that the quartic in (3.7.29) has no real roots. Consider next a linear system of the form
duet) -at = allu(t) + a12v(t) + bllu(t dv(t) -;]t
711) + b 12 V(t
-
712) 3.7.30
= a21 u(t) + a22v(t) + bZ1 u(t -
72d
+ b Z2 v(t -
722)
where aij, bij (i,j = 1,2) are real numbers and tij (i,j = 1,2) are nonnegative real numbers. The characteristic equation associated with (3.7.30) is
which on expansion becomes,
+ a22)A + (all a22 - a'2 a 21) - A [b ll e-'Tn + b22 e-'T" 1 + a22bUe-.l.Tll + allb22 e-.l. Tll - a21b12e-.l.T12 - a12 b 21 e -.l. + b l1 b22 e-.l.(Tll+T22) - b12b21e-.l.(T12+T2d = o. 3.7.31
PiA) = A2 - (all
T21
We assume that lently,
aij,
bij in (3.7.30) are such that P(O)
t=
0 in (3.7.31) or equiva-
3.7.33
For convenience in the following we let al =
-(au
+a22)
i31
= -bl l
i3z = -b 22
81
= b11 b22
82 = -b Zl b12
/'11
= a2Z bll
/'21
= -a12 b21
/'12
= -a21 b12
/'22
=
aU b22
The condition in (3.7.32) becomes 2
n
az
+
L i,j=l
lij
+ L 8i t= O. i=1
3.7.34
§9.7. Stability switches
250 We let>.
=
iw (w being a real number) in (3.7.31) and derive that
= (iw)z - (all + azz)(iw) + (aUa22 - a12aZl)
P(iw)
+ b11(-iw)e-iwTll + anbzze-iwTll
+ bzz(-iw)e-iwT22
+ allbzze-iwT22 + bllbzze-iw(Tl1+T22)
- aZ1 b1Ze-iwT12 - a1ZbZ1 e- iWT21 - b21 blze-iw(T12+T21) = 0
or equivalently [_w z + (allan - a12az1)] - i[w(all +w
[b
ll
+ an)]
e-i(WTll +rr/Z)
+ bzze-i(WT22+7r/Z)1
+ aZZb ll e- iWTll + all bzze-iwTn - aZ1bIze-iwT12 - alzbzle-iwT21
+ bll bzze-iw(Tll +T22) 3.7.35
Separating the real and imaginary parts in (3.7.35), w 2 - (alla2Z - a12a2J) = w{b l1 COS(WTll
+ anbll
COSWTll
+ 7r/2) + b22 COS(WT22 + 7r/2)} + allb22 COSWTZ2
+ bllbn COS{W(Tll
+ T22} -
a21b12 COSWT12
- alZb21 COSWT21 - bIZ b21 cos{w(rIz
w(all
+ a22) =
-
[w{b ll sin(wTl1
+ T2J)}
+ "/2) + 1>,2 sin(WTn + ,,/2)}
+ an b11 sinwT11
+ an bzz
sinwTz2
+ bl1 bZ2 sinw( 1'11
+ TZ2) -
a21 bI2 sinwT12
- a1Z b21 sinwT21 - b12 b21 sin{w( 1'12
+ T21)}]'
Squaring and adding the respective sides of (3.7.36) and (3.7.37),
{w 2-(alla22 - a12aZl)}2 +wZ[all
3.7.36
+ a221 2
= w 2{b ll Z + bzz Z + 2bl1 bZ2 COS(WTll + 7r /2 - WTZZ + (anbll)2 + (allb 22 )z + (a21blz)2 + (b ll bn )2 + (a21 blZ)z + (a 1z b2d 2 + (b 12 b2d z
- 7r /2)}
3.7.37
251
§3.7. Stability switches
+ 2w [{ blla22bll + bl l bl l b22
cos{WTll
- bu a,Z "'" COs(WTll
+ {b 22 a 22 b ll
-
7r /2 + w( T11
WTll)
+ bllallb22
+ T22)}
COS(WT22
WTll - WT22) -
b 22 a 12 b 2I COS( WT22
+ 7r /2 -
WT21) -
+2[a22bllallb22 COSW(Tll -
T22)
COSW(Tll -
T12) -
-
a22bllb12b2I
COS{W(Tll -
T12 -
-
all b 22 a 21 b 12 cos{w( T22 - T12)} -
-
allb22b12b21 COS{W(T22 -
+ T22 -
- bll b 22 a 12 b 21
cosw( Tll
+ a21b12a12b21
COSW(T12 -
+ a12b21b12b21
COS{W(T21 -
T21)
T11 -
COSW(Tll -
T2t)
bllb22a21b12
bll b22b12b21
+ a21b12b12b21
T12 -
cosw( T11
WT12) WT2d}
T22)}
T11 - T22)}
T2d}
COSW(Tll
COS{W(TlZ -
T2J)}].
+ 7r /2 -
COS{W(T22 -
all b 22 a lZ b 21 cos{w( T22 -
T2d -
WT12)
WT22)
+ 7r /2 -WT12 -
COS{W(Tll -
+ all bZ2 b ll b 22
T12 - T21)} -
WT22)
+ 7r /2 -
+ IT/2 -
Cos(WT22
b22b12b21 COS( WT22
a22 b ll a I2 b 21 T2d
COS(WT22
b 22 a2I bI2
+ a22bllbllb22
- a2zblla2Ib12
+ 7r/2 -
COs(WTu - WT,Z - WT21) }
+ 7r/2 ~ WTll) + b22allb22
+ IT /2 -
cos(WTl1
bl l a2I b I2 cos(WTll
-
+ 7f /2 - WTZ d -b ll b,Z ""1
cos(WT22
+ b 22 bll b 22 -
+ 7r/2 -
COS(WTll
+ T22 -
+ T22 -
T12 -
T12)
T12 -
T21)
T21)}
3.7.38
Let the right side of (3.7.38) be denoted by few). For arbitrary real w, we have from (3.7.38),
few) ~ w 2 {lb ll l + Ib22 1}2
+ [Iazzbll l + lallbzzl + Ibubzzl + laZ,b,zl + la'Z"",1 + Ib12b21rf + 21wI [(Ibn l+ Ibzz l) (lazlbIZi + la,zbzd) + Ibubzzllall - azzl + (Ib u l+ I""zl) (Ib u bzzl + Ihz"",l) ]. If we denote the right side of (3.7.39) by M, then
3.7.39
1
§3.7. Stability switches
252
where
+ Ib22 ( = lazzblll + lall bZ2 + la21 b12 + la12 b211 5 = Ib ll b2Z ! + jb 12 b21 I·
f3 = Ibul
1
1
}
1
3.7.41
A sufficient condition for the. nonexistence of a real number w satisfying (3.7.31) can now be obtained from (3.7.38) - (3.7.41) in the form
w' + ("'; - 2"'2 -
fP)w2 -
21w1 [,9 (la21 b12 I + la12b,d)
+ Ibl l b,2l1all - a221 + ,98] + "'~ -
(7 + 8)2 >. 0
3.7.42
in which
The inequality (3.7.42) is of the form w4
+ awZ -
bw + c > 0
3.7.43
where
3.7.44
If a and c in (3.7.43) are positive numbers, then we can write (3.7.43) as follows:
w4
+ a(w -
bj2a)2
+ (c -
bZ j4a) > O.
3.7.45
One can see from (3.7.45) that a set of sufficient conditions for the nonexistence of a real number w satisfying P( iw) = 0 in (3.7.31) are given by a > 0 and
c - b2 j4a > O.
3.7.46
Thus, we conclude from (3.7.46) that a set of sufficient conditions for the nonoccurrence of stability switching in (3.7.30) are given by 3.7.47
253
§3.7. Stability switches
{ (bul+lb:22I) [laZlbl21 + lalzb2l 1+ Ibllbzzliall + (I blll + Ibzz l)(lb11 b22 + Ib12 bzll)] 1
r
azzl
< [ail + a~z + 2alZaZl - (Ibll l + Ibzzl) Z] [ (all azz _ a12 aZl ) z - (la21 blll + la21 bI2 1 + la12 b211
+ lall b22
1
+ Ibubzzl + Ib12bzll + Ibubz,l + Ib12bzll) ']. 3.7.48 It is not difficult to apply the above technique for the derivation of sufficient conditions for the nonoccurrence of stability switching in systems with arbitrary delays such as
?= b1ju(t - + ?= b2jv(t - r2j) dv(t) -;It = a21 u(t) + a22v(t) + ?= CljU(t - 6j) + ?= C2jV(t - e2j). duet)
-;It = auu(t) + a12v(t) +
n
n
rlj)
J=l
J=l
n
n
J=l
J=l
3.7.49
The interested reader can examine (3.7.49) with respect to stability switching as well as stability (for more details see Freedman and Gopalsamy [1988]). 3.B. Oscillations in linear systems
In this section we consider delay induced oscillations (not necessarily leading to periodicity) in linear vector - matrix systems. In particular, we obtain a set of sufficient conditions for all bounded solutions of a linear system of differentialdifference equations of first order to be oscillatory ( defined below) when the system has a one or more delays. For results related to this section we refer to Gopalsamy [1984c, 1986a, 1987]. We first consider systems of the form
t > 0, i
= 1,2 ...
,n
3.8.1
where aij and r are real constants with r > O. If we denote the colunm vector {Xl(t), ... ,xn(t)}T by x(t), then we can rewrite (3.8.1) in vector matrix notation
254
§3.8. Oscillations in linear systems
as follows: dx(t) A Xt-7; -( ) ---;u-=
t>O
3.8.2
where A denotes the n x n matrix of constants {aij, i,j = 1,2 ... ,n}. If (3.8.2) is supplemented with initial conditions of the form 3.8.3 where cjJ : (-7,0] I-t Rn, cjJ is continuous, then one can show that solutions of (3.8.2) - (3.8.3) exist on [-7, (0); in fact, we have from (3.8.2) - (3.8.3), x(t)
= .t dt.
It will follow from elementary properties of Laplace transforms that
X(A) = [A1+
["*lr) forsomej E {1,2, ... ,n} ~
rlO'jle
for somej E {1,2, ... ,n}.
3.8.15
But (3.8.15) contradicts (3.8.11) and hence (3.8.1) cannot have a bounded nonoscillatory solution when (3.8.10) - (3.8.11) hold and the proof is complete. [] Let us consider next, a linear delay-differential system of the form
d~~t) = Bx(t) + Ax(t -
r);
t>O
3.8.16
257
§3.8. Oscillations in linear systems
where A and B denote real constant n X n matrices with elements aij , bjj (i, j = 1,2, ... , n) respectively and r > is a constant. We adopt the following norms of vectors and matrices:
°
n
/lx(t)/I
n
n
IXi(t)li., IIAII = m~ L /aijli
= L i=1
J
IIBII = m~x L Ibjjl·
i=1
J
i=1
The measure fJ(B) of the matrix B is defined by
fJ
(B) = lim III +BBII-l 8-0+ 8
which for the chosen norms reduces to
p(B)
=
l~j'In [b
jj
+
t.lbijl]. i¥:j
(For more details of the measure of a matrix we refer to Vidyasagar [1978]).
Theorem 3.8.2. Assume the following for the system (3.8.16);
°
(i) detA # 3.8.17 (ii) fJ( B) + HAil # 3.8.18 (iii) (IIAllre) exp ( - rlfJ( B)I) > 1 Then all bounded solutions of (3.8.16) corresponding to continuous initial conditions on [-r,O] are oscillatory on [0,00).
°
Proof. Let us rewrite (3.8.16) in component form
dx.(t) ---;it = LbijXj(i) + Lajjxj(t-r), n
n
j=1
j=1
i
= 1,2, ... ,n
3.8.19
and suppose that there exists a solution say yet) = {Yl (t), . .. Yn(t)}T of (3.8.19) which is bounded and nonoscillatory on [0,00). It will then follow that there exists a t* > 0 such that no component of yet) has a zero for t > t* + r and as a consequence we will have for t ~ t* + 2r, 3.8.20
§9.8. Oscillations in linear systems
258
and hence
duet)
at
~
p(B)u(t) + 1\ A Ilu(t - 7),
t
2:: t* + 27
where u(t) == L:~=1 IYi(t) I; by the above preparation, we have u(t) t* + 7. Consider the scalar delay differential equation
dv(t)
dt = p(B)v(t) + II A Ilv(t - 7), with v(s) = u(s),s E [t*,t*
+ 7].
u(t)
~
t 2:: t*
3.8.21
> 0 for t 2::
+ 27
3.8.22
It is left as an exercise to show that
vet)
for
t 2:: t*
+ 27.
3.8.23
We claim that all bounded solutions of (3.8.22) are oscillatory on [t* + 2r, 00). Suppose this is not the case; then the characteristic equation associated with (3.8.22) given by 3.8.24 A = pCB) + II A II e- AT will have a real nonpositive root say). **. It will follow from (3.8.17) that A** Thus A** < 0 and we have from (3.8.24)
IA** 12:: IIAII elA**rl -lp(B)I·
=/: O.
3.8.25
It is now a consequence of (3.8.25), that
1~
(II A II .1'''1 T) I{W'I + IJl(B) I}
1}
H
~ {II A II .-1 p( 8) ITT} { exp [ (I >." I + IJl(B) I) T I [{I >." I + II 1'( B) I
3.8.26 The last inequality contradicts (3.8.18). Hence, our claim regarding the oscillatory nature of v on [0,(0) is valid; now since v has arbitrarily large zeros, u will have arbitrarily large zeros which means that I:~=1 IYi(t) I is oscillatory implying that yet) is oscillatory; but this is absurd since ]jis a nonoscillatory vector. Thus, there cannot exist a bounded nonoscillatory solution of (3.8.19) when the conditions of the theorem hold and the proof is complete. [] The following result deals with oscillations in linear systems of equations with a multiplicity of delays.
259
§3.8. Oscillations in linear systems
Theorem 3.8.3. Let aij , Tij (i,j = 1,2, ... , n) denote real constants such that aij =I- O,Tij > O(i = 1,2, ... ,n) andT;j ~ 0, (i,j = 1,2, ... ,n; i =l-j) and consider the system
t > O.
3.8.27
If aj and Tij of (3.8.27) satisfy n
and
det A = det(aij) =I- 0,
!aii ITii e > 1 + e .2: !aij !Tije,
3.8.28
j=l
j#i
then all bounded solutions of (3.8.27) corresponding to continuous initial condiTi]' are oscillatory on [0,00). tions denned on [-T, 0], T = l Ia"le Ti ;l6 1 - L....t ~ Ia··1 e Tij f]
-
H
j=l
j#i
1
6
1,
3.8.30
§.9.8. 08cillation.3 in linear .3Y.3tem8
260
we derive that
n
!81 + L
!aii!eTiilol2:: laiileTiiO.
j=l
J#-i
Rearranging terms in the above,
and this leads to (1+e
t
lij! aij
I) 2:: I
ajj
llii e
for somei E {I, 2, ...
,n} .
3.8.31
)=1
i#-i
But (3.8.31) contradicts (3.8.28). Thus (3.8.27) cannot have a bounded nonoscillatory solution when the conditions of the theorem hold. [] The following corollaries are of interest by themselves. Corollary 3.8.4. Suppose that the coefficient matrix A = (aij) in (3.8.1) has at least one real negative eigenvalue say f3 which is such that 0
0 for each x E IR n , x =f 0 E IRn . We note that the verification of (3.9.30) is nontrivial due to the explicit dependence of (3.9.30) on B. In the following result we derive a sufficient condition explicitly in terms of A and H rather than directly through Bas in (3.9.30).
Theorem 3.9.3. If the elements
aij
of A and Hjj of H satisfy
3.9.31
then every solution of (3.9.28) satisfies n
L xHt) i=l
--t
0
as t
--t 00.
3.9.32
§S.9. Simple stability criteria
272
Proof. Consider a Lyapunov functional V = V(x)(t) defined by
V =
t
[X:C t ) +
,=1
t Jot (1
00
I Hij(u,s) IdU)
t
)=1
X~(S)dS] .
3.9.33
Calculating ~~ along the solutions of (3.9.28), dV n [ dt = ~ 2xj(t)
+
{nj;aijxj(t) + j; J.t Hij(t,S)xj(s)ds } n
t, ([I
-t
t {2a
i;
,=1
Hij(U,t) IdU) xJ(t)
([IHi j (t,S)I X 1(S)ds)
)=1
$
0
1
3.9.34
0
+ L (Ia jd + la;j
I)
}=1
jf;i
+
t,[1 Hij(t, s) Ids +t, [0 IHji (u, t) Idu }x;(t)
3.9.35
n
~ -11-
L x;(t).
3.9.36
i=1
One can now see that (3.9.36) leads to
from which the uniform boundedness of both II x(t) II and II x(t) II for t 2:: 0 will follow. An application of Lemma 1.2.2 of Barbalat (see Chapter 1) implies that /I x(t) II -+ 0 as t -+ 00 and this completes the proof. []
273
EXERCISES III 1. Assuming that a, b, I, aj, Ij (j = 1,2, ... ,n) are positive constants, prove that solutions corresponding to positive continuous initial values of the following remain positive and exist for all t ~ 0:
(i)
d~~t) = x(t - I)[a - bx(t)].
(ii)
d~~t) = x(t - I) - bxZ(t).
(iii)
d~~t) =
Ej=l aix(t - Ii) - bx 2 (t).
Prove that the trivial solution of each of the above equations is unstable while the nontrivial steady state is asymptotically stable with respect to positive initial values. Also examine the absolute stability (independent of delay) of the non trivial steady states.
2. Can you generalize the result of problem (1) above, to the following integrodifferential systems: (i)
d~~t) = (fooo k(s)X(t-S)ds)[a-bx(t)]. oo
(ii) d~~t) = a Jo
k(s)x(t - s)ds - bx 2 (t).
(iii) d~~t) = aJ; k(s)x(t - s)ds - bxZ(t). State your assumptions on the delay kernel k(.). 3. Discuss the stability and instability of the trivial and nontrivial steady states of the following scalar systems (assume a, b are positive constants and 11,12 are nonnegative constants).
(i) d~~t) = ax(t - It) - bx(t - 11)X(t - 12)X(t). (ii) d~~t) = x(t - 1I)[a - bx(t - IZ)X(t)].
(iii)
d~~t)
(iv)
d~~t) = x(t) (a - b[ 10
=
10
00
k.(s)x(t - s) ds 00
[a - bx(t)jooo k,(s)x(t - s) dS].
k( s)x(t - S)ds] ').
(State your assumptions on k1 , k2, k in (iii) and (iv) above).
Exercises III
274
4. Let b, c, r be real constants and let P denote the class of all nonnegative
solutions of
dy(t)
d:t
= by(t - r)[l - yet)] - ey(t) ,
for t E [0,(0). Assume b > 0, c ~ 0. Prove that the trivial solution is asymptotically stable within the class P if b < c and the nontrivial constant solution yet) == 1 - (c/b) is asymptotically (locally) stable if e < b. ~ [0, 1], is continuous}. Prove that ife ~ b > 0, then the trivial solution of the equation in problem 4 above, is globally asymptotically stable with respect to S. If c < b, then show that yet) = 1 - (c/b) is globally asymptotically stable for all initial conditions in S with (s) > O,S E [-r,01.
5. Let S = {I: [-r,O]
°:;
6. Generalize your discussion of problems 4 and 5 above for systems of the form
d~~t)
=
(b 1= k(s)y(t - S)dS) [1 -
yet)] - cy(t).
State your assumptions on the delay kernel k(.). 7. Let the nonnegative function y denote a solution of the difference inequality:
yet) ::; ay(t - ret)) + bexp( -,Bt) y(t)::; (t),
° °: ; ret) ::; r*. °
where a ~ 0, b ~ 0, ,B > and exist constants a > and N >
°
t E [-r*,O]
yet) ::; N exp( -at), where a
< min{,B,a o }
If a < 1, then prove that there
such that
t
~
°
and a o is the unique positive root of
and
N=
sup
1(s)l+b[l-aexp(ar*)]-I.
sE[-r· ,0]
What type of generalization to vector - matrix systems can be developed? (for more details see Xu [1989].)
Exercises III
215
8. Let 71, 72 ,73 ,7 be nonnegative constants such that 7 Show that the set G = { O?
dx(t) -;It
= x(t)[rl - allx(t) - a12y(t - 7)] }
(1)
dy(t)
-;It = y(t)[T2 - a21x(t - 7") - a22y(t)]. dx(t) = x(t) dt
[1 - kl + ay(t x(t) - 7")
dye t)
[
l)
1
y( t) -;It = yet) 1 - k2 + f3x(t - 7") .
(2)
Exercises III
276
dx(t) = X(t){kl a:t
}
x(t) - ay(t - T)}
(3)
dy(t)
dt = y(t){ -k2 + f3x(t -
d~~t) = x(t)[al -
T) - Yet)}.
a2x(t) - f3y(t - T)] )
1
-dy(t) _ ()[ hy(t) - ry t 1 - ( ). dt x t - T dx(t) = x(t a:t dy(t)
dt
TI)[rl
+ allx(t) -
(4)
al2y(t - T2)] } (5)
= yet - T3)[r2 - a2lx(t - T4) - a22y(t)].
11. Let aI, 0:2, {31, (32 be positive constants. Prove that for each positive constant e, the system
(1 - Xl) - e(xI - X2) dX2(t) =a 2x 2(1- X2) -e(x2 -Xl) dt f32 dXI(t) = alxI dt
f3I
has a positive steady state (xr ,xi) with state (xr, xi) of
xr > 0, xi > 0. Prove that the steady
dx~?) = ""X, (1 -;:) - ex, + e [= k'2(t dx;?) = "2 X2 where kl' k2 : [0,00)
1.
!---;.
(1 -;:) - eX2 + e['= k
21 (t
S)X2(S )ds
- S)Xl( s)ds
[0,00) are continuous and integrable on [0,00),
00
kj(s)ds=l;
1.
00
skj(s)ds < 00,
i=1,2
is locally asymptotically stable. Under what additional conditions can you derive a similar result for a system of the form
dx~?) = ""X, (1- ;:) - ex,(t) [= kll(t - S)Xl (s )ds
['= k (t - S)X2( S)ds "2 2 (1 -;:) - eX2(t) [= k 22 (t - S)X2(S )ds +e
dx;?) =
12
X
+E
[=
k2,(t - S)x,( s )ds.
277
Exercises III
Formulate sufficient conditions on kij ( i,j = 1,2) for your result. (For more results of this type see Gopalsamy [1983c]) 12. Assume that f is a continuously differentiable function such that there exist positive constants x* and H satisfying
x* f(x*) - H
= o.
Obtain sufficient conditions for the local asymptotic stability of the steady state x* of the following scalar equations:
(a)
d~~t) = x(t)f(x(t - 1'» - H.
(b)
d~~t) = x(t - 1')f(x(t - 1'» - H.
(c)
d~~t) =x(t- 1'df(x(t- 1'2»-H.
(d)
d~\t)
= x(t)f (
(e)
d~\t)
=
J~= k(t -
[f(J~=
s )x( S)ds) - H.
k.(t - s)x(s)dS)]
J~= k2(t -
s)x(s)ds - H.
where 1',1'1,1'2 are nonnegative constants and k, k1, k2 : [0,00) 1-4 [0,00) are piecewise (locally) continuous on [0, 00) such that for i = 1, 2
13. Assume that the function f in problem 12 is given by the following (a, b, n are positive constants, n ~ 1).
(a)
f (x) =
(b)
f(x) = a - blogx
(c)
a - bx
f(x) = a - bxn. Do the same as in problem 12 for these
f.
Exercises III
278
14. Let
h ,12 below be continuously differentiable functions of their arguments.
Assume that Tij (i, j = 1,2) are nonnegative constants and suppose there exist positive constants xi, hi (i = 1,2) such that X;fi(X~
,x;) = hi
;
i
= 1,2.
Derive sufficient conditions for the local asymptotic stability of (xi, xi) in the following:
Do the same as in (1), (2) above if
hex, y) =
an X - a12Y
Tl -
hex, y) = T2 - a21 x - a22Y where
Ti,aij
(i,j = 1,2) are positive constants.
15. Can you formulate and analyse the local asymptotic stability of (xi,xi) in the systems of problem 14, when the delays are continuously distributed over an infinite interval? 16. Assume that kl , k2 , b1 , b2 are positive constants. Under what conditions the system dx(t) = kl _ b x(t) dt 1 +y(t) 1
dy(t)
d1 = k2X(t) -
b2y(t - T)y(t)
has a steady state (x*, y*), x* > 0, y* > O? Derive sufficient conditions for the local asymptotic stability of (x*, y*). Generalize your result to a system of the form
dx(t)
kl
-dt- = 1 + y( t dy(t)
d1
TI)
-
b1x(t - T2)X(t)
= k2X(t - T3) - b2y(t - T4)y(t).
279
Exercises III
17. Under what conditions the following scalar systems will have locally asymptotically stable positive steady states; examine stability switching also;
dx(t)
---;It dx(t)
=
AX(t - .r) a+xn(t-r) -,x(t).
(i)
Ax(i)
-at =
a
sup xes). + xn(t) - ,xCi), xCi) = sE[t-r,t]
( ii)
( iii)
d~~t)
= -,xCi)
+ ;Jx([iDe-ax([tj).
(iv)
;J
dx(t)
-at = -,x(t) + 1 + x(t -
(v)
r)
(A,a,r,rl,r2,r3 are positive constants). 18. Derive sufficient conditions for the existence of a locally asymptotically stable positive (componentwise) steady state of the system
dx(t)
a
---;It = 1 + ;Jx'Y(t - rd
AX(t) 1 + J-l y6(t - r3)
dy(t) = AX(t - r4) dt 1 + tLy6(t _ r5) - wy(t) where rl, .. ,r5 ,a,;J",8,J-l,A,W are positive constants.
Wheldon [1975]
19. Formulate and examine the asymptotic behavior of integrodifferential analogues of the systems in problems 17 and 18 above. 20. Examine the local asymptotic stability of nonnegative (componentwise) steady states of the following systems:
dx(t) = x(t) [ b -at
1
+ aux(t) + a12 Jo[00 k 12 (s)x(t - s)y(t - s) ds
1) 1
dy(t) = yet) [ b + a21 Jo[00 k (S)X(t - s)y(t - s)ds + a22y(t) . ----;It"" 2 21
Exerci.'3e.'3 III
280
dx(t) = dt
roo 10
kn ( s )x(t +
dy(t) dt
= roo JO
s) ds
a121°° k
[b
I
+ allx(t)
12 (S)X(t
- s)y(t - s)ds ]
k22 (S)y(t-s)ds [b 2 +a 21 [00 k 21 (S)X(t-s)y(t-s)ds
Jo
+ a22y(t) ]. Formulate your conditions on the various delay kernels appearing in the above systems. 21. Consider the scalar system
dx(t)
---;It
= -ax(t)+ Jot
where a is a positive number, k : [0,(0)
/.00 k(s)ds
I b I). It) + cx(t - 12),
I a I > I b I + I c I).
(2)
Exercises III
284
26. Consider a vector matrix system
dx(t) --;u= Aox(t) + AIX(t -
t>O
r)j
in which x(t) ERn; T > OJ A o, Al are real matrices. Let Q = II Ao II; f3 = 1/ Al " denote operator norms of the matrices consistent with some norlillR n • Assume that Ao is such that
for some constants a ~ 1; b > O. Let (J" = 2(-. If (J" < 1 then prove that for any r > 0 the system (*) is asymptotically stable and the following estimate is valid:
II x(t) 1/
::;
a{ sup "x(s) sE[ -r,rj
lI}e-
8t
;
t
~0
where 6 is the unique solution of the equation b - J.L = af3ep.r.
27. Let
~
denote the characteristic quasi polynomial defined by m
0, m
L IQiCiy) I < IP(iy)l· j=l
=0
285
Exercises III
28. Suppose p, q are real numbers such that p + q < O. Prove that there exists a real number say 8 = 8(p, q) > 0 such that all the roots of
..\ = p + qe-).r have negative real parts if 0 < I given by 8(p, q) < 7r[lp I + I q IJ/8.
< 8. Prove also that an estimate for- 8 is
29. Let all the roots of
D("\) = det[ aij
+ bij -
..\8ij J = 0
have negative real parts. Prove that there exist two positive numbers 8 8( aij, bij ) > 0 and € = €( aij, bij ) > 0 such that all the roots of
satisfy
~e(..\) ~ €
provided 0
~ lij ~
8.
=
Qin Yuan-Xun et al. [1960]
30. Prove that positive constants € and '" can be suitably selected so that the trivial solution of (for details see Qin Yuan-X un et al. [1960])
dx(t) -;It =€x(t) + yet) + ",[y(t) - yet - I)] dy(t) -;It = €y(t) - x(t) - ",[x(t) - x(t - I)] is unstable for
I
= 0 but is asymptotically stable for some positive I.
31. Obtain a set of necessary and sufficient conditions for the trivial solution of
to be asymptotically stable for all
I
~
0.
32. Assume that all the parameters are positive constants in the following population systems. (a) derive a set of sufficient conditions for the systems to have a positive steady state.
Exercises III
286
(b) obtain the variational systems corresponding to a positive steady state. (c) examine whether the trivial solution of the variational system can be asymptotically stable in the absence of delays. (d) whenever the trivial solution of the variational system is asymptotically stable in the absence of delays, examine whether a delay-induce~ switching from stability to instability can take place. (e) if delay induced switching from stability to instability cannot arise, can you prove that the positive steady state of the full nonlinear system is globally asymptotically stable with additional assumptions?
I: aj x(t - rj) - bx (t). n
dx(t) dt
-- =
2
dx(t)
-;It = axm(t- r) - bx(t)j
mE [1,00).
dx(t) = /X{l - X(t)} - bx(t)y(t) dt K
d~~t)
d~~I)
(1)
j=l
1
= ')'x(l) { 1-
x~)} _ bX(I)ym(l) (4)
r)ym(t - r) - 8y(t)
by(t) dx(t) = X(t){l _ X(t)} _ ax(t) dt /. K a + x(t) b + ym(t)
=c
dx(t)
--;It
(3)
= cx(t _ r)y(t - r) - 8y2(t).
dy(t) --;u = cx(t -
dy(t) dt
(2)
ax(t - r) by(t - r) _ 8y(t) a+x(t-r)b+ym(t-r) = x(t)[b1
-
(5)
allx(t) - a12y(t)]
dy(t) --;u = b2x(t -
rl)y(t -
dz(t) --;u = b3y(t -
z rz)z(t - r2) - a33 z (t).
2
rl -
d~~t) = ')'x(t) { 1 - (x~)
a22Y (t) - a23y(t)Z(t)
r} -t,
aj
X(I)yi' (I)
dYj(t) ] ] (t - r·) ] - d·y ] ]·(t) dt = c'a ] ]·x(t - r·)y~i j=1,2,···,nj
0
0, there exist tt, t 2 , (tl = tl(c!), t2 = t2(C2» such that
x(t) S (bja)
x(t)
~
+ CI
for
t ~
(bja) - cz for t
~
tl
tz .
CI
> 0 and
4.1.4
We have been able to note (4.1.4) directly from the solution (4.1.2). We can also make the same observation, if we show that the positive steady state (ajb) of (4.1.1) is globally asymptotically stable by other methods, which do not need an explicit knowledge of the solution.
§4.1. Some preliminaries
293
As a second example, let us consider the scalar equation dx dt
x) = AX (1·;.... k -
a
ax
+xA
4.1.5
where A, k, a, A are positi~re constants. One can see that, if A > A, then (4.1.5) has a Wlique positive steady state say x· and that x(O) > 0 will imply x(t) > 0 for all t ~ O. We consider the Lyapunov function v where vex)
=x -
x* - x*log(x/x*).
4.1.6
Calculating ~; along the solutions of (4.1.5), we have 4.1.7 As a consequence of (4.1.7) one can derive the following for (4.1.5).
"If the positive constants A, k, a, A in (4.1.5) are such that A > max(A, Ak/a), then every solution x(t) of (4.1.5) with x(O) > 0 has the asymptotic behavior lim x(t) = x*
t-oo
4.1.8
where x* is the unique positive solution of
x* [A (1 - x*) - ~l = 0" . K a+x·
4.1.9
Consider now the scalar equation dx dt
=~-Dx b + xm
where b, f3, D and m are positive constants. The above system can be put in the form d:i; = ~ {(f3 - D)(b/D) _ xm}. 4.1.10 dt b + xm
If f3 > D, (4.1.10) has a unique positive steady state x; also solutions of (4.1.10) corresponding to x(O) > 0 exist for all t ~ 0 and satisfy x(t) > 0 for t ~ O. If we consider a Lyapunov function v defined by 4.1.11
§4.1. Some preliminaries
294
then we note
dv(x(t)) = _ (xm _ xm)2 dt
4.1.12
showing that, every solution of (4.1.10) has the following asymptotic behavior (details are left to the reader)
x(O) > 0 ~ x(t) > 0 and x(t)
-P
X as t
4.1.13
-P 00.
From the foregoing examples one can observe the following: if in an equation of the form
dx dt = K - f(x),
4.1.14
where K is a positive constant and f : [0,00) f--+ [0,00), f is continuously differentiable and monotone increasing such that f(O) = 0, f'(x) 2 c > 0 for x ~ 0, then (4.1.14) has a positive steady state say x such that f(x) = K. The reader should be able to verify that (4.1.14) has the following behavior:
x(O) > 0
~
x(t) > 0 for t
~
0 and
x(t)
-P
X as t
-P
00.
4.1.15
It has been relatively easy to verify the existence of positive steady states in the above systems due to their scalar nature. When we consider the dynamics of multispecies population systems described by nonscalar systems of differential equations, the problem of ascertaining the existence of positive steady states becomes difficult; it is not uncommon to assume that such steady states exist and then proceed to analyse the asymptotic behavior of the relevant systems. However, in a number of multispecies model ecosystems such as the Lotka-Volterra competition equations, it is possible to propose sufficient conditions for the existence of positive steady states. It is found that the same set of sufficient conditions which guarantee the existence of a positive steady state, sometimes can also guarantee the global attractivity of such a steady state. The following result is of the above type and is due to Kaykobad [1985].
Lemma 4.1.1. Suppose 'xi, aij (i,j
that aji
aij
'xi > E7:~ J .,.'
>0 20 aij('xj!ajj)
= 1,2, ... , n) are nonnegative constants such i = 1,2, ... ,n i,j = 1,2, ... ,n i = 1,2, ... ,no
4.1.16
295
§4.1. Some preliminarie3 Tben tbe Lotka-Volterra competition system i = 1,2, ... ,n
bas a componentwise positive steady state x· tions
= (xi, ... , x~)
4.1.17
satisfying "tbe equa-
n
L aii x; = Ai
and
xi > 0;
i
= 1,2, ... , n.
4.1.18
i=l
Proof. Let AD denote the n x n diagonal matrix; AD = diag( au, a22, ... ,ann)' Then (4.1.16) will imply that AD is nonsingular and that (AD)-l > 0 in an elementwise sense. Define an n X n matrix B as follows: B
= A(AD)-l -
I
4.1.19
where I denotes the n x n identity matrix. We note that B is nonnegative (elementwise) and also that A
= (I + B)AD;
4.1.20
The assumptions in (4.1.16) will imply that the components of a column vector c = col. { Cll C2, ••• , cn } defined by
c= (I -
B).,
4.1.21
satisfy the condition Ci > 0, i = 1,2, ... , n. Since Ai > 0, i = 1,2, ... , nand B 2:: 0 (elementwise), it will follow from (4.1.21) and the componentwise positivity of c that p(B) < 1, (p(B) being the spectral radius of Bj see for instance Berman and Plemmons [1979}, Ch. 6). Let p = p(B)j by the Perron-Frobenius theorem there exists a vector J = col.{ d l , d 2 , ••• ,dn } , d j 2:: O,j = 1,2, ... ,n such that
BT being the transpose of B. Since Ai > 0, Ci > 0, i (J)Tc> 0; but we have from (4.1.21) and (4.1.22),
=
1,2, ... , n we have
§4.1. Some preliminaries
296
which implies that 1- P > OJ a consequence of this is that both (I -B) and (I +B) are nonsingular. The nonsingularity of A = (aij) now follows from (4.1.20) and that of AD. We have
A-IX = (AD)-I(I + B)-IX
= (AD)-I(I + B)-I(I = (AD)-I(I2 _ B2)-le
B)-Ie 4.1.24
= (AD)-l
(tB2i) C
(since pCB)
< 1)
)=0
( componentwise )
[J
and the proof is complete. The set of sufficient conditions n
Ai >
L aii(>\i/aji),
i = 1,2, .. . ,n
j=l j ¢:i
of (4.1.16) has an interesting ecological interpretation; in fact, the motivation for the derivation of Lemma 4.1.1 has come from an analysis of Lotka-Volterra competition equations. We remark that the conditions (4.1.16) are only sufficient conditions for the conclusion of Lemma 4.1.1. One can argue, that by means of Cramer's rule, it is possible to give necessary and sufficient conditions for (4.1.18) to have a componentwise positive solution; such conditions are analytic and unintuitive with respect to the system (4.1.17). It will be found below that the conditions (4.1.16) are also sufficient to make the steady state x* of (4.1.17), a global attractor with respect to solutions of (4.1.17) with Xi(O) > 0, i = 1,2, ... ,n. The proof of the following result (see Gopalsamy [1980], Gopalsamy and Ahlip [1983]) is similar to that of a somewhat more general one to be proved in the next section. One of the implications of the following result is that whatever the size of the delays, nonconstant periodic solutions cannot exist for the system considered and this is contrary to the commonly held expectation of the influence of delays in model ecosystems.
Theorem 4.1.2. Assume that the conditions (4.1.16) of lemma 4.1.1 hold. Let Tij 2:: 0, (i,j = 1,2, ... ,n;, i =1= j). Then every solution of the delay differential
297
§4.1. Some preliminaries system
dUi(t) = Ui(t) [ Ti -;uUi(S) = xi > x; > 0. If E* exists, then one can ask, under what additional conditions (if needed), the steady state E. : (xi,xi,xi) is globally attractive in the sense that
Xi(O) > 0 =} lim Xi(t) t-oo
= xi ,
i
= 1,2,3.
4.2.4
An intuitive examination of (4.2.1) together with (4.2.4) suggests the following: if both the consumers do not overexploit or "overkill" the resource (as measured by the consumption (or predation) parameters al2 and aI3), and if the resource can reproduce itself sufficiently (as measured by the potential regeneration rate parameter bI ) so as to withstand consumption pressure, then the three species community described by (4.2.1) not only can "persist" in the sense
Xi(O) > 0 =} lim t-.oo inf Xi(t) > 0 , i = 1,2,3, but also satisfy (4.2.4). We proceed to establish a set of sufficient conditions under which all solutions of (4.2.1) with positive initial values will converge as t -+ 00 to the positive equilibrium E. of (4.2.1). The sufficient conditions will be of such a type, one can intuitively foresee. Precisely we prove the following:
§4-2. Competition
300
Theorem 4.2.1. Assume that the constants bi, aii (i = 1,2,3) are positive and aij ~ 0 , i =f j , i,j = 1,2, ... , n. Suppose the following hold: ( i)
(ii) ( iii)
b2 hI b3 bi bi > a 1 2 - - - a 1 3 - a22 all a33 all b2 bi b3 bI ] 1 b2 .[ bi - a I 2 - - - a 1 3 - - > a22 all a33 all au b3 [ bI
b2 bi
b3 bI
- aI2-- - a13-a22 all a33 all
]
-
1
au
4.2.5 b3 hI
a23-a33 all
2 bI > a 3b2 --' a22
all
4.2.6 4.2.7
Then the equilibrium point E.: (xr,xi,xi) of (4.2. 1) defined by (4.2.2) exists and all solutions of (4.2.1) have the following behavior:
Xi(O) > O:::} lim Xi(t) = t-oo
xi
j
i
= 1,2,3.
4.2.8
Proof. We first note that (4.2.5) - (4.2.7) are kept in a form in which it is easy to interpret, rather than in a compact and simple form. It is found from (4.2.5) ( 4.2.7) that b3 b2 b2 > a23- and b3 > a32a33
a22
which will imply the existence of E. : (xi, xi, xi), xi > 0, i = 1,2,3. It is easy to establish that all solutions of (4.2.1) are defined on [0,00) and Xi(t) > 0 on [0,00) when Xi(O) > 0, i = 1,2,3. Using such a positivity of the solutions of (4.2.1) and the property of logistic growth one can show that for any C 1 > 0 there exists a tl > 0 satisfying
ui 1) X2(t) < ui 2)
Xl(t)
0 such that
[b b3 [b
b2
bi l i -
-
a12U?) - aI3U?)
?) - a I3 U?)] a!t 3 aI2 U ?) - a I3Ui )] a!t a12 U
>0
> a23 U ?) > a32 U ?)
} .
4.2.10
§4.2. Competition
301
The possibility of such a choice of Cl > 0 satisfying (4.2.9) - (4.2.10) is guaranteed by (4.2.5)-(4.2.7). Having selected C1 > 0, tl > 0 we choose C2 > 0 small enough to satisfy
and
4.2.11
The possibility of choosing C2 > 0 satisfying (4.2.11) is a consequence of (4.2.10). It will follow from (4.2.1) and (4.2.9) that
leading to the existence of a t2 > tl for which
Xl(t) > Lil)
=
X2(t) > Li 2 ) X3(t) >
Li
3
t > t2 •
4.2.12
)
It is a consequence of (4.2.1) and (4.2.12) that
dXl(t) < Xl(t) ({ bi
~
-
a12Ll(2) - a13Ll(3)} - allXl(t) ) t
> t2'
4.2.13
We also note
>0
(by (4.2.5».
4.2.14
§4.2. Competition
302
One can now show from (4.2.13) - (4.2.14) that there exists a 0, C3 < min { c2} such that
t3
> t2 and
C3
>
t,
Xl (t)
< UJI)
U
= {bi
(2) Z
=
X3(t) < UJ3)
=
X2(t)
.,
t3'
4.2.15
The positivity of the estimates U?) and U~3) are verified as follows:
b2U2(1)
-
a23Ll(3) = b2 [ ( bl - a23 [
> b2 [ (b i > bz [ >
°
(b
(3»)
(2) - a12 L I - a13 L I
(b 3LP) -
a12 Li
2
a~3
a3z U?») ) -
a 13L
+ C3 ]
-
c; 1
i3») 2-]a2a~LiI) all a33
i
3
a12Ui2) - a I3U »)
l -
1
all
2-]- a23~~ all
aaa all
(by the second of (4.2.11»;
4.2.16
and similarly,
I baU2(1) - a32 L (2)
= ba [ (b I - a32 [
b[(b
1 I aI2Ll(2) - a13 L (a») ~
-
(b 2Lil) -
+ c3 ]
i3») a~I - c; 1 a 13 U?») 2-]- a32~~ all a22 all
a 2aU
> a
l -
>0
(by the third of (4.2.11).
a12Ui2) -
4.2.17
Now using the upper estimates in (4.2.15), we derive a set of lower estimates as before; first we need the following verification: bi
-
(2) (a) a12 Uz - a13 Uz
= bi
-
aiZ -
> bi
> bi >0
[
(1) bU
an [
Z
2
-
(3»)
a23 L l
1
a22
cal + 2"
(b U?) - LF») a~3 + "; 1 3
a 32
-
aI2
(~u?) + ~) a22 2
-
a 12
U?) - alaU~a)
al3
(by the first of (3.16»,
(~U?) + C1 ) aa3 2 4.2.18
303
§4.2. Competition •
(1)
(2)
(3)
in which we have used e3 < e1' Let us define £2 '£2 '£2 1 } [ b1 - a12U2(2) - a13U2(3)] ~ (1) U(3)
b2 £2 b3 £2(1)
-
a23
-
an
as follows:
.
2
t > t2 •
'
U(2)
4.2.19
2
Using (4.2.18) we derive that
4.2.20 Similarly,
4.2.21 It will follow from (4.2.18)-(4.2.21) that there exists a positive e4 satisfying £~1) _ e4 > 0 } { b2 (£~1) - e4) - a23U~a)} a~2 - T > 0 .
{b3(.e~1) - e
4) -
32
a UJ2)}
< min {i, e3} 4.2.22
a;3 - T > 0
Now using the upper estimates in (4.2.15), we have
dXl(t)
~
(2)
(3)
> Xl(t) { [b 1 - a12 U2 - a13U2 ] - allXl(t) } t > ta
with which and (4.2.22) one can show, there exists a t4
X1(t) > L~l)
=
X2(t) > L~2)
=
X3(t) > L~3)
=
[b 1
-
a12 UZ(Z) - a13 U(3)] Z
1 Ci7i'
C4
4.2.23
> ta such that
>0)
- a U(a)]..L - ~ > 0 [ bZ L(l) 2 23 2 a22 2 - a32 U(2)]..L - ~ >0 [ b3 L(l) 2 2 a33 Z
. '
t > t4 • 4.2.24
§4.2. Competition
904
The positivity of the estimates L~i), i We thus have
= 1,2,3 is a consequence of the choice of C4.
i
< Xi(t) < U?); L~i) < Xi( t) < U~i);
i = 1,2,3; i = 1,2,3;
Li )
4.2.25
At this stage let us compare the respective lower-and upper estimates: for instance, U2(1)
-
UI(1)
= { bI < c3 -
(2) (3)} a12 L l - a13Ll -
1
all
CI
a22
0;
a further similar analysis will lead to
Thus, we have from the above
U?) }
LP) < L~I) < XI(t) < UJ1) < L~2) < L~2) < X2( t) < U?) < U?) LP) < L~3) < X3(t) <
0, 1,2, ... , n; i =f j; furthennore, let
aji
> 0, (i = 1,2, ... ,n) and
aij
2:: O,i,j =
n
bi >
L
i
aij(bj/ajj);
= 1,2, ... ,no
j=l j#i
Then it will follow from the result of Lemma 4.1.1 that (4.3.1) will have a steady state x* = (xi, ... ,x~) with xi> 0, i = 1,2, ... ,n. If we let i
= 1,2, ... , n
4.3.2
in (4.3.1), then dYi(t) ~
* (t = - ~ L...J aijXjYj
Tij
)
- Yi
( t ) L...J ~ aijXjYj * (t -
j=l
Tij
j=l
i
) 4.3.3
= 1,2, ... ,n
together with
Yi(S) = [' is any root of (4.3.6), then it will follow from Gershgorin's theorem (Franklin [1968]) that n
>. + a"x~e-..\rii 1 < ""'" a ··x~e-..\rji U I L....J }I I I
4.3.7
j=1 j 'Fi
for some i E (1,2,3, ... ,n). As a consequence of (4.3.7), complex constants kj = kj(>'), I kj(>') I < 1, j = 1,2,3, ... , n will exist, so that for any root>. of (4.3.6), n
i;x~ >. + a"e-"\r H I
+~ L....J aJI"x~ kJ·(>')e-..\r = 0 j ;
I
j=1
i'Fi
4.3.8
for some i E (1,2, ... , n) . The following result provides a set of sufficient conditions for all the roots of (4.3.6) to have negative real parts. Lemma 4.3.1. Assume the following: (HI) the real constants 7"ij ~ 0 (i,j = 1,2, ... , n) satisfy n
7"ii
~ ffi;in
1 0 so that NI (t) ~ mast -+ 00 where m ~ bdbll and hence. there exists a number si satisfying
This proves (4.3.40) for solutions of (4.3.36) which are nonoscillatory about (bi /b ll ).
§4.S. Competition and cooperation
914
If Nl is oscillatory about bl/b 11 , then (4.3.40) is established by considering
a local minimum of NI and arguments are similar to the corresponding details carried out in Chapter 1 for the delay logistic equation. We omit these details. [] We shall proceed to discuss the global attractivity of the positive equilibrium (Ni,Ni) of (4.3.21). We let
NI(t) == N;[l
+ YI(t)],
N2(t) == N;[l
+ Y2(t)]
4.3.43
and derive from (4.3.21) that YI and Y2 are governed by
dYl (t) -;[t
= -[all(t)Yl(t -
1'11) + aI2(t)Y2(t - 1'12)]
dY2(t) -;[t = -[a21(t)Yl(t - T2d
4.3.44
+ a22(t)Y2(t - 1'22)]
where for t > 0,
a11(t) = bllNI(t);
a22(t) = b22 N 2(t)
al"2(t) = bI2 (N; /N;)NI(t) ;
a2I(t)
= b21 (N; /N;)N2(t).
4.3.45
We can conclude from the results of the above lemmas that there exists a number (J > 0 such that for all t > (J, mll ::; all
(t) ::;
Cll;
m22 ::; a22 (t) ::; C22 j
a12(t) ::;
CI2
a2I(t)::; C21
for
where
t;:::
(J"
I
b11 Md Tll] } m22 = b2 exp[(bz - b22 M 2)T22] mll
= bI exp[(b i
-
r1
Cll
=
TI e
Tll
C22
= T2er2T22
CI2
= b12(N;/N;)(Tl/bll)erlTll
4.3.46
4.3.47
4.3.48
C21 = b2I (N; /N;)(T2/b22)er2T22. For convenience we define two numbers J.Li and J.Li as follows:
J.Lr
= bI exp[(bI -
-
bllMt)Tll]
+ C12 T12) + CZI (C21 1'21 + CZ2 T22) + CllTll(Cll + CI2) + C2I T21(C21 + C22)] [Cll (Cll 1'11
4.3.49
J-l; = b2 exp[(b2 - b22 M 2)T22] -
+ C12 T12) + C21 (C21 1'21 + CZZT22) + CIZ T12( Cll + C12) + C22 TZ2( C21 + cn)]. [C12( Cll 1'11
4.3.50
315
§4.9. Competition and cooperation
The following result provides a set of sufficient conditions for the global attractivity of the positive equilibrium (Ni, NZ) of (4.3.21). Theorem 4.3.6. Assume the following conditions hold:
(i)
ri, bij, Tij E
i,j
[O,oo)j
= 1,2.
(ii) bl =
rl -
b12(r2/b22)eT2T22 > 0
b2 = r2 - b21(rI/bn)eT1Tll
4.3.51
>0
(iii) JL~
4.3.52
> 0,
(iv) The quadratic form
= [Yl
Q(y}' Y2)
Y2] [ mn C12 + C21
C12
+ C21 1[Yl 1
m22
Y2
is nonnegative on the set
Then all positive solutions of (4.3.21) satisfy 4.3.53
Proof. We define afunctional V = V(Yl,Y2)(t) = VI + V2 where VI and V2 are as in the case of (3.9.8) and (3.9.12) of Chapter 3. We estimate the rate of change of V similar to that in Theorem 3.9.1 of Chapter 3. On using (4.3.45)-(4.3.50) and assumptions (i)-(iv), we will be led to an inequality of the type
V(Yt,Y2)(t)
+ J.1.
itot[Yi(s) + y~(s)] ds ~ V(Yl,Y2)(tO)
where JL = min{J.L~,JL;}. The remaining details of proof are similar to those of Theorem 3.9.1 of Chapter 3 and we omit these details. Thus, we conclude that lim YI(t)
t--oo
= 0;
lim Y2(t) =
t-oo
o.
[]
§4.3. Competition and cooperation
316
We proceed to derive an alternative set of sufficient conditions for the validity of (4.3.53). Let us rewrite (4.3.44) as follows;
dVl(t) --;It =
-all (t)Vl(t)
- a12(t)V2(t)
+ al1(t)J.~", dV2(t) --;It
rileS) + a,2(t)
J.~r" il2(s)ds 4.3.54
= -a21(t)VI(t) -
a22(t)V2(t)
+ a21 (t) J.~,." il, (s) ds + a2'( t) J.~r" !i,( s) ds , For any fixed t ~ to ~ (J' (see (4.3.46), we can without loss of generality assume that VI(t) 2:: 0, since otherwise, for that t we can consider -Vl(t). Thus, for fixed t the sign pattern of (Vl(t),V2(t» can be
(+,+) ,
(+,-)
and we can write [to, 00) such that
[to, 00) = J 1 U J2 J 1 = {t ~ to IVI(t) ~ 0, Y2(t) ~ O} J 2 = {t ~ to !V1(t) ~ 0, V2(t)
< O}.
We recall that
aij(t) > 0 for
t
~
to
and m"I)
< a' ·(t) < c··'
-
I)
_
i,j = 1,2
')'
from (4.3.46) .
Now for t E J 1 , the system (4.3.54) simplifies to
~ dt
[!IY2VI I(t)I(t)] -< P [!IY21(t)I(t)] + c [I11f21(t) 1ft-l(t)] YI
1
where PI =
c_[
cil III
[-;11
+ CI2 1 12 C21 + C22122C21
C21 121 Cll
Cll III C12
+ C12 1 12 C22]
C21 121 C12
+ C~2T22
4.3.55
917
§4.9. Competition and cooperation
Uhl(t)=
sup
IY21(t) =
IYII(s),
sE[t-r,t]
sup sE[t-r,t]
I Y21(s)
If we assume that the matrix -( PI + C) is an M- matrix then by the result of Tokumaru et al. [1975J (see section §3.6 of Chapter 3) it follows that there exist positive numbers kP), k~2), 61 such that 4.3.56 For t E J 2 one can similarly show that (4.3.54) leads to
.:i [I YI let)]. < P. dt
IY21(t) -
2
[IIY21(t)I(t)] + [IIY21(t)I(t)] C
YI
YI
4.3.57
where C12
]
-m22 Again if the matrix -(P2 + C) is an M-matrix, as before there exist positive numbers k~I), k~2), 62 such that 4.3.58 This discussion leads to a sufficient condition for the global attractivity of the positive steady state of the competition system (4.3.21). We summarise the result as follows:
Theorem 4.3.7. If the matrices -(PI
+ C)
and
- (P2
+ C)
4.3.59
are both M -matrices, then solutions of (4.3.54) satisfy lim YI(t) = 0;
t-oo
lim Y2(t) = O.
t-oo
Proof. Proof is an easy consequence of (4.3.56) and (4.3.58) .
4.3.60
[J
We recall that there are simple criteria for verifying whether or not a given matrix is an M -matrix (section §3.6 of Chapter 3). We ask the reader to carry
318
§4.3. Competition and cooperation
further and simplify the conditions of Theorem 4.3.7 in order to obtain these conditions in terms of the parameters of the competition system (4.3.21). We shall now consider the asymptotic behavior of models of cooperation and in particular "facultative mutualism"; as an example, we study the following model of "hypercooperation";
4.3.61
where at, a2, K l , K2 E (0,00); ; al > K 1 , a2 > K2 j 81 and 82 are odd positive integers. When 81 > 1 or 82 > 1, (4.3.61 ) is a model of hypercooperation; models of hypergrowth have been discussed by Turner et. al. [1976], Turner and Pruitt [1978] and Peschel and Mende [1986] where some evidence of the relevance of hypergrowth models to reality is illustrated. Basically hypergrowth models correspond to situations where in the early phases, a population system flourishes with exponential growth and near saturation, the rate of saturation slows down in nonlinear way. It is this nonlinear slowing down near saturation, that makes hypergrowth different from the other well known growth models considered in mathematical ecology and biology. For instance if 81 ~ 3 or 82 ~ 3, the positive equilibrium of (4.3.61) (for details of this see Chapter 3) is not linearly asymptotically stable and therefore the local asymptotic stability of the positive equilibrium of (4.3.61) cannot be studied by linearization (or variational) methods. Dynamical behavior of cooperative systems without time delays has been discussed by Krasnoselskii [1968], Selgrade [1980], Hirsch [1982-85, 88a, b] and Smith [1986a, b, c]. Cooperative systems with time delays have been considered by Martin [1976, 1978, 1981]' Banks and Mahaffy [1978a,b], Ohta [1981]. One of the crucial assumptions used by Martin [1981] is that the growth rates are dominated at 00 by an affine function (assumption F4) and that all the eigenvalues of the matrix of such an affine function have negative real parts. It is the opinion of the author, that the existing results on the global convergence of time delayed cooperative systems are implicitly based on the following assumption: "the corresponding linear variational system has a negative stability modulus". This is equivalent to the assumption of linear asymptotic stability of positive equilibrium of (4.3.61).
§4.9. Competition and cooperation
919
We show below that even if the linear variational system associated with a unique positive equilibrium of a cooperation model is not asymptotically stable, such an equilibrium can be a global attractor with respect to all other positive solutions. For instance, if 81 2 3 or f)z 2 3, the unique positive equilibrium of (4.3.61) is not linearly asymptotically stable; we show, however, it is a global at tractor. The following lemma is due to :NIartin [1981J and establishes the property of preservation of upper bounds.
Lemma 4.3.8. Let
~
= ('PI, 'Pz) and P = (PI, pz) satisfy tbe following:
= 1,2
Pj E C([-Tj, O] ,R+);
j
'Pj E C([-Tj,OJ, R+);
j = 1,2
'Pj(O)
> 0;
j
s E [-Tj,O];
= 1,2
j = 1,2.
corresponding to j = 1,2, then
Nj(t, if?) s;. pj(t) Proof. Let satisfies
€
for
t 2 0, j = 1,2.
4.3.63
be an arbitrary fixed positive number. First we show that p}E)(t)
j = 1,2
where
4.3.64
§4.3. Competition and cooperation
320
dp~: (I) = F, ((P~') (I), p~,) (I -
dP~(t) = F2 ((p;'\t p)E) (t) = pj(t)
T,),v\') (I) )
+ €,
Suppose there exists s >
T2))
[p,( I) -
F, (p, (I), P2(1 - T2))]
+€
+ [P2(t) -
F2(p, (t - T,),P2(tll]
+€
+
t E [-7"j, 0],
j = 1,2.
4.3.65
°such that
p(E)(t) 2:: N(t, if!) for all t E [O,s]
4.3.66
and either (i) piE\s) = NI(s, 4» or (ii) p~e)(s) = N 2 (s, 4»; if (i) holds, then
pid(s) - NI(s, 4» = FI(pie)(s), p~E)(S - 7"2)) + {PieS) - F1(PI(S), P2(S - 72)) + €} - FI(Nt(s, if!), N 2 (s - 7"2,4>))
> FI(piE)(s),p~E>Cs - 7"2)) - Ft(NI(s, 4», N 2 (s - 72,4»
>
°
4.3.67
°
by the quasimonotone property of FI since %f; 2:: (verify this). From (4.3.67), it follows that p~E)(t) > NI(t, 4» for t E (8, S + 8) for some small positive 8. Now letting € ~ 0, we have PI(t) 2:: NI(t,4» for t E (0, S +8); if necessary, one can repeat this argument to conclude PI(t) 2:: NI (t, 4» for t 2:: 0. Similarly, one proves p2(t) 2:: N 2(t, 4» for t 2:: 0. [] The next result deals with the preservation of lower bounds. Lemma 4.3.9. Let 4> = ('PI, 'P2) be as in the case of Lemma 4.3.8. and let q( t) = (qi (t), q2 (t)) satisfy the following:
qjEC([-7"j,O],R+)
;
j=1,2
qj(s) S 'Pj(s) , s E [-7"j, 0], j = 1,2 dqI(t) < (t) [KI +atq2(t-7"2) _ (t)] 8 dt - ql 1 + q2 (t _ 7"2) qt
1
4.3.68 2
dq2(t) < (t) [K2 + a2qI(t - 7I) _ (t)] 8 dt - qz 1 + q1 (t - 7"1 ) q2
921
§4.9. Competition and cooperation
If N(t) = {N1(t),Nz(t)} = {N1(t,cp),Nz(t,cp)} denotes the solution of (4.3.61) corresponding to
Nj(s,1!) == 'Pj(s), S E [-rj,O], j
= 1,2,
then
Nj{t,1!)
~
qj(t)
for
t
~
0, j
= 1,2.
4.3.69
Proof. Details of proof are entirely similar to those of Lemma 4.3.8 and therefore [J are omitted.
The next result is an analogue of Kamke's Theorem (see Coppel [1965]) for delay differential equations and has been established by several authors in many different forms (Mikhailova and Podgornov [1965J, Sandberg [1978], Ohta [1981J, Martin [1981] and Smith [1987]). Theorem 4.3.10. Let
'ljJ2(S); S E [-r2,0]; 'ljJ2(0) > O. Assume also 'PI, 'ljJ1 E C([-rl,O],R+) and 'PZ, 'l/Jz E C([-rz,O],R+). Then the solutions N(t,1!) = {N1(t, 1!), Nz(t, cPj(s),
F2 (p; ,pi) < S
°
E [-lj,O] , j = 1,2.
4.3.74
Then the following limits exist: lim Nj(t,p*) , j = 1,2,
t-+oo
where
p*
= (* Pl,P2*) ,
4.3.75
324
§4-3. Competition and cooperation
is the solution of (4.3.61) satisfying S
E
[-Tj,O].
Proof. By choice, Fl (pi, pi) < 0, F2(pi, pi) _< 0; if we choose pj( t) == pj for all t ~ -Tj , j = 1,2 then
'PI (t) = 0 > Fl (pr ,p;) = Fl (PI (t),P2(t - T2» P2(t) = 0 > F2(pr ,pn = F2(Pl (t - Tl) ,P2(t».
4.3.76
By Lemma 4.3.8, it follows 4.3.77 By the semigroup and order preserving properties of solutions of (4.3.61), 4.3.78
for all t, h ~ O. Thus Nl (t,p*) is non-increasing and bounded below; hence limt-+oo Nl(t,P*) exists. The existence of limt_oo N2(t,P*) follows by similar arguments. O. Lemma 4.3.13. Suppose there exist numbers qi, qi such that
qi > 0, qi > 0 and 4.3.79
Then the following limits exist: lim Nl (t, q*), t-+oo
j
= 1,2
where {N1(t,q*) , N2(t,q*)} = N(t,q*) is the solution of (4.3.61) satisfying S E [-Tj,
Proof is similar to that of Lemma 4.3.12.
0], j
= 1,2. []
The next result shows that the unique positive equilibrium of the hypercooperation model is a global attractor with respect to all positive solutions of (4.3.61).
925
§4.9. Competition and cooperation
Theorem 4.3.14. Let N(t,p) = {NI(t,p), N 2(t,p)} be tbe solution of (4.3.61) corresponding to tbe initial condition
wbere
PI(S) ~ 0, PI(O)
P2(S)
~
> 0, PI E C([-rI,O], R+) 0, P2(0) > 0, P2 E C([-r2,0] , R+).
Tben 4.3.80 P roof. Choose posi ti ve numbers PI , P2, ql , q2 such that
(Pl,P2) > (N;,N;); Fj(PI,P2) < 0; qj < pj(s) < Ph
< (N;,N;) Fj(qI,q2) > 0, j = 1,2 (ql,qZ)
4.3.81
s E [-rj,O], j = 1,2.
4.3.82
It is not difficult to verify that such a choice of qI, q2, PI, P2 is always is possible. We have from the above Lemmas and Corollary, that
{N1 (t,q),N2(t,q)}
~
{N I (t,p),N2 (t,p)} for
~
t~
By Lerruna 4.3.12, there exist positive numbers
{N1 (t,p),N2 (t,p)}
°
O"j ~
j
Nj, j
= 1,2
4.3.83
= 1,2 such that
4.3.84
and therefore by Lemma 1.2.3 of Barbalat (see Chapter 1)
4.3.85
showing (0"I,0"2) = (N;,N;) since (N;,N;) is the unique positive solution of (4.3.85). Similarly we conclude that lim Nj(t, q) = NJ;
t--oo
j = 1,2.
4.3.86
§4.9. Competition and cooperation
926
The conclusion (4.3.80) now follows from (4.3.83), (4.3.84), (4.3.85) and (4.3.86). The proof is complete. [] Dynamical systems modelling cooperation have been considered by Matano (1984] who has assumed that the flow generated by such systems is "eventually monotone" . A sufficient condition for the generation of such a flow has been obtained by Hirsch [1982, 1984} in terms of irreducibility of the Jacobian matrix of the vector field modelling the cooperative dynamics. A consequence of the result of Theorem 4.3.14 is that if the cooperative system (4.3.61) is stable without time delays, then a delay induced instability leading to a Hopf-type bifurcation to periodic solutions is not possible; in short, delay induced stability switching in cooperative systems is not possible if the time delays appear only in cooperative interactions. We wish to emphasize that our result on global attractivity of the equilibrium in the hypercooperation model is obtained with minimal hypotheses on the system compared with other relevant results in the literature. One of the reasons for the specific choice of the model has been to make the results more transparent for applications. The reader can examine the global attractivity of the positive equilibrium of each of the following models of cooperation:
dx(t) -;It
[Kl
= rlx(t) 1 + e-y(t)
dy(t) --;tt =
1
x(t) ;
i(t) =
[K2 1 r2y(t) 1 + e-x(t) -yet) ;
ii(t) =
-
J.' J.'
x(s)ds y(s)ds.
I
§4.9. Competition and cooperation
where ()
= 1,3,5, .. etc.
and
]{l
>
al
> 0,
I{2
d~~t) = TIX(t)[Kl(l _ d~~t) = T2y(t) [K2(1
>
a2
> 0,
927
T
E [0,00) , 8 E [1,00);
e-y(t-T») _ x(t)] }
I
_ e-X(t-T») - y(t)].
dYl(t) = -a Y (t) + b y~(t - Tm). dt I I I 1 + y~ (t - Tm) ,
yeO) > 0,
dYj(t) = -ajYj () n ( ). -;ut + bjYj-l t - Tj-l; J = 2,3"", m Yj(S) = 0; i
= 1,2, ... , nj
together with the following assumptions:
(AI) the delay kernels k ij (i, j = 1,2, ... , n) , kij : [0,00) on [0,00) and normalised such that
/.00 k
ij ( s)
ds = 1 ;
/.00 Ik
ij ( s)Ids
< 00 ;
~
( -00,00) are integrable
/.00 slk
ij ( s)Ids
< 00
4.4.5
i,j=1,2, ... ,n.
(Az) the real constants b i , aij, bij, Cij (i, j = 1,2, ... , n) are such that there exists a solution x* = (xi, xi, ... , x~) with xi > 0 (i = 1,2, ... , n) of the linear system n
L(aij+bij+Cij)xj+bi=O; i= 1,2, ... ,nj
4.4.6
j=1
the discrete delays Tij ~ 0 (i, j = 1,2, ... , n) are constants such that 0; i,j = 1,2, ... ,n. (A3)
the real parameters
bi, aij, bij , Cij
satisfy
laid +
[b;;[
a;; < 0; [a;;[ >
t t t +
[a;;[
/.=
[k;;(s)[ds
bijTij
1=
4.4.7
i = 1,2, ... ,no
We note that x* is unique by virtue of (4.4.5) and (4.4.7). Along with (4.4.5) (4.4.7), we consider initial conditions of the form Xi(S)='Pi(S)~O; SE(-oo,O); 'Pi(O»Oi SUpl'Pi(S)I 0 ; i = 1,2, ... , n. The following result provides a set of sufficient conditions for the asymptotic stability (stability in the large or global attractivity) of X* (Gopalsamy [1984aJ). Theorem 4.4.1. Assume that the hypotheses (AI) - (A3) hold for (4.4.4). Then all solutions of (4.4.4) corresponding to the initial conditions in (4.4.8) satisfy
lim Xi(t) = xi; i = 1,2, ... ,n.
4.4.10
t-+oo
Proof. Consider the Lyapunov functional vet) = V(t,X1(.), ... ,x n (.» defined by
v(t) =
t.
[JIOg{X;(t)fxi}J + t.Jb;;J tr;; Jx;(s) - xjJds
+ t.Jc;;J {' Jk;;(s)J (t,Jx;(U) for
xjJdU) dS]
4.4.11
t ~ O.
It is easy to see from (4.4.11) and the type of initial conditions that
+
t
;=1
!cijl (sup lepj(s) - xii)
~ Vo
8=:;0
[00 Ikij(s)ls dS] io
< 00 for some positive number
Vo.
4.4.12
§4.4.
330
Lyapunov functionals
and n
vet) 2:
2:: Ilog{xj(t)jx:Jl·
4.4.13
i=l
Calculating the upper right derivative D+ v of v along the solutions of (4.4.4) and simplifying,
D+v(t) ::; -
t, [I +
a;; 1-
{ t.1aj;1 + t.1b;;1
t leji/ Jof=
j
,;:i
Ikji(s)1
dS}] I Xi(t) - xi I
J=l n
~
-62:: IXi(t) -
xii
4.4.14
i=l
where
o < fJ =
l~ifn [ 1a;; 1- t.1aj;l- t. (Ibjd + lejd [0 Ikj;(s )ldS) ]. j
?!i
It can be shown that (4.4.14) will imply (4.4.10); we leave the rest of the details of proof to the reader as an exercise (see Gopalsamy [1984a] for details). []
The next result provides a "mean-diagonal dominance" type sufficient condition for the convergence of all positive solutions of (4.4.4). Theorem 4.4.2. Suppose the hypotheses (AI) and (A2) hold for the system (4.4.4) and assume that in addition the following holds: (A4) aii < 0; i=1,2, .. ,n
(A5)
4.4.15
for i
= 1,2, ... , n.
Then all solutions of (4.4.4) and (4.4.8) satisfy (4.4.10).
§4.4.
331
Lyapunov functionals
Proof. Consider a Lyapunov functional v( t, x(.), ... , X n (.» defined by
v(t) =
t,[
(x;(t) - xi - xilog(x;(t)/Xi))
+ "2 ?= Ibijl 1
n
it
}=1
+~
t, f le;;1
for
(Xj(U) -:- xj)2du
t-r"
4.4.16
I)
Ik;;(s)1
(E.
(x;(u) - Xj)'dU) dS]
t > O.
Calculating the rate of change of v in (4.4.16) along solutions of (4.4.4) we have
d~~t)
=
~ n
[
a;;(x;(t) - xi)'
+ ~ a;; [x;(t) - xi] [x;(t) - xi] n
i't"i
n
+L
bij [Xj(t - Tij) - xj] [Xi(t) - xi]
j=l
t 1
00
+
Cij
j=l
+~
t
kiiCs) [Xj(t - s) - xj] [Xi(t) - xi] ds
0
Ibijl { (Xj(t) - xj)2 - (Xj(t - Tij) - xj)2 }
j=l
+~
t,
Ie;; I['" Ik;;( s)1 { (x;(t) - xj)' - (x;(t - s) - xj)' } dS]
: ; t, [
-la;;I(x;(t) - xi)'
t
laijl {(Xj(t) - xi)2
+ (Xj(t) - xi)2}
2
+~
t
Ibijl {(Xi(t) - xi)2
+ (Xj(t) - xj)2}
+~
j=l j't"i
j=l
+~ ~
t,
le;;I!.oo Ik;;(s)1 {(x;(t) - xi)'
+ (x;(t) - xj)'} dS]
n n[ laiil- 2I(n .I ~(Iaijl + lajd) + 2 ~(Ibijl + Ibjd)
-t;
j't"i
§4 ·4.
332
Lyapunov functional.3
n
2:)Xi(t) - xif
.:; -p,
where
4.4.18
j=1
In
o < p, = 1 0 by (4.4.15), op.e can show that (4.4.18) will imply (the reader should supply the additional arguments) the convergence in (4.4.10) and the proof is complete.
o
It is found from Theorems 4.4.1 and 4.4.2 that if the instantaneous (nondelayed) responses in (4.4.4) dominate (see for instance (4.4.7) or (4.4.15)) all delayed responses, then the positive steady state x* of (4.4.4) is a global attractor. One is now entitled to ask the following: if the system (4.4.4) is such that there are no instantaneous responses in the average growth rates, then one has a system of the form
dY~;t) = Yi(t) ( bi -
t.bij
i'=
t>0; i
kij(t -
S
)Yj(s) dS)
4.4.20
= 1,2, ... , n
(b i , Cij being real constants) where the linear system
4.4.21 for
i = 1, 2, ... , n
has a solution y* = (y;, ... ,y~) with yi > 0, i = 1,2, ... , nj under what conditions the steady state y* of (4.4.20) will be a global attractor with respect to nonnegative initial conditions? It should be noted that the system (4.4.20) does not contain delay independent stabilizing negative feedbacks as in the case of (4.4.4). We will derive an answer to the above question using the concept of positive definite kernels defined as follows:
§4.4-
999
Lyapunov functionals
Definition. Let K denote the n x n matrix of elements (K)ij = bij k ij where bij (i,j = 1,2, ... , n) are real constants and k ij : [0,(0) 1-+ (-00,00) are such tbat
1.= kij(s)ds = 1 ;
1.=
Ikij(s)lds < i
00;
1.T(kij(S))2dS
0, tbere exists a positive constant J.l such that
r;,1. n
T
fi(t)
(t 1. f,;.bijkij(s)!;(t-S)ds n
21' {
{tfM}
)
dt 4.4.23
dt
for each finite positive number T. The following result is not new to the existing literature.
Theorem 4.4.3. Assume tbat the delay kernel K = (k ij ] in (4.4.20) is positive definite satisfying (4.4.23) and
1
00
slkij(s)lds < 00; i,j
= 1,2, ... ,n.
4.4.24
Let y* = (y;, yz, . .. , y~) , yj > O,j = 1,2, ... , n be a positive steady state of (4.4.20). Let (4.4.20) be supplemented witb bounded continuous initial conditions of the form
Yj(s)
= 0 and derive from (4.4.34),
vet) ::; -p
1.' {t. (yi[ + 1.' /I /I + 1.' /I Y
e ;(') -1
M
::; v(O)
h(s)
M
J
y}
ds
v(s)ds + v(O)
4.4.36
/I v(s)ds
4.4.37
h(s)
which by Gronwall's inequality and (4.4.30) implies vet) ~ v* for some constant v*
>0
4.4.38
showing that vet) is bounded uniformly in t for t~ O. By continuation, it will follow that solutions of (4.4.28) exist for all t ~ 0 since we have from (4.4.31), n
I::Yi
[eY;(t) -
Yi(t)
-1] ~ v*
4.4.39
i=l
and v* is independent of t. We have from (4.4.36),
t.
yi[ eY;(t)
-
Y;(t) - 1 J + JL
1.' {t,(y;(S) - vi)' }
ds
4.4.40
~ v(O) + Mv* ].00 II h(s) II ds ~ N < 00. From the boundedness of E~=l IYi(t)1 on [0,00) and the hypotheses, the boundedness of for t > 0 and i = 1,2,3, ... , n (see (4.4.28) and note that
%
§4.4.
996
2:7:11hi(t)!
~ 0 as
t
Lyapunov functionals
~ 00). It will then follow that
2::=1 lY;(t)1
is uniformly
continuous on [0,00). Thus, we have from (4.4.40) that (i)
(ii)
yn 2:7=1 {Yi(t) - yn
2:7=1 {Yi(t) -
2
is uniformly continuous on [0,00).
2
E LI[O, 00). which together imply 2:~1 !Yi(t) - yil ~ 0 as t ~ 00 and this completes the
0
~~
An immediate question now is the following: are there verifiable sufficient conditions for a matrix kernel K = [kij] to be positive definite? The answer is yes and a result for this purpose is formulated below whose proof is similar to that of the scalar case treated in Chapter 1 and is left to the reader as an exercise. Proposition 4.4.4. Let K = [kijl ,i,j = 1,2, ... , n be such that (4.4.22) holds. Then K is positive definite satisfying' (4.4.23) if the matrix k where
K = K(i7J) = ?Re(Kij(i7J» =?Re
1.
00
kij(t)e il1t dt ; T/ E (-00,00),
4.4.41
is a positive definite matrix whose eigenvalues are bounded below by a positive constant J.l. Let us consider briefly a class of integrodifferential equations with a special type of delay kernels:
dx.(t) (n n jt T = Xi(t) b + ~ aijXj(t) + ~ ,Bija i
-00
e-a(t-s)xj(s)ds
i = 1,2, ... ,n ; t
) 4.4.42
>0
where bi , aij, ,Bij (i = 1,2, ... ,n) are real constants and a is a positive constant. The linear "chain trick" introduced by Fargue [1973] and used by Worz-Busekros [1978], MacDonald [1978], Post and Travis [1982] for analysing (4.4.42) is as follows: define a new set of variables xn+i,j = 1,2, ... , n so that xn+j(t)
="
1'=
>0
4.4.43
j=1,2, ... ,n.
4.4.44
e-a('-')Xj(s)ds ; t
and immediately derive
§4 ·4·
337
Lyapunov /unctiona13
Thus, the system (4.4.42) of n-integrodifferential equations becomes a system of 2n autonomous ordinary differential equations i = 1,2,"', nj
4.4.45 j = 1,2, .. . ,n.
If x* = (xi,xi, ... ,x:),xt > 0, i = 1,2, ... ,n is a solution of n
L(aij
+ f3ij)xj + b
j
= OJ i = 1,2, ... ,n;
4.4.46
j=l
* * ... ,xn,xn+l, * * *) ,xn+i * *. = 1, 2 then, ( xl,x2, ... ,x2n = Xj') , ... ,n'IS a componen-t wise positive steady state of (4.4.45). Asymptotic stability of (xi, ... , xin) for the system (4.4.45) is equivalent to that of (xi, ... , x:) for (4.4.42). We formulate our next result in terms of M-matrices; for properties of M-matrices we refer to Chapter 3 (or Araki and Kondo [1972], Plemmons [1977]). The following result concerned with the stability of the system (4.4.42) is due to Post and Travis [1982].
Theorem 4.4.5. Corresponding to the system (4.4.45) define a 2n x 2n matrix B as follows:
B12] B 22
;
Bij (.. Z,)
( B ll ) ..
')
(B 12 )ij=-If3ijlj
=,1 2) are n x
= {la ii1i
-Iaiil;
. n rna t rIces.
i =j i=fj
i=1,2, ... ,nj j=n+1,n+2, ... ,2n
(B21 )ij = diag( -a) (B22)ij
= diag(a).
The positive steady state x* = (xi, ... , x:) of (4.4.42) is globally asymptotically stable if B is an M-matrix and aji < OJ i = 1,2,3, ... , n. Proof. Consider a Lyapunov function
§4.4.
338
Lyapunov junctiona13
defined by 4.4.47 where d 1, d2, . .. ,d2n are positive constants to be chosen suitably. Calculating the derivative of V along the solutions of (4.4.45) and simplifying one can verify that (see Post and Travis [1982]), 4.4.48 where
x-
X* = col.{(Xl - xi), (X2 - x;), ... , (X2n - x;n)}
D = diag.{dI, d 2 , ••• , d2n }.
Since by assumption B is an M-matrix, there exists a positive diagonal matrix D such that DB + BT D is positive definite and hence we have from (4.4.48) that, dd~ calculated along the solutions of (4.4.45), is negative definite from which the [J result will follow. If we let (3ij = 0, i,j = 1,2, ... , nand Q = 0 in (4.4.45), then (4.4.45) will simplify to a system of n ordinary differential equations
i=1,2, ... ,n
4.4.49
yilog(yi/yi)}
4.4.50
for which a Lyapunov function of the form n
V(Y1, Y2,···, Yn) =
I: di{Yi i=l
where Y; > 0 and 2:'}=1 aijYj = .Ai, i = 1,2, ... , n has been used by numerous authors (see Harrison [1979] for a narration). A calculation of ~~ in (4.4.50) along the solutions of (4.4.49) leads to 4.4.51 4.4.52
§4.4. in which
339
Lyapunov functionals
= diag( d 1 , d 2 , ••• , d n ) ; A = {aij} y* = col{(y - y;), (Y2 - y;), ... , (Yn D
y-
y~)}.
It will follow from (4.4.52) that a sufficient condition for the global asymptotic stability of y* = {Yi, ... , Y:} for (4.4.49) is that there exists a diagonal matrix D = diag(dll ... ,dn ) such "that DA + ATD is positive definite. It has been noted by Krikorian [1979] that the algebraic problem of finding necessary and sufficient conditions, for the existence of a positive diagonal matrix D such that DA + ATD is negative definite for a given square matrix A, remains unsolved (see also Barker, Berman and Plemmons [1978]). Furthermore, the negative definiteness of (D A + AT D) demands that all the diagonal elements aii (i = 1,2, ... , n) of A be negative (a condition which we have extensively used); if possible such a requirement is worth relaxing. In many cases, it is not difficult to find a positive diagonal matrix D so that (DA + ATD) is positive semi-definite; in such a case although Lyapunov's stability theorem is not applicable, the following extension (see LaSalle and Lefschetz [1961], Barbashin [1970]) of Lyapunov's stability theorem can be used: "if ~~ in (4.4.52) is negative semi-definite, then every solution of (4.4.49) approaches as t ~ 00, the largest invariant subset of the set of points in Rn for which ~~ = 0". For instance, consider the example of Krikorian [1979]; dx}
dt
= Xl('x1 - allxl - a12 x 2)
dX2
dt = X2( -'x2 + a2l x l dX3
dt
= X3( -'x3
a23 x 3)
4.4.53
+ a32 x 2).
Consider a Lyapunov function v = vex!, X2, X3) for (4.4.53) defined by 3
V(Xl,X2,X3) =
?= ai [Xi - xi - xil09(Xi/x n]
4.4.54
.=1
where al, a2, a3 are positive constants to be selected suitably. Computing ~~ for (4.4.54) along the solutions of (4.4.53) we have
~~
= -alall(xl - x;)2
+ (a3 a32 -
+ (a2a21
- ala12)(x2 - X;)(X3 - x;)
a2 a 23)(X2 - X;)(X3 - xi)·
Suppose we choose aI, a2, a3 such that
§4.4.
Lyapunov functionals
4.4.55 Now ~~ = 0 only when Xl = xi. Let us look for invariant (with respect to (4.4.53» sets of the form
E = {(x}, X2, x3)lxI = x~ , xz'> 0 , X3 > o}. If E is invariant with respect to (4.4.53), then we have the following implications: Xl
= x~
= 0 =::} ).1 -
=::}
Xl
=::}
Al -
all xi
=::}
-A2
+ a21 x I -
=::}
X3 = x;.
anXI - al2x2
=0 = x; =::} X2 = 0
= 0 =::} Xz a23 x 3 = 0 =::} -A2 + a2lxi -
- alZ x 2
a23x3
=0
Thus, the only invariant subset of (4.4.53) is the point (xi, xi, xi) which is a positive steady state of (4.4.53) whose existence is assumed. We can conclude by LaSalle's extension of Lyapunov's stability theorem that (xi, xi, xi) of (4.4.53) is globally asymptotically stable. Other examples solvable by this technique are listed in the exercises. 4.5. Oscillations in Lotka - Volterra systems In competitive and cooperative model systems with no time delays, solutions can converge to equilibria monotonically with time; our discussion in Sec. 4.2 illustrates this phenomenon. The introduction of time delays in model equations, has been to produce certain observed fluctuations in the population densities both in controlled and field environments; furthermore, time delays are natural in many population systems due to maturation processes among many others. It is in this spirit one is interested to examine whether or not delay induced oscillations exist in model systems. Also a knowledge of fluctuations in population densities can prove useful in devising appropriate feedback control strategies. The results of this section are from Gopalsamy [1991]. We discuss the oscillation of solutions about the equilibria of delay differential equations of the type
dx~~t) = Xi(t)[b i -
taijXj(t-rjj)],i = 1,2, .. ,n; J=l
bi,aij E (0,00), i,j = 1,2, ... ,n.
4.5.1
§4.5. Oscillations in Lotka . Volterra systems
341
We have seen in Chapters 1 and 2 that time delays have a tendency to produce oscillations in otherwise nonoscillatory systems. A familiar example of this aspect is provided by the scalar equation with a single delay
duet) dt
= u(t)[b - au(t - r)]
4.5.2
°
which is nonoscillatory if r = where a, b are positive constants and is oscillatory about its positive steady state if (ber) > 1. Usually together with (4.5.1) we consider initial conditions of the form Xi(S)= 0, j = 1,2,3, ... , n if every solution x = {Xl, X2, ••• , x n } of (4.5.1) corresponding to (4.5.3) has at least one component, sucb tbat [Xj(.) - xj] is oscillatory on [0,(0) for some j E {1,2,3, ... ,n}. Tbe system (4.5.1) is said to
342
§4.5. Oscillations in Lotka - Volterra systems
= {xi, ... , x~}
be nonoscillatozy about its steady state x*
(4.5.1) has at least
if
one solution corresponding to (4.5.3) such that the vector x(.) - x* = {Xl(.) -
xi , X2(') -
x;, ... , x n (.)
x~}
-
is nonoscillatory on [0, 00 ). We remark that the above definitions constitute one of several possible directions of generalizing the concept of oscillatory and non-oscillatory scalar systems to the case of finite dimensional vector systems.
=
Let us now consider (4.5.1) by relaxing the requirement aij ~ 0, bi > 0, (i,j 1,2, ... ,n) and examine under what conditions all positive solutions of (4.5.1) will be oscillatory about a positive equilibrium.
Theorem 4.5.1. Suppose the parameters of (4.5.1) satisfy the following: bi, aij (i,j = 1,2, ... , n) are real constants such that aii > 0, i = 1,2, ... , nand the system (4.5.1) has a componentwise positive steady state . and 1 pro> -
where
e
p
4.5.4 4.5.5
= l~~n (xi [a ii -
-
tI
aji
j=l j 1'0
I]).
Then every nontrivial nonconstant solution of (4.5.1) and (4.5.3) existing on = {xi,x 2, ... ,x~}.
[-r,oo) is oscillatory about the steady state x*
Proof. First we show that every nontrivial and nonoscillatory solution of (4.5.1) and (4.5.3) converges as t -+ 00 to the positive steady state x*. For instance, suppose x(t) = {Xl (t), X2(t), . .. ,xn(t)} is a nonoscillatory (about x*) solution of (4.5.1) and (4.5.3) on [-r,oo). As a consequence there exists a t1 > 0 such that
Xi(t) -xi
=f. 0
for
t ~ tl ; i
= 1,2,; .. ,no
4.5.6
We can rewrite (4.5.1) in the form
d dt Ui(t)
n
=-
Laid xj(t-rjj) - xj] j=l
t
> 0 ; i = 1,2, ... , n
4.5.7
§4.5. Oscillations in Lotka - Volterra systems in which
Ui(t)
= log[xi(t)/x:J
; t > O.
We have from (4.5.6) and (4.5.7) that :t I Ui(t) I $ -a .. I Xi(t - Tii) -
xi I +
t,1
aij
IIXj(t -
Tjj) - xj
I
i~i
4.5.8
> tl + T
t
and hence
~ {t, IUi(t)l} $ - t, [aidxi(t - TiO) - xil-
t, j
$ -
t>
t [(a t .
T .. ) -
XiI}.5.9
+T
tl
lajd) IXi(t -
ii -
1=1
lajdlxi(t -
7f:i
TiO) -
xi I].
4.5.10
J=1
i#i
It follows from (4.5.5) that P=
.':2ifn [a .. - t,1 aji I] > 0
4.5.11
i#i and therefore d (
dt
t; IUj(t) I n
)
::; -p
t; IXi(t - Tii) - xi I· n
An integration of both sides of (4.5.12) over [t2
+ T, t]
4.5.12
leads to 4.5.13
One can conclude from (4.5.13) that E?=llui(t)1 is bounded on [0,(0) and hence the derivative of this sum is also uniformly bounded. From these it will follow as
t
-+
00;
i=1,2, ... ,n.
4.5.14
To proceed further we now rewrite the system (4.5.7) in the form d
-d Ui(t) t
=-
n
I: aiixj[exp{Uj(t-Tjj)} -1] . )=1
i = 1,2,3, ... , n ; t > t3
4.5.15
§4.5. Oscillations in Lotka - Volterra systems
344
and show the existence of
~jj = ~jj(t)
on [t3
exp{Uj(t - 1'jj)} - 1 =Uj(t -
+ 1', 00), i,j
= 1,2, ... , n such that
1'jj)exp{uj(~jj(t))}
4.5.16
i,j = 1,2, ... ,n ; t > t3 +1'.
Let t, t 1 be such that
We note 4.5.17
i,j=1,2, ... ,n
where uj(B jj ) lies between Uj(t - 1'jj) and Uj(tI). Considering the limiting case of (4.5.17) as tl -1- 00, we derive
exp{Uj(t - 1'jj} -1 = Uj(t i,j
1'jj)exp{uj(~jj(t))}
for some ~jj(.)on [ta +1',00) such that ~jj(t) -1- 00 monotonically as t 1,2, ... ,no Using (4.5.18) we rewrite the system (4.5.15) in the form
d
dt Ui(t)
4.5.18
= 1,2, ... ,n ; t > t3 +1' -1-
00 ;
i,j =
n
=-
I.: aijxjUj(t - 1'jj)exp{uj(~jj(t))} j=l
= -aiixi(t -
1'ii)exp{ui(~ii(t))}
n
- L aijxjUj(t - 1'jj )exp{ Uj(~jj(t))}
4.5.19
j=l j¢i
for
i = 1,2,3, ... ,n ; t
> ta + r.
As a consequence of the facts Ui(t) -1- 0 and ejj(t) -1- 00 as t -+ 00 (i,j == 1,2, ... ,n), it follows that there exists a t4 2: ta + r such that (4.5.19) leads to
n
+
L laijlxjluj(t -
rjj)lexp{luj(~jj(t4))1}
j=l j¢i
t > t4 ; i == 1, 2, ... , n
4.5.20
§4.5. Oscillations in Latka - Volterra systems with the implication
+
t j
lajilxilui(t - Tii)leIU;«;;(t. m]
4.5 ..21
'Fi
We can simplify (4.5.21) to obtain
-t
lai ;le/ ui ({ii(t 4 »/]
xii Ui(t -
'Tii) I
4.5.22
}=l
For convenience let us set 4.5.23
From (4.5.22), 4.5.24 Using the facts Ui(t) -7 0 (since Xi(t) sides of (4.5.24) on (t, 00),
xi) as t
-7
-7
00,
and integratiing of both
4.5.25 which will lead to Wet)
~
0:
where
1.:. n
Wet) =
W(s) ds
I: IUi(t)l· i=l
4.5.26
§4.5. Oscillations in Lotka - Volterra systems We let
F(t) =
fr
1:.
4.5.27
W( s) d.s
and derive
F(t)
= -aW(t -
To)
< -aF(t - To).
4.5.28
It follows that the scalar delay differential inequality (4.5.28) has an eventually positive solution. Since pTo > ~, it is possible to choose t4 large enough so that aTo
1 e
>-
(from)
pTo
1
> -.
4.5.29
e
It is well known that when (4.5.29) holds, (4.5.28) cannot have an eventually positive solution. This contradiction proves the assertion. [] We remark that (4.5.5) provides a sufficient condition for the oscillatory (not necessarily periodic) coexistence of the n-species Lotka-Volterra system (4.5.1). The nonoscillation of competition systems of the type
(L: n
dXi(t) r-I dt- -- x-(t) Z
) • a-Z]-x ] ·(t - 1"-) I]'
i = 1,2, ... ,n
i=l
has not been considered except when
Tij
=
l'
(see Gopalsamy et al. [1990a]).
4.6. Why positive steady states? An n-species population system modelled by coupled integrodifferential equations of the form dx -(t) (n n jt -it = Xi(t) Ai + [; aijXj(t) + [; bij
-00
kij(t - s)xj(s)ds
) 4.6.1
t> 0 ; i = 1,2, ... , n is said to be capable of equilibrium coexistence if and only if each solution x(t) = {Xl(t)"",xn(t)} of (4.6.1) with
Xi(O) > 0 ; Xi(S)
= 0, 1,2, ... , n, then the system is said to be a leaky system.
Ui?: 0,
z=
It is a simple exercise to show that corresponding to an initial condition of the form
i=1,2,:···,n
4.7.4
where c = col. {Cl , .•• , cn } is a (componentwise) nonnegative constant vector, solutions of (4.7.3) - (4.7.4) are defined for all t > 0 and are such that Xi(t) ;::: o for t?: 0 , i = 1,2, ... ,n. We first consider the following problem which is of interest in drug administration. Assume that (4.7.3) is a leaky system with no input (i.e. Ui = 0, i = 1,2, ... , n) and the state of the system is impulsively altered at a specified sequence of time points so that the modified system is described by
dXi(t)
~
--;It = LtaijXj(t)j
4.7.5
j=l n
Xi(tm
+ 0) -
Xi(tm - 0)
= L Cij(tm)Xj(tm -
0)
4.7.6
j=l
i = 1,2," ',n;
m = 1,2,3""
o = to < tl < t2 < ... < tm
-+ 00
as
m -+
00
where Cij(t m ), i,j = 1,2,"" nj m = 1,2",' are real nonnegative constants. Intuitively one expects that in a leaky system, if the impulsive perturbations are not "too frequent" and if the perturbations Cij(t m ) are not "too large" then the impulsive system (4.7.5) - (4.7.6) should eventually lose all the substance from the system as t -+ 00. The following result provides a set of sufficient conditions under which the above intuitively expected result holds.
Theorem 4.7.1. Let A denote an n x n matrix wi th elements aij, i, j = 1,2, ... , n. Suppose there exist positive constants a, f3 and Co such that (i) tm - t m - 1 ?: f3 > OJ m = 1,2,3, ... , 4.7.7 (ii) 0 ~ ciiCtm) < Co for i,j = 1,2"", n m = 1,2,3",' , 4.7.8 (iii) J.l(A) + log(l + nco) = -a < 0, 4.7.9 where J.l( A) denotes the matrix measure induced by the matrix norm
!
\I A \I
t
= l:$J:$n m?-x laijl i=1
where J.l(A)
t
= l:$J:$n m?-X {an + I aij I}. i=1 i:f;j
957
§4- 7. Dynamics in compartments Tben all solutions of (4.7.5) - (4.7.6) satisfy n
II x(t) II =
L
IXj(t) I:::; II x(to) Ile-o(t-to)
t ~t6·
for
4.7.10
i=l
Proof. Let X(t) denot-e the flUldamental matrix eAt. ~y direct calculation one can derive that for t in the open interval (tk, tk+I), k = 0,1,2"",
x(i) = X(t - i.) { [I + C(i.) I X(i. -
i.-I)} {[I + C(i.-I)][ X(i'_1 - i.-
... { [I+C(tl)][X(tl-tO)] }x(t o)
2 )]}
4.7.11
where x(t) = C01.{Xl(t), ... , xn(t)},C(tk) denotes the n x n matrix with entries Cij(tk) i,j = 1,2, ... ,nj k = 1,2,3, ... and I denotes the n x n identity matrix. Maintaining the order of the terms on the right side of (4.7.11), we can rewrite (4.7.11) compactly in the form k
x(i)
= X(i -
i.) { }] ([ I
+ C(ij) ][X(tj -
ij_I)
I) }X(io);
i
E
(i •• ik+J). 4.7.12
It will follow from (4.7.12), on using the fact
II X(t) II = II eAt II :::; eJl(A)t j
t~O
that
4.7.13
k
II x(i) II : 0. On [O,T) no component of the solutions of (4.7.14) - (4.7.16) can become negative; for instance, suppose xp for some p E {I, 2, ... , n} is the component which becomes negative not later than any other component of a solution; that is there exists a t* E (0, T) satisfying
Xp(S) < 0 for
s E [0, i*];
for some
€
s E (i*,i*
+ €)
> 0. This will mean that dXp(t) dt
and
I
< 0·
t=t.
'
but we have from (4.7.14) that
which contradicts the definition of t*. Thus nonnegativity of solutions of ( 4. 7.14)(4.7.16) will follow. Now let x(t) = C01.{Xl(t), ... ,xn(t)} and yet) = COl.{Yl(t), ... ,Yn(t)} be any two solutions of (4.7.14) on a common interval of existence corresponding to nonnegative initial conditions. We have then d
dt[Xi(t) - Yi(t)] = - [fOi(Xi(t» - fOi(Yi(t»] n
- I:: [!ii(Xi(i»
-!ii(Yi(i»]
j=1 j-¢i
4.7.19
n
+ I:: [fij(Xj(t»
- J;j(Yj(t»]
j=l
j-¢i
for
i
= 1,2, ... , n .
Consider a Lyapunov function vet) = v(t,x(t),y(t» defined by n
vet) =
I:: I Xi(t) - Yi(t) I; i=1
t E [0, T).
4.7.20
§4.7. Dynamics in compartments
960
Calculating the upper right derivative n+v of v and using the monotonicity of fOi, we can derive that n
D+v(t) ~ -
L
4.7.21
Ifoi(xi(t» - fOi(Yi(t»1
i=l
which shows that if one of the solutions x(t) or yet) is defined on [0,00) and remains bounded on {O, 00), then the other is also defined on [0,00) and remains bounded on [0,00). Thus, either all solutions of (4.7.1) remain bounded or no solution remains bounded on [0,00). Suppose (4.7.14) has a solution x(t) = col. {Xl (t), ... , xn(t)} such that (4.7.17) holds. Then every solution of (4.7.14) is bounded on [0,00) showing that the system (4.7.14) has a compact convex invariant set in Rn. As a consequence of Brouwer's fixed point theorem, it will follow that such an invariant set must contain at least one (fixed point) steady state x* = (xr, ... ,x~) of (4. T.1). Since x* lies in a bounded closed set of nonnegative octant of Rn, x* is nonnegative componentwise.
IT we choose now Yi( t) == xi (i = 1,2, ... , n) in (4.7.20), we then have
D+
(t,
Ix;(t) - Xii)
~-
t,
4.7.22
Ifo;(Xi(t» - fOi(xill .
We can show that (4.7.22) and the monotonicity of fOi (i = 1,2 ... ,n) will imply (4.7.18). Suppose (4.7.18) does not hold. We note that Xi(t) is bounded since Xi is bounded and hence the right side of (4.7.14) is bounded on [0,00) implying that Xi is uniformly continuous on [0,00). IT t:~ Xi(t) =:f xi or if Xi(t) does not converge to xi as t -+ 00 for one or more i E (1,2 ... , n), then we can find a sequence {tk; k = 1,2, ... }, to < t1 < ... , tk -+ 00 as k -+ 00 such that
k
= 1,2, ...
4.7.23
for some positive number e. As a consequence of (4.7.23) and the uniform continuity of X for t ~ 0, it will follow that there exists a constant Tf > such that t 1 - Tf > to , and
°
4.7.24
961
§4.7. Dynamics in compartments We have from (4.7.22) - (4.7.24) that n
LI
n
Xi(tj)':"
xi I::; -(€/2)j 1] + L I Xi(tO) - xi I
;=1
i=1
which shows that L:~=1 I Xi(t) - xi I can become ~egative for large t and this is impossible. Thus our assertion (4.7.18) holds and the proof is complete. [] Let us consider next compartmental systems which incorporate "transport delays" (for numerous examples related to this, see Gopalsamy [1983c]). For instance, in the place of (4.7.3) we consider a delay-differential system of the form 4.7.25
i = 1,2,3, ... , n
for
t>0
;
t=
j) are nonnegative constants. It has been an where iij(i,j = 1,2, ... ,n; i implicit assumption in (4.7.3) that the transit time for material flux between any two compartments is negligible. In several physiological systems involving the transport of tracers of blood from one compartment to another, there is usually a finite time iij required for the transport of material from compartment j to the i-th compartment (from right ventricle to left ventricle etc.). Thus, it is worthwhile and perhaps necessary to consider (4.7.25) to be a generalisation of (4.7.3). Detailed mathematical analysis of compartmental systems with transport delays has been done by Lewis and Anderson [1980a,b], Gyori and Eller [1981]' Krisztin [1984] and Gyori [1986]. In the following, we first consider the effects of delays in (4.7.25) on the asymptotic behavior of solutions of (4.7.25) as t ~ 00. Theorem 4.7.3. Suppose the constant parameters of (4.7.25) satisfy the following: iij ~
aOi
> 0;
i,j'= 1,2,3, ... ,nj
0;
aii
=
aOi
+L
a ji
>
i=1,2, ...
0;
,n.
j=1 j#i
Then all solutions of (4.7.25) corresponding to initial conditions of the type
t E [-i, 0];
i
=
max
l$i,j$n i#j
i'"
')'
4.7.26
§4.7. Dynamics in compartments
962
satisfy
Yi(i) where x*
~
0
=
for
i
~
(xr,xi, ...
lim Yi(t)
O· and ,x~)
t-co
= xi;
i=1,2,,,.,n
is a steady state of (4.7.25) with xi
4.7.27 ~
O,i ==
1,2,3, ... ,no
Proof. We note that the existence of x* is not a part of the assumptions. Define a sequence y(k)(t) = {yi k)(t), y~k)(t), . .. , y~k\t)}, k = 0,1,2,3, ... as follows: (0)
_
Yi (t) -
{c/>i(t) for t E [-T; 0] c/>i(O) for t > 0
i = 1,2,,,.,n
4.7.28
c/>i(t) for t E [-T,O]
+ .I: aij
1
j=l
0
n
e-a"t c/>i(O)
+ Ui
1.t
t
e-a,,(t-s)y;k\ s -.:. Tii )ds
4.7.29
ii:-i
t > 0;
e-a,,(t-')ds;
i = 1,2, ... ,n.
It can be shown that the sequence {y(k)(t); t ~ -T} converges as k ~ 00 to a limit function y*(t) and the convergence is uniform on bounded closed subsets of [-T, 00). It will then follow from (4.7.29) that
c/>i(t) for t E [-T,O]
yi(t) =
4.7.30
A consequence of (4.7.30) is that ~
= 0,l,2,,,.,n:::} yi(i) ~ 0,
i=l,2, ... ,n
t
~ -T.
Consider now the linear system (in the unknowns ml, m2, ... , m n ) of the algebraic equations n
Laijmj +Ui j=1
= 0
i
= 1,2, ... ,no
4.7.31
363
§4.7. Dynamics in compartments
It follows from our assumptions on the coefficients aij in (4.7.31), that the matrix A = (aij) is diagonal (column) dominant with ajj < 0, i = 1,2, ... , nand aij ~ 0, i,j = 1,2, ... ,n; i t'j. It is known (see Araki and Kondo [1972]) that (-A) is a stable M-matrix such that (-A) is nonsingular and the elements of (_A)-l are nonnegative. Thus, the linear system (4.7.31) in the unknowns ml, m2, ... , mn has a nonnegative .solution i
= 1,2, ...
,no
We conclude that leaky compartmental systems have unique nonnegative equilibrium states. Let us for convenience, suppose x· = {xI, xi, ... , x~} is the nonnegative steady state of (4.7.25). To prove the convergence in (4.7.27), we let
i=1,2, ... ,n and derive that
i=1,2, ... ,n.4.7.32
Consider the Lyapunov functional v(t,w(.) for (4.7.32) defined by
v(t, w(.»
n ( n =~ I w;(t) I +j;, a;j f.-TO) I w;(s) Ids t
)
;
t > 0.
4.7.33
j#i
One proceeds to show that the upper right derivative D+ v of v along the solutions of (4.7.32) satisfies n
D+v :::; -
L aOil Wj(t) I,
t>O
4.7.34
i=l
from which it can be shown (the reader should' try this) that Wi(t) 00, i = 1,2, ... , n and this completes the proof.
-t
0 as t
-t
[]
Since (4.7.32) is a linear autonomous system of delay differential equations, one is entitled to ask the following; does it follow from the assumptions of a leaky compartmental system that all the roots of the associated characteristic equation have negative real parts? The following result contains an affirmative answer to this question.
§4.7. Dynamics in compartments
364
Theorem 4.7.4. Assume that
aij
2: 0,
Tij
2: 0, i,j
= 1,2, ... , n; i t= j
and
n
aii
= aOi + 2::: aji > 0;
i = 1,2, ... , n.
j=l j#i
Then all the roots of 4.7.35 where parts.
Oij
= 1 if i
Proof. Let
Z
=j
and
Oij
= 0 for i
=f j , i,j = 1,2, ... , n have negative real
be any root of (4.7.35). By Gershgorin's theorem of matrix theory
we know that n
I Z + aii I:::;
2:::
aije-
ZTij
for some
i E (1, 2, ... , n)
4.7.36
j=l
j#i
or equivalently, there exists Mi = Mi(Z) , I Mi(Z)
I :::; 1 such that
Z is a root of
n
Z + aji
+ Mi(Z) 2::: aije-
ZTij
=0
for some
i E (1,2, ... , n).
4.7.37
j=l j#i
It is enough to show that (4.7.37) has no roots with nonnegative real parts. Define Ii and gi as follows:
for
Since aii > 0, fi(Z) has no zeros Z with region ~e(z) ~ 0 , we have
~e
as in
~
Z
(4.7.37) .
4.7.38
0 and on the boundary of the
n
I fi(Z) I = IZ + ail I ~ aii > I:: aji ~ Igi(Z) I; j=l j#i
4.7.39
965
§4.7. Dynamics in compartments
hence IIi(z) I > I gi(Z) I on the boundary of ~e (z) ~ O. Since fi(Z) t- 0 for ~e (z) ~ 0, it follows from Rouche's theorem that fi(Z) + gj(z) i 0 for ~e (z) ~ 0 and this completes the proof. [] We find that although the result of Theorem 4.7.4 provides a set of sufficient conditions for the asymptotic stability of the trivial solution of (4.7.32), this result has not exhibited in any way, the effects of the transport delays on the mode or rate of convergence of solutions of (4.7.25) to its steady state. One expects from the result of Theorem 4.7.4 that the above convergence should be exponential. The following result is concerned with an examination of the effects of transport delays on the convergence of solutions of (4. 7~25) to its steady state. Theorem 4.7.5. Let r be any fixed positive number; let A and matrices such that aj;
is a fixed positive constant. (iii) Fij : [0, T] ~ [0,1] is continuous from left, monotone nondecreasing and Fij(O) = 0; Fij(T) = 1; i,j = 1,2" ... , n. (iv) Ui, i = 1,2, .... , n are known nonnegative constants. One of the immediate questions is to ask whether (4.7.44) has a nonnegative equilibrium state and whether such an equilibrium is globally attractive of all other nonnegative solutions. First we derive the following result which is a generalization of Theorem 4.7.2.
°
Theorem 4.7.6. Assume that Pij, Fij, T are as in (i) - (iv) above. Suppose a set of nonnegative initial conditions
t E [-T,O];
0 v(t) =
t. [I
Xi(t) - Yi(t)
I 4.7.46
967
§4.7. Dynamic3 in compartments
The next result (due to Krisztin [1984]) asserts the existence of a globally attractive constant steady state of (4.7.44)j a consequence of the following result is the nonexistence of nbnconstant periodic solutions for (4.7.44) Theorem 4.7.7. Suppose the assumptions of Theorem 4.7.6 hold. Assume further that P~i (i = 1,2, ... , n) are strictly monotonic. Then all bounded solutions of (4.7.44) (if such solutions exist on [-T,oo» are such that they converge to a nonnegative equilibrium state of (4.7.44). Proof. Let xCi) = {xl(i), ... ,xn(t)} be a solution of (4.7.44) which is bounded on [- T, 00) satisfying
o ::; lim sup Xi(t) = Mi < 00 t-oo
i
o ::; lim t-oo inf Xi(t) = mi < 00 Define a function G : IRn
1-4
= 1,2, ...
,no
Rn as follows:
n
n
G i (ZI, Z2, .• ' ,zn) = - LPij(Zi) + LPij(Zj)+Uii j=O
i
= 1,2, ...
,no
4.7.47
j=l
First we show that i
= 1,2, ...
4.7.48
,no
If (4.7.48) is not true, then there exists an io E {1, 2, ... ,n} such that
Let a that
= Gio(Ml, M 2 , •••
,Mn). Since Pij are continuous there exists
T
> 0 such
n
n
- L Pjio(Mi o - €) + L Pioj(Mj j=O j=l Now choose
€
such that if t ~
T,
sup x }·(t)
f~r-T
+ e) + Uio < aj2.
4.7.49
then
-< M·} + eo,
j=1,2, ... ,n.
We have by the monotonicity of Pij,
dXi o n - d (t) ~ - LPjio(Mj - e)
t .}=o
n
a
+ LPioi(Mj + e) +Uio < 2" < 0 . }=l
4.7.50
§4.7. Dynamics in compartments
368
t 2: rand Xio(t) E [Mio - €, Mio + €].
for
This contradicts the definition of G io proving i = 1,2, ... ,no
By similar arguments we have Gi(ml, m2,'" , m n ) above and the equality
0, i
= 1,2, ... , n. From the
n
n
L Gi(Zl,."
~
, zn) =
L[Ui - POi(Zi)),
i=1
;=1
it follows that n
n
o ~ LGi(M1,M2, ... ,Mn)- LGi(ml,m2, ... ,m n) i=1
;=1
4.7.51
N
= - 2:.)Poi (Mi )
-
POi(mi)] ~ 0
i=1
showing i = 1,2, ... ,no
By the strict monotonicity of POi, we have that
Mi = mi = tlim Xi(t); --00
i
= 1,2, ... ,n.
It is now easy to see that x* = (xr,x;, ... ,x~), xi = Mi = mi, i = 1,2, ... ,n is a steady state of (4.7.44). The nonnegativity of xi, i = 1,2, ... , n is a consequence of that of Xi(t) on [-T, 00), i = 1,2, ... ,no To prove that x* = (xi, ... ,x~) is attractive of all other nonnegative solutions, one can proceed as in the case of no delays, using the Lyapunov functional proposed in the proof of Theorem 4.7.3 with Yi == xi, i = 1,2, ... ,n. 0 Some examples of compartmental systems are listed below; it is left to the interested reader to investigate the convergence, persistence and oscillatory characteristics of the following:
dx~it) = -.\;Xi(t) + ~ aij l~TJ Xj(s) ds + Ui; i = 1,2,"" n.
l
969
§4.7. Dynamic3 in compartment3
dXi(t) d- = t
\
-/\iXi
() ~ ajjxj -()d t + L...J S S + Ui; i=l i~i
Xj(t) =
sup
i = 1,2,000,
Xj(s),
no)
sE[t-rj ,t]
dXi(t) = -AiXi(i) + ~ aijXj([iJ) ds + Ui; ) -----;[t
f;r
z = 1,2, .. ·,n.
dx~?) =
-AiXi(i) +
~ aijX j(jJjt) + Hi;
0< Jli < 1,
i
)
= 1,2,,,,, n.
dx i ( t) = -AiXi(i) \. -----;[t + ~ aij log{Xj(t )}.+ Hi )
f;r
i = 1,2, .. ·,n.
We remark that the techniques developed for the analysis of compartmental systems can also be used in studying the dynamics of neural networks modelled by (for details of neural networks see Marcus et al. [1991] and the references cited therein) systems of the type
in which f3i, ajj , a i E R, i = 1, 2, ... , n and the responses.
Tji
correspond to delays in neuronal
970
EXERCISES IV 1. In the following equations, prove that solutions corresponding to positive initial conditions remain positive and are defined for all t > O. Derive sufficient conditions for the existence of a positive equilibrium and for its global attractivity. Investigate the P?ssibility of delay induced bifurcation to periodic solutions and obtain sufficient conditions for the oscillation of all positive solutions about the positive equilibria (assume that all the parameters are positive constants).
dN (t) dt
= r N (t) [J{ - N (t -
J{ + N (t - r)]. r)
dN(t) = -rN(t) + ae-PN(t-r). dt dN(t) = -rN(t) + aN(t _ r)e-PN(t-r). dt dN(t)
(3
---;It = -rN(t) + 1 + N(t - rf dN(t) = -rN(t) (3N(t - r) dt + 1 + N (t - r) dN(t) dt dN(t) dt
d~~t)
= -rN(t)
+
(3N(t)Nn(t - r). (}n+Nn(t-r)
= rN(t)[l- N(t -
. J{ r)]_ 1 +N2(t) N2(t)
= - r N(t)+ aN(t - r)
[1+ 11(1 - Nn~;: r)) ].
Investigate the convergence and oscillatory characteristics of the above systems when N(t - r) is replaced by N(t) , N([t]) and Nt respectively where
N(t) =
N(s),
sup sE[t-r,t]
and
Nt
=
it
N(s) ds
t-r
[t] = greatest integer contained in t.
Exercises IV
971
2. In the previous exercise replace N(t - r) by the integral tenn
1
00
K(s)N(t-s)ds
with a suitable nonnegative delay kernel Kj examine the stability (local and global) of equilibria; examine the oscillations and persistence of the resulting integrodifferential systems. 3. Discuss the stability of equilibria, persistence and oscillations about the equilibria in the following systems:
dx(t) = X(t)[7'l - anlog(x(t - r)] --;It dy(t) -;It
= yet) [7'2 + a21 1og[x(t dx(t) = 7'l X(t) dt
[1 -
+ a12 log[y(t - r)J]
lj 1 lj
r)] - a22 log[y(t - r)]]. x(t - r)
al
+ Ilty(t -
r)
dy(t) _ 7' (t)[lyet - r) dt - 2Y a2 + f32X(t - r) . dx(t) dt dy(t) --;It
~X(t)[k-X(t)-
=y
()[ x(t - r) ( )1 t 1 + x(i _ r) - ay t .
dx (t) _ (.) [ _ x (t - r) dt - 7'X t 1 K - ax(t)y(t) [1 dy(t) --;It
y(t-r)
1 + x(t - r)
1 1
-
exp { - /3x(t - r)}
= -8y(t) + y(t)[l -
exp { - /3x(t - r)}.
dx(t) = a - (3x(t)
+x
dt
d~~t) = ax2(t _ r) -
2
(t - r) yet - r)
by(t).
j
}
Exercises IV
372
4. Consider a Lotka - Volterra system
[2: a··x ·(t)1 n
dXi(t) r·, - . dt- = x·(t) '
'1
1
,
i
= 1,2,··· ,no
1=1
Assume that solutions of this system are uniformly bounded in the positive orthant of Rn and there exist positive constants Pl,P2,··· ,Pn such that
holds for all equilibria on the boundary of the positive quadrant of Rn. Then prove (for details see Jansen [1987] and Hofbauer and Sigmund [1987]) that the above Lotka Volterra system is persistent where A denotes the matrix with entries aij. Can you extend this result to delay and integrodifferential system of equations? Derive that a necessary condition for the persistence of the above system is the existence of a positive equilibrium point. 5. Assume that solutions of the Lotka - Volterra system of the previous problem are uniformly bounded in the positive orthant and there exists a differentiable function P : Ri- Ho R with the following properties:
(i) P(x) = 0 for x on the boundary of the positive orthant of Rn; P(x) > 0 for x in the interior of (ii) The function (~) extension to R+.
Ri-.
= 1/1
defined on the interior of
Ri-
has a continuous
(iii) For all x on the boundary of Hi- there exists aT> 0 for which 1 {T T Jo tjJ(x(t))dt > O. Prove that the Lotka - Volterra system of the previous problem is persistent (for more details see Kirlinger [1986], Hofbauer and Sigmund [1988]
373
Exercises IV
and the literature cited therein). Can you obtain a similar result applicable to systems with time delays and integrodifferential equations. 6. Consider the functional differential equations for i = 1, 2, ... , n
oo
where"\ > 0, aij < 0, J.lij : [0,00) 1-+ R is of bounded variation, Jo /dJ.lij( s)/ = 1 and 7]ij( s) = bij( s) - 8jj e->'s obeying 7]ij( s) = constant for s > T. Assume that the nonlinear system has a positive steady state X* = (xi, ... , x~), xi > 0, i = 1,2, ... , n. Suppose there exist constants d j > 0, with n
ddaid >
n
L dj/aij/ + L dj(/bij/ + 18ij !), 1;~
i = 1,2, ...
,n.
j=l
Prove that the steady state x* is locally asymptotically stable. Prove or disprove the statement: x* is globally asymptotically stable with respect to positive solutions (see Busenberg and Travis [1982]). 7. Consider the retarded functional differential equations for i = 1,2, ... ,n
with
aii
/.00 d7]ii( s) + 8ii < 0; fIX> Id7]ij( s)/ = 1 o ~ .
aij7]ij( s) - 8ije ->.s = constant for s > T and ,.\ > Suppose the system has a positive steady state x* 1,2, ... , n and there exist positive constants dj >
°
i, j = 1,2, ... ,n .
= (xi, ... ,x~), xi
> 0, i
=
°, j = 1,2, ... , n such that
Exercises IV
374
d;la;;
1~ cos(vs )dry;;(s) + 8;; J.~ cos(vs),\e -,. d·1 >
t.
dj (la;;1
+ 18;;1)
j~i
for all real v satisfying
Prove that x* is locally asymptotically stable (Busenberg and Travis [1982]). 8. Consider the linear system dXi(t) --;It =
?=k aijXj(t) + Lk [fO_ Xj(t + s) ( L dT/ijm(S) 1 .= 1,2, .. , k f.
J=l
J=l
h
)
j Z
m=l
where aij ~ 0 for i =j:. j and dT/ijm(S) are nonnegative measures on [-h,O]. Prove that a necessary and sufficient condition for the asymptotic· stability of the trivial solution is that the following hold (for more details see Obolenskii [1983]) f.
an
+~
1. 0
0;
(-It anI
+ 2:~=I J~h d1Jn1 m
ann n = 1,2, ... , k. ••••••
+ 2:~=l J~h d1Jnnm
9. Consider a biochemical system modelled by the autonomous ordinary differential equations dXl(t) _ al _ b x (t) dt - 1 + kl Yn ( t ) 1 1
dYl (t)
--;It
= Q:'lXl(t) - (31Yl(t)
dXi(t) dt dYi(t)
=
aj
_
1 + k i Yi-l(t)
bjXi(t)
--;It = Q:'iXi(t) - (3iYi(t)
for i = 2,3, ... ,n. Assume that all parameters appearing above are positive constants. Obtain sufficient conditions for the existence of a positive steady state and for its global asymptotic stability with respect to positive solutions.
375
Exercises IV
If there are time delays in the above model so that
for i = 2,3, ... , n, examine whether there exists a delay induced instability leading to persistent oscillations (Banks and Mahaffy [1978a, bJ). 10. Derive a set of sufficient conditions for the existence of a componentwise positive steady state and its global attractivity with respect to positive solutions in the following (assume all parameters are positive constants).
(1)
(2)
i = 1,2, ... , n.
(3)
i=1,2, ... ,n. (4)
(5)
i = 2,3, ... ,no
Exercises IV
376
11. Consider a competition system modelled by
- i = 1,2, ... , n; where bil aij, Tij are nonnegative constants. Assume n
bi
> Laij(bj/ajj),
i = 1,2, ... ,n.
j=l j~i
Show that the system has a globally asymptotically stable positive steady state. Develop a similar result for a system of the form
= 1,2, ... ,n.
i Do the same for the systems
dXi(t) = x;(tl { b; - a;;Iog[x;(tl] --;It
fu~
a;j log[xj(t - T;jl] } ;
i
= 1,2, ... ,n;
i
= 1,2, ... , n.
12. Suppose there exists a positive steady state x* = (xi, ... , x:), xi 1, 2, ... , n for the Lotka-Volterra system
dx- =x- { b-+"a--xn } _, dt t 1 L....t I] ] j=l
i = 1,2, ... ,no
>
0, i =
377
Exercises IV
Suppose further, there exists a constant positive diagonal matrix C such that CA+ATC is negative definite where A denotes the n x n matrix (aij). Prove that the steady state x* is globally asymptotically stable. Under the same assumptions as above, examine whether x* is (i) locally; (ii) globally asymp.totically stable for the time delayed systems i = 1,2, ... ,n
i
= 1,2, ... ,n.
(1)
(2)
13. Investigate the asymptotic behavior of the dynamics of a prey-predator system modelled by
Y- - c:x -dx = x [ ab - dt 1 + ax dy -d t
dx- = Y [ -c + 1 + ax
fLY
/.00 kl (s )y( t 00
s )ds
1
0
+
1 0
1
k 2 (s)x(t - s)ds .
14. Derive sufficient conditions for the existence of a globally asymptotically stable positive steady state of
i = 1,2, .. . ,n;
where r, fJi, ki' aij ; (i, j = 1,2, ... , n) are nonnegative constants. Consider the cases 0 ~ fJi < 1 and fJi > 1, i = 1,2, ... , n. If there are time delays in the above system so "that Xi () t xi (t ) e· ' . ---;u= .\iXi(t) 1- ( T ) - ~aij [
d
n
(
x·(t 1 k
j
Ii') ) J
e.] J
•
;'
= 1,2, ... ,n,
then under what additional conditions, the global asymptotic stability of the positive steady state holds?
Exercises IV
978
15, Derive a set of sufficient conditions for the existence of a globally asymp-
totically stable positive steady state of an integrodifferential system of the form
d:~t) = x(tl(a -
bx(t) - ey(t)
_/,T k,(s)x(t - s)ds
_/,T k2(s)y(t _ sjdS] d~~t) = yet) [ _ d +px(t) - qy(t) + /,T k,(s )x(t - s) ds
_/,T k (s)y(t-S)dS]. 4
16. Consider the autonomous ordinary differential system 1 dy· dt
= Yi [');
(6)
dx(t) dt
(7)
dx(t)
d:t = x(t) [a - bx(logt)]; t > l. dx(t)
-;It = x(t) [a -
(8)
blog{ x(t)}].
(9)
d:t = x(t)[a - blog{x(At)}].
(10)
dx(t)
36. Discuss the asymptotic behavior of the following prey-predator system due to Anvarinov and Larinov [1978] :
d:~t)
= X(t)[,,->.y(t)->'
1.= K,(s)y(t-s)ds
-1.= 1.= R, (s, u)y(t - s )y(t - u) du dS] d~~t) = yet) [ -
j3 + J1.x(t)
+ 37. Let aij
~
0, i
=f j
+ J1.1°O I(2( s )x(t -
s) ds
1.= 1.= R2(S,U)X(t-u)x(t-S)duds].
, i,j = 1,2, ... , n. Prove that the trivial solution of i = 1,2, ... ,n
is asymptotically stable if and only if
> O.
387
Exerci3e3 IV Under the same conditions prove that the trivial solution of
Yi(t) = ('taijyrj+I)2mi+l;
i = 1,2, .. ,n
}=l
where kj and mj are nonnegative integers, is also asymptotically stable. Generalize (see Martynyuk and Obolenskii [1980]) the above result to systems of the form
38. Investigate the convergence and oscillatory characteristics of the following models of cooperation:
()]3]
= Xl3()[I.t)] -log[X3(>.t)]]
dX;?) = rl Xl (t)
in which [t] denotes the integer part of t and 0
< A < 1.
CHAPTER 5
MODELS OF NEUTRAL DIFFERENTIAL SYSTEMS 5.1. Linear scalar equations
Consider a linear neutral integrodelay differential equation of the form
x(t) + t,b;X(t-U;)+fJ
J."" K (s):i:(t-s)ds 2
+aox(t) + ~a;X(t-T;)+a
J."" K.(s)x(t -s)ds =0
5.1.1
in which x(t) denotes the right derivative of x at t. (Throughout this chapter we use an upper dot to denote right derivative and this is convenient in writing neutral differential equations systematically). Asymptotic stability of the trivial solution of (5.1.1) and several of its variants have been considered by many authors. There exists a well developed fundamental theory for neutral delay differential equations (e.g. existence, uniqueness, continuous dependence of solutions on various data; see, for instance, the survey article by Akhmerov et al. [1984]); however, there exist no "easily verifiable" sufficient conditions for the asymptotic stability of the trivial solution of (5.1.1). By the phrase "easily verifiable" we mean a verification which is as easy as in the case of Routh-Hurwitz criteria, the diagonal dominance condition or the positivity of principal minors of a matrix etc. Certain results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [1986] that a linear neutral differential equation can have unbounded solutions even though the associated characteristic equation has only purely imaginary roots. (see also Snow [1965], Gromova [1967], Zverkin [1968], Brumley [1970], and Datko [1983]); such a behavior is not possible in the case of ordinary or (non-neutral) delay differential equations. It is known (Theorem 6.1 of Henry [1974]) that if the characteristic equation associated with a linear neutral equation has roots only with negative real parts and if the roots are uniformly bounded away from the imaginary axis, then the asymptotic stability of the trivial solution of the corresponding linear autonomous equation can be asserted. However, verification of the uniform boundedness away from the imaginary axis of all the roots of the characteristic equation is usually difficult. An alternative method for stability investigations is to resort to· the technique of Lyapunov-type functionals and functions; this will be amply illustrated in this chapter.
§5.1. Linear scalar equations
394
Let us consider (5.1.1) with the following assumptions:
(Hd
ao,
0,
a, {3, ajTj
are real numbers such that ao > 0, Tj ~ 0, j = 1,2,3, ... n (or m as the case may be).
aj, bj, Tj, (Jj
=I 0,
=I 0,
bj(Jj
(H2) ,KI, K2 : [0,(0)
J-lo
(Jj ~
(-00,00) are piecewise continuous on [0,(0) such that
(H3) A set of initial conditions x(t) = (S + r)x(S)dS] = -aCt
+ r)x(t) -
5.1.21
b(t)x(t - 0-),
§5.1. Linear 3calar equation3
398
and consider a Lyapunov-type functional vet) = v(t,x(.)) defined by
vet)
= vet, x(.)) = [x(t) -
t r a(s
r
+ T)x(s)ds
+ t r a(s + 2T)([ a(u + T)X 2 (U)du )dS + t r jb(s + T)j ( [ a(u + T)X 2 (U)du )dS
5.1.22
+ 2 t.jb{s + uljx 2 (slds + 41~T Ib(s + r + a)!a 2(s + r)x2(s) ds. Restricting our attention only for t 2: T, we calculate (5.1.15) and simplify it so as to obtain
D+v lit
~
-
( 2 x (t)a(t
D;t
V
along the solutions of .
+ r)[2 - J.Ll(t)] 5.1.23
+ :i;2(t -
a)lb(t)![l - J.L2(t)]).
As in the case of the proof of Theorem 5.1.1, it can be shown that x is uniformly continuous on [T,oo) and furthermore aCt + r)x 2 (t) E Ll(T,oo). By Barbalat's lemma (Lemma 1.2.2) lim aCt + r)x 2 (t)
t-+oo
Since
= 0.
5.1.24
I!:: a( t + r) > 0, the result follows from (5.1.24) and the proof is complete.
Corollary 5.1.3. Let a, b, rl, r2 be real numbers such that a > 0,71 > 0, r2 2 If
arl
o.
+ Ibl < 1,
then the trivial solution of :t [X(t l - bx(t is asymptotically stable.
T2l] + ax(t -
T.) =
0
5.1.25
399
§5.1. Linear scalar equations Proof. Proof is based on the Lyapunov functional
t
[x(t)-bX(t-r2)-a Jt-Tl X(S)dsj2
+ albl
Lr,
+ a'
x'(s) ds
Lr, ([
x'(u) dU) ds ..
[]
We omit the details. If r = 0 and aCt) is a constant functional
v(t,X(.),x(.))
==
a in (5.1.15), then one can consider the
= x2(t) +
.!.It a
x2(s)ds
5.1.26
t-cr
and, derive that
vet)
= -ax2(t) - (.!.a )[1 ~ -ax2(t) if !b(t)!
Once again it is possible to show that if Ib(t)! of (5.1.15) satisfy (5.1.19).
b2(t)Jx 2(t - a)
< 1.
< 1 and aCt) == a > 0, then solutions
The result of Theorem 5.1.2 can be used to obtain sufficient conditions for the global asymptotic stability of the positive equilibrium of
N(t)
= rN(t) [1 - N(t -
r)
K
+
aN.(t - a)
1 + N2(t - a)
j.
For more details of this analysis, we refer to Gopalsamy [1992].
5.2. Oscillation criteria In this section we derive sufficient conditions for the oscillation and nonoscillation of first order neutral equations of the form k
x(t) + px(t - r) +
L qiX(t - ai) = O.
5.2.1
i=l
Some of the conditions we obtain are easily verifiable when the parameters are known. We note that the asymptotic stability of the trivial solution of (5.2.1) is not necessarily determined by the negativity of the roots of the characteristic equation. However, the oscillatory nature of (5.2.1) is determined by the roots of the characteristic equation. The following is due to Kulenovic et al. [1987b].
§5.2. 03cillation criteria
400
Theorem 5.2.1. Consider the scalar neutral differential equation (5.2.1) and assume that pER, T 2:: 0, qi > 0 and CTi 2:: 0 for i = 1,2, ... k.
Then a necessary and sufficient condition for the oscillation of all solutions of . (5.2.1) is that the characteristic equation k
).. + p)..e- AT +
L qje-
AU
,
= 0
5.2.2
i=l
has no real roots. Proof. The proof of the necessary part is simple. Suppose that every solution of (5.2.1) oscillates. IT the characteristic equation (5.2.2) has a real root )..0, then (5.2.1) will have the nonoscillatory solution
yet) = e Aot • But this contradicts the hypothesis that every solution of (5.2.1) oscillates. The proof of the sufficiency part is quite involved and therefore we shall restrict here to the special case where p E (-1,0); the reader is referred to the original article of Kulenovic et al. [1987b] for other cases where p ~ -1 and p 2:: o. Let p E (-1,0) and suppose (5.2.2) has no real roots. Let k
F()")
= ).. + p)..e- AT + L qje- AU,.
5.2.3
i=l
Then F(O) =
2::=1 qi >
o~
< ...
0) k
zn(t) - zn(t - a)
+ L qj i=l
1 t
Zn-l(S - O'j)ds
=0
t-o
and this will imply
Thus, for a
= 0' k
we get k
0= Zn(t)
+ L qiZn-l(t -
-
(1i) < Zn(t)
i=l
+ !!..Zn(t) O'k
which proves that
iJ
4
A2 = - = 3 2 (1k (1kqk is an upperbound of An. This completes the proof of Pl' We now prove Pz with J1. = m. Let A E An and set Then
§5.2. Oscillation criteria
405
and so 'Pn(t) is a nonincreasing function for any A E An satisfying (5.2.4). We note that
Zn+I(t)
+ (A + m)Zn+I(t) k
=-
L qiZn(.t -
ai)
+ (A + m)Zn(t) + peA + m)zn(t -
T)
i=I k
= e- At [ -
L qieACTi T, P < 1 and lim t-oo
i
t
t-(u-r)
1-p
Q(s)ds> - e
E
§5.2. Oscillation criteria
408 then, every solution of
x(t) - px(t - 7) + Q(t)x(t - (1)
= 0, t > to
is oscillatory". The next result (due to Gopalsamy and Zhang [1990J) provides an alternative and somewhat weaker condition for all solutions of
x(t) - ex(t - 7) + p(t)x(t - (1)
=0
5.2.33
to be oscillatory. Theorem 5.2.6. Assume the following:
(i)
e, 7, (1 are positive numbers, 0 < e
(ii)
p E C(R ,R+), pet + 7)
(iii)
Ro > 1=.£. e'
= pet),
< 1,
(1 ;:::
7 ;:::
O.
tER,
fLrP(S) ds = Po.
5.2.34
Then all nontrivial solutions of (5.2.33) are oscillatory. Proof. Suppose the conclusion does not hold. There exists a nonoscillatory solution x(t) which we shall assume to be eventually positive; then, there is a T > 0 such that x(t) > 0 for t 2: T. We have from (5.2.33),
d dt [x(t) - ex(t - 7)] :::; 0 for t > T
+ (1 = T 1.
Now there are two possibilities:
(i) x(t) - ex(t - 7) :::; 0 for t > Tl (ii) x(t) - ex(t - 7) > 0 for t > T1 • We first show that (i) is not possible. If (i) holds, we have for some constant
8> 0,
x(t) - ex(t - 7) :::; -8 for t > Tl and leading to
x(t) :::; -8 + ex(t - 7)
:::; -8 + e[-8 + ex(t - 27)] :::; -8[c + e 2 + ... + en]
+ en+1x(t -
(n
+ l)e).
§5.2. Oscillation criteria
409
If we let
II 'to II =
su p
tE[Tl-T,Td
I'to (t) I,
then for t 2:: Tl and sufficiently large n, 5.2.35 Since 0 < e < 1, (5.2.35) implies that x(t) will be negative and this contradiction shows that x(t) - ex(t - r) ::; 0 for t 2:: Tl is not possible. Let us then suppose x(t) - ex(t - r) > 0 for t 2:: T and define
wt= ()
x(t-r)-ex(t-2r) >l. x(t)-ex(t-r) -
5.2.36
Dividing both sides of (5.2.33) by [x(t) - ex(t - r)] and integrating,
log[w(t)] = It ( p(s)x(s t-T xes) - exes = lt p(s)[x(s - 0') t-T
0') )dS - r) - exes - 0' - r) + exes - 0' - r)] ds xes) - exes - r) t lt p(s)ex(s-O'-r) 2:: l t-T p(s)w(s)ds + t-T xes) _ exes _ r) ds.
5.2.37
Using the periodicity of p in (5.2.37),
lt xes - r) - exes - 2r) t log[w(t)] 2:: t-T p(s)w(s)ds - e t-T xes) _ exes _ r) ds l = lt p(x)w(s)ds - elt w(s): {log[x(s - r) - exes - 2r)]}ds. t-T
t-T
s
5.2.38
Let t* be a number such that t - r < t* < t and t
I
*
t-T
p(s)ds
R = -2°,
1t
p(s)ds
t*
R = ~.
2
We show that wet) is bounded above. On integrating (5.2.33) over (t*, t),
x(t) - ex(t
~ r) -
[x(t*) - ex(t* - r)]
+
t p(s)x(s - O')ds =
it-
0
§5.2. Oscillation criteria
410 which implies
x(t*)
~ ex(t* -
~ t p(s)x(s it·
r)
>
t
it·
O')ds
p(s) [x (:5 - 0') - exes - 0' - r)]ds
~ [x(t -
0') - ex(i - 0' - r)]
it
5.2.39
p(s)ds
t*
Po
= [xCi - 0') - ex(i - 0' - r)1"2 .
Integrating (5.2.33) over [t - r, t*],
x(tO) - ex(t' - r) - [x(t - r) - exit - 2r)] + l~'/(S)X(S - u)ds
=
o.
As a consequence of the previous equation,
x(t - r) - ex(t - 2r)
~
l-r
~
[x(t* - 0') - ex(t* - r - 0')] ~o
t*
p(s)[x(s - 0') - exes - 0' - r)]ds 5.2.40 •
Since x(t) - ex(t - r) is decreasing, we can combine (5.2.39) and (5.2.40) so as to have
x(t*) - ex(t* - r)
~ (x(t -
r) - ex(i -
2r)](~O)
~ [x(i* -
r) _ ex(t* _ 0' _
r)](~O 2)
~ (x(t* -
r) - ex(t* - 2r)]P! .
Thus w (t
for any i*
~
*)
=
x(t*-r)-e.x(t*-2r) 4 <x(i*) - ex(i* - r) - Pi
5.2.41
T. vVe let
lim inf wet) t-co
and note that f
p ( l_c)log(f) £ _ 0 which implies
(I-C) e --
~
Po =
it
p(s)ds
t-r
and this contradicts (5.2.34); the proof is complete. Corollary 5.2.7. [f0 < c < 1, cr Po T
~
T,
[]
pet) == Po > 0 and if,
> [1 - c,B( c) ]2/ ,B( c)
where ,B( c) is a solution of
1- d
= log[e],
then every solution of
x(t) - cx(t - T)
+ Pox(t - cr) = 0
is oscillatory. Proof. Most of the details of proof are similar to those of the previous theorem and we shall be brief. We have from (5.2.44)
(1 - d) lO~(f)
~ PoT.
We define F as follows:
F(f)
= (1 -
d) log(f) f
5.2.46
§5.2. Oscillation criteria
412
and note that pI (e) = 0 leads to 1 - c1
= log(l).
It is found that j3( c) is a zero of this equation and so
13(0)
= e,
F"(j3(C»
1
< j3(c)
~ e
= -(1 + j3(C)]jj33(C) < O.
It follows that
F(j3(c» = supF(e)
= [l-cj3(c)J2jj3(c)
f;:::l
and hence (5.2.46) implies PoT ~
F(j3(c»
which contradicts our hypothesis.
(]
Theorem 5.2.8. Suppose tbe following bold: (i) c, T, cr be nonnegative numbers, 0 < c < 1,
T
2:: 0, cr > OJ
(ii) P E C(IR, R+), pet) 2:: Po > 0, t E Rj
(iii) poo-e > 1 - c ( 1 + f!!:;: ).
5.2.47
Then every solution of
x(t) - cx(t - T)
+ p(t)x(t -
cr) = 0
5.2.48
is oscillatory. Proof. We shall show that the existence of a nonoscillatory solution of (5.2.48) leads to a contradiction. Suppose y is a nonoscillatory solution of (5.2.48); we can assume that there exists aT> 0 such that yet) > 0 for all t 2:: T. (If yet) < 0 eventually the procedure is similar). One can show that nonoscillatory solutions of (5.2.48) tend to zero as t ~ 00 due to (i) and (ii). Thus we have from (5.2.48),
1 + 1 00
yet)
= cy(t -
T)
+
p(s)y(s - cr)dSj t 2:: T
+ T = to 5.2.49
00
2:: cy(t - T)
Po
yes - cr)ds,
t
> to.
413
§5.2. Oscillation criteria It is not difficult to show from (5.2.49), that
yet)
~
for large t where
=_
cy(t - r) => yet)
(10:( c») ;
II. r iO:
~
o:e-p.t
( /)
= y( to)e p.to
T
5.2.50
5.2.51
•
Define a sequence {Yn(t)} as follows:
Yo(t) == yet) t ~ to cYn(t - r) + Po ftoo Yn(s - (1)ds; Yn+l(t) = { yet) - y(to) + CYn(to - r) + Po ftC: Yn(S - (1) ds;
5.2.52
t:::; to.
It follows from (5.2.52) that
Yn+l(t) :::; Yn(t)
~
... :::; yo(t); t
~
to.
5.2.53
Furthermore from (5.2.50),
Yo(t)
~
o:e-p.t
which implies Yl(t) ~ ae-p.t leading to Yn+l (t) ~ ae-p.t, n = 1,2,3, .... Thus we have from (5.2.53),
ae-p.t :::; Yn+l(t) :::; Yn(t) :::; ... :::; yo(t), t ~ to. By the Lebesgue's convergence theorem, the pointwise limit of {Yn(t)} as n exists and 00
ae-p.t :::; y*(t)
= cy*(t -
r) + Po
1
y*(s - (1)ds
5.2.54 ~
00
5.2.55
where
y*(t)
= n-+oo lim Yn(t).
Thus, y*(t) is a nonoscillatory solution of
x(t) - cx(t - r) + Pox(t - (1) =
o.
5.2.56
But by Theorem 5.2.4, the equation (5.2.56) cannot have a nonoscillatory solution when (5.2.47) holds. This contradiction proves the result. [J We have seen that (5.2.1) can have a nonoscillatory solution when the associated characteristic equation has a real root. It is, however, desirable to obtain
414
§5.2. Oscillation criteria
verifiable sufficient conditions in terms of the parameters of (5.2.1) for its characteristic equation to have a real root. Also in certain cases, such as (5.2.43) when pet) ¢ a constant, the method of characteristic equation is not applicable. We shall now derive sufficient conditions for (5.2.1) and (5.2.43) to have nonoscillatory.solutions. We need the following lemma which combines both the Banach contraction mapping principle and Schauder's fixed point theorem. Lemma 5.2.9. (Nasbed and Wong [1969J) Let X be a Banach space; n be a bounded closed convex subset of X; A, B be maps of n into X sucb tbat Ax + By E n for every pair x, yEn. If A is a strict contraction (i.e. it satisfies tbe condition tbat for all x, yEn,
IIAx - Ayll :::; ,lIx -. yll for some" 0 :::; , < 1) and B is completely continuous (B is continuous and maps bounded sets into compact sets), tben tbe equation
Ax+Bx bas a solution in
=x
n.
Theorem 5.2.10. Assume tbat tbere exists a positive number J1. satisfying ce llr
peller
+ -J1.- < 1. -
5.2.57
Tben (5.2.22) bas a nonoscillatory solution wbicb tends to zero as t
~
00.
Proof. Let C = C([ -T, 00), IR ) denote the Banach space of all bounded functions defined on [- T, 00) with values in III = (-00,00) where T = max( 0", r)j the space C is endowed with the sup-norm. Let n be the subset of C defined by e- 1l1t
:::;
x :::; e- 1l2t j
cx(t - r) < Dx(t);
J1.1
> c
J1.2
> 0 on [-T,oo)}
< D < 1 for t
where J1.2 satisfies (5.2.57) and J.ll > J.l2' Define a map S:
S(x)(t)
= Sl(X)(t) + S2(X)(t)
~
0
n~C
5.2.58
as follows: 5.2.59
415
§5.2. Oscillation criteria where
S1(X)(t) = cx(t - T)
1
00
S2(X)(t)
=
px(s - O')ds.
It is easily seen that the integral in S2 is defined whenever x E n. It follows also from (5.2.58) that 51 is a contraction (due" to c < D < 1) and that 52 is completely continuous. The set
n is
closed, convex and bounded in C. We show that for every pair
x,y E n, For instance, we have for any x, y in
S1(X)(t)
n,
1.
+ 5 z(y)(t) S ce- JL2 (t-r) + p = e- JL2t [ceJL21'
00
e- JL2 (s-cr)ds
+ P~Z2cr 1
5.2.60
S e- JL2t where 112 by assumption satisfies (5.2.57). Also
Sl(X)(t) + 5 2 (y)(t)
~ ce- JLdt - r ) + p ~
ce- JL1 (t-r)
~
e- JL1t
provided 111 is large enough. For any x E n,
c5(x)(t - T) = c[cx(t - 2T) +
S c[Dx(t - T) +
1
00
e-JL1(s-cr)ds 5.2.61
I.:
1:
1
px(s - O")ds] px(s - 0") ds]
5.2.62
00
< D [cx(t":" T)
+
px(s - 0') ds]
= DS(x)(t). From (5.2.60) - (5.2.62), it follows that
S}(x)
Sen)
n.
+ Sz(y) E n if (x,y) E n.
n
n
Thus c By Lemma 5.2.9, the map S : -+ C has a fixed point in which is a nonoscillatory solution of (5.2.22) and the proof is complete. []
§5.2. Oscillation criteria
416
Corollary 5.2.11. Assume that one of the following holds: po-e s:; 1 - ce(r/u) (i) (ii) pTe(r1/r) s:; 1 - ceo Then (5.2.22) has a nonoscillatory solution which tends to zero.
5.2.63 5.2.64
Proof. The conclusion follows from Theorem 5.2.10 for the choices of f-L = ~ and f-L = ~ respectively in (5.2.57). []
In the equation
x(t) -
(21e)X(t - 1) + (;e )x(t -
1) = 0
the condition (5.2.63) of Corollary 5.2.11 is satisfied since po-e
= (1/2),
1 - ce( rJu) = (1/2).
This equation has a nonoscillatory solution x(t) = e- t • Theorem 5.2.12. Let c, T, 0- be nonnegative numbers, 0 < c < 1, T ~ 0, 0- > O. Let p E C(R +, IR +) and pet) -+ Po > 0 as t -+ 00. If there exists a positive number f-L satisfying
5.2.65 then
+ p(t)x(t -
x(t) - cx(t - T)
0-)
= 0
5.2.66
has a nonoscillatory solution.
Proof. Details of proof are similar to those of Theorem 5.2.10 and we will be brief. Define a map S: n -+ C([-T,oo),R) where n is defined in Theorem 5.2.10 for a suitably selected positive number T; let S be as follows: S(x)(t)
= cx(t -
T)
= Sl(X)(t)
+ /.00 p(s)x(s -
+ S2(X)(t)
(say).
o-)ds
5.2.67
417
§5.2. Oscillation criteria To show that Sl(X)(t)
+ S2(y)(t) E n
Sl(X)(t) + S2(y)(t)
for (x, y) En, we have
~ ce- Jl2 (t-r) +
1
00
p(s)e- Jl2 (S-U)ds
eJl2U = ce- Jl2 (t-r) _ 112
1
00
p(s)d(e-Jl2S) 5.2.68
t
= e-p,t [cep'T + PO::'"] ~
e- Jl2t
for all t ~ T where T is sufficiently large (we have used a limiting form of the mean value theorem of integral calculus in the last step in the derivation of (5.2.68». The other details of proof are similar to those of Theorem 5.2.12 and hence we omit them. [) Theorem 5.2.13. Assume that c, 7, a are nonnegative numbers and p E C(R+, R+); also suppose pet) ~ Po. If there exists a positive number 11 satisfying
5.2.69
then (5.2.66) has a nonoscillatory solution. Proof. Let yet) be a nonoscillatory solution of
x(t) - cx(t - 7) + Pox(t - a)
=0
which exists by virtue of (5.2.69) and Theorem 5.2.10. {xn(t),n = 0,1,2, ... } for t E [-T,oo) as follows:
5.2.70 Define a sequence
Xo(t) = yet) CXn(t - 7) + !tOO p(s)xn(s - a)ds; t>T Xn+l(t) = { roo yet) - YeT) + cXn(T - 7) + JT p(s)xn(S - 7) ds;
5.2.71
t
~
T.
Since y is a nonoscillatory solution of (5.2.70), by Theorem 5.2.10 yet) t -4 00 and hence
1
~
0 as
00
yet)
= cy(t -
~ cy(t -
7) +
1
poY(S - a)ds
j
t> T 5.2.72
00
7) +
p(s)y(s - a)ds ; t > T.
§5.2. Oscillation criteria
418
One can now show that {xn(t)} has a pointwise limit for t > T say x*(t) satisfying
x*(t) = cx*(t - r)
+
1=
p(s)x*(s - r)ds;
t >T
5.2.73
and x*(t) 2': ae- JLt for some positive numbers a and J1.. Since x* is a nonoscillatory solution of (5.2.66), the proof is complete. []
5.3. Neutral logistic equation In this section we consider the behavior of solutions of
_duet)
&
= ru(t)
[(1 _u(tK- r)) + ci& (1 _----,--u(tK-----:..r))]
5.3.1
in which c is a real number and r, r, K are positive numbers. It is shown in Pielou [1977], that a modification of the well known logistic equation
d~;t)
= rx(t)
[1 - x~) 1
leads to an equation of the form
dN(t) dt
[N(t)
- - = rN(t) 1-
+ cdN(t) ] dt K
5.3.2
where c, T, K are positive numbers; the modification itself is based on a model of Smith [19631 (for more details see Pielou [1977, p.38-40)). It is possible to consider (5.3.1) to be a generalisation of (5.3.2) incorporating a single discrete delay; it is also possible to generalise further with several discrete and continuously distributed delays. The following analysis of (5.3.1) is based on the results of Gopalsamy and Zhang [1988]. We first consider the asymptotic stability of the positive steady state K of (5.3.1). We assume that together with (5.3.1), initial conditions of the type
u(s)
= IAI on ~e(A) = O. Also IH2 (A)llRe).=o ::; IAI(rTl
5.3.30
1)/ ATI) . ~e('\)
+ rlcl)
1 -
riel.
5.3.36
Then evezy solution of (5.3.35) is oscillatozy. Proof. Suppose the result is not true. Then, the existence of a nonoscillatory solution of (5.3.35) implies that the characteristic equation 5.3.37 has at least one real root and this root cannot be nonnegative by virtue of (5.3.36). We let ,\ = -/-L in (5.3.37) and there exists a positive number /-L satisfying 5.3.38
It is easy to see from (5.3.38) that /-L
> /-Lrlel + rel'T
which implies
(1 -
re JlT
riel) > -
~ rrej
5.3.39
/-L
but (5.3.39) contradicts the second of (5.3.36) and hence the result follows.
[]
425
§5.9 Neutral logistic equation
Corollary 5.3.5. Assume that the conditions of Proposition 5.3.4 hold. Then all solutions of d 5.3.40 dt [vet) + rcv(t - T)] + rv(t - T) = 0 are oscillatory. Proof follows from that of Proposition 5.3.4 if we let
(J
=
T
in (5.3.37).
[]
Corollary 5.3.6. Let r, T, c be positive numbers. Then there exists a nonoscillatory solution of (5.3.40) which tends to zero as t ~ 00. Proof. The characteristic equation associated with (5.3.40) is h()") = A + Arce-).r We note
h(O)
= r > 0,
+ re-).r
1 c
=
o.
5.3.41
1 c
h( -- ) = -- < O.
It follows that (5.3.41) has a real negative root corresponding to which (5.3.40) has a nonoscillatory solution which tends to zero as t ~ 00. [] Proposition 5.3.7. Let r, T be positive numbers; c be a nonpositive number. Then every bounded nonosci1latory solution of
r))
~ (log{l + yet)} + cry(t limsup ly(t)1
0 log[l + y( t)] - lelry( t - T) < 0 log[l
eventually eventually.
}
5.3.43
§5. :3 Neutral logi.3tic equation
426
In case (i), log[l + yet)) - Iclry(t - r) is decreasing and bounded below; hence, there exists a 2:: 0 such that a = lim {log[l t-oo
+ yet)] - Iclry(t - r)}.
An integration of (5.3.42) on (T, (0) leads to
!roo y( s ) ds
0 for t ;:::: T and y has a bounded derivative. Thus, y is uniformly continuous on [T, (0). By Barbalat's lemma (Lemma 1.2.2) the result follows. In case (ii), we let z(t) = log[l + yet)] ~ rlcly(t - r) and note that since z is decreasing, by the boundedness of y, it will follow as before that yet) ~ 0 as
t
~ 00.
Suppose next y is eventually negative; note that y( t)
> -1
for t ;:::: 0 implies
y is bounded. As before we have two possibilities namely
(iii) (iv)
log[l + yet)] - Iclry(t - r) > 0 log[l + yet)] - Iclry(t - r) < 0
eventually } eventually.
5.3.45
In case (iii), log[l + yet)] - Iclry(t - r) > 0 and is increasing and hence tends to a finite limit as t ~ 00 due to the boundedness of y; the remaining details are similar to the cases (i) and (ii) treated above. [] It is an open problem to remove the boundedness hypothesis in the formulation of the previous result as well as in the next one.
Theorem 5.3.8. Assume that the hypotheses of Proposition 5.3.4 hold. Then every nontrivial solution of the neutral equation d dt[log{l
+ yet)} -Iclry(t -
a)]
= -ry(t -
r)
5.3.46
is oscillatory if both y and if remain bounded on (0,00). Proof. Assume the result is not true; then there exists a nonoscillatory solution y of (5.3.46), which we shall first suppose to be eventually positive. Such a solution satisfies limt_oo yet) = O. We have by integration of (5.3.46)
1
00
log[l
+ yet)] = Iclry(t - a) + r
yes - r)ds
5.3.47
427
§5.3 Neutral logistic equation and this implies that
yet) > Iclry(t - a) + r Define a sequence 0, Ni > 0 satisfying 5.4.3 Since solutions of (5.4.1) - (5.4.2) remain positive for t
~
0, we can let 5.4.4
and obtain for i
= 1, 2
§5.4. A neutral Lotka- Volterra system
431
It is known from Theorem 9.1 of Hale [1977, p.304), that the trivial solution of 5.4.5 is linearly asymptotically stable if the trivial solution of the linear system
5.4.6
obtained from (5.4.5), where Pij = f3ijNj , aij = aijNJ ' i,j = 1,2 is asymptotically stable. To derive sufficient conditions for the trivial solution of (5.4.6) to be asymptotically stable we proceed as follows; we first rewrite (5.4.6) in the form
~ [XI(t) + P11XI(t -
all
(111) +PI,X,(t - 0,
a22
> 0.
5.4.9
§5.4. A neutral Lotka- Volterra system
492
Calculating the rate of change of VI along the solutions of (5.4.7),
dVI dt
= 2 [ Xl(t)
-all l
+ PllXl(t -
0"11) + P12 X2(t - 0"l2)
t
Xl(S) ds - a121t
+ 2 [x2(t) + P21 X1(t - a211t
X2(S) dS] [- allx1(t) - a12 X2(t)]
t- T 12
t-Tll
.
0"2I) + P22 X2(t - 0"22)
x1(s)ds - a221t
t-T21
X2(S)dS] [- a21xl(t) - a22X2(t)].
t-T22
5.4.10 One estimates the right side of (5.4.10) and derives
dV1 dt
~ -[Xl(t) X2(t)] +
[all a12 + a21
a12 +a21 ] [Xl(t)] a22 x2Ct)
xil- an + an{lPn 1+ IPl21) + an{anTn + lal2 h2) + Ia21 1{Ip21 1+ Ip221) + Ia21 In all hI + a22T22)]
+ x~ [ - a22
+ a22(1 P21 1+ Ip22 I) + a22(1 a21 1721 + a22 7 22)
+ Ia12I(IPn 1+ IPlll) + Ia'21( all Tn + Iall h2)] +8 in which
8 = alllplllxi(t - 0"11) + alllp12Ix~(t - 0"12)
+ ail 1t
xi(s) ds
+ all Ia1211t t- T 12
t-Tll
+ 1a12IIPll Ixi(t +laI2Ia111t
x~Cs) ds
O"ll) + 1al21lpl2lx~Ct - 0"12)
Xi(s)ds+ai21t
t-Tll
x~(s)ds
t - T l2
+ a22lp21l x iCt - 0"2d + a22Ip22Ix~(t - 0"22)
+ a221 a21 11t
xiCs) ds
t-T21
+ Ia21 IIp21 Ixi(t +
a~11t
t- T 21
+ a~21t
x~Cs) ds
t-Tn
0"2t} + I a21 Ilp22Ix~(t - 0"22)
xrCs) ds
+ 1a21 la221t t- T 22
x~(s) ds.
5.4.11
§5.4. A neutral Lotka- Volterra .sy.stem
We choose another functional
V2
=
V2(Xl(,),X2(.»(t) such that
V2(XI(')' X2(.»(t)
= WI
where
WI =
and
W2 =
[au1pu 1+ 1a12l1pu
+ W2
5.4.12
I] 1~.H x~(s) ds
+ [anIP" 1+ 1a" lip" I] 1~."
x;(s) ds
+ [au1plZ 1+ 1a12 IIp12 I] 1~."
x~(s) ds
+ [ani Pnl + Iaullpnl] 1~."
x~(s) ds
(a~1 + au Ia121) LTH ([ x;(u) dU) +
433
ds
(a~1 + a221 a21 I) LT" ([ X~(U)dU) ds
LTU ([ x~(u)dU) ds + (I Ian + a~2) LT" ([ x~(uldU) ds. + (aul al21 + a12) a21
The functional V = V(Xl, X2)(t), defined by
V( Xl, X2)(t)
=
Vi (Xl, X2)(t) + V2(Xll Xz)(t)
5.4.13
;. has the following properties: if {Xl (t), X2(t)} denotes any solution of (5.4.7), then
Vex}, xz)(t)
~
0 for
t
~
0
dV(xt,xz)(t):::; -fxI(t) X2(t)] [ all dt a12 + a2l + J-llxi(t) + Jl2X~(t)
5.4.14
in which 1-'1 = [ -
au + all(lpu 1+ IPnl) + au( a","" + Ial2 !rI2)
+ la21lClp211 + IPzz /) + laZII(l a21l Tzl +aZz T 2z) + IPll I(all + Ial2!) + IpZl l(aZ2 + IaZl I) + au '"u (a" + 1al21) + 1a21 !rz,(1 a21 1+ a22l]
5.4.15
§5.4. A neutral Lotka- Volterra system
and j.l2
= [-
a22
+ a22(1 P21 I + IP221) + a22(1 a21 IT21 + an T22)
+ Ia121(lpll 1+ Ip12 I) + Ia121(allTl1 + Ia12I T12) + Ip121(all + Ial21) + Ip221(a22 + Ia21 I)
5.4.16
+ I aI2I T12(all "+ Ial2!) + a22 T22(1 a21 1+ a22)]' The foregoing preparations enable us to derive the next result: Theorem 5.4.1. Suppose the following are satisfied: (i) The quadratic form 2
L
aijXiXj
= (Xl
i,j=l
(ii)
is nonnegative for {Xl,X2} E IR X IR. aij, Pij , Tij (i,j = 1,2) satisfy all
where
j.ll
> 0,
a22 > 0,
. j.ll
< 0,
j.l2 < 0
5.4.17
and j.l2 are defined by (5.4.15) and (5.4.16) .
(iii) Iplli II ( Ip211
IP 12 I) Ip22 I
+(
allTll la211T21
5.4.18
Then the trivial solution of (5.4.1) is locally asymptotically stable in the sense that all solutions of (5.4.6) satisfy lim
t-+CXl
[xi(t) +x~(t)]
5.4.19
= 0.
Proof. We consider the Lyapunov functional V where
defined by (5.4.8) and (5.4.12) and note that V(Xl, X2)(t) tion, we have
~
O. From our prepara-
5.4.20
435
§5.4. A neutral Lotka- Volterra system
Since the quadratic form in (5.4.20) is nonnegative, (5.4.20) leads to 5.4.21 in which 5.4.22 As a consequence of (5.4.21) and the definition of V, 2
2
I Xi(t) I s ~ Ipij II Xj(t -
O";j)
t
.
I + [;, Iaij 11_r;; I Xj(S) Ids + (Vo)~ .
5.4.23
We let
mi(t) =
sup
IXi(S) I
5.4.24
sE[-Il,t]
and note that (5.4.23) implies
m1(t)] [ m2(t) where Q _ -
Since
~ Q [m1(t)] + (Vo)~ m2(t)
[1] 1
[IPll I IP121] + [ aUi'll Ip21 I Ip22 I la211i'21
5.4.25
5.4.26
IIQII < 1, we derive from (5.4.26), 5.4.27
The boundedness of XI(t), X2(t) for t 2:: 0 follows from (5.4.27) and (5.4.24). The boundedness of Xl and X2 on [0,00) is verified similarly. For instance, we let
Zi(t)=
sup
IXi(s)1
5.4.28
sE[-Il,t]
and, note from (5.4.6) that
Since IIQII < 1, we have IIPI! = II [pij] 3;2 remain bounded on [-tt, 00).
I! < 1. It follows from (5.4.29)
that Xl and
§5.4. A neutral Lotka- Volterra system
436
It is now easy to see from (5.4.21) that xHt)+xHt) E L 1 (0, 00); the boundedness of the derivatives of Xl and X2 on (0,00) will guarantee the uniform continuity of Xl and X2 on (0,00). By Barbalat's lemma (see Lemma 1.2.2), it follows that
xi(t) + x~(t)
-+
0
as
t
-+
00
and this completes·· the proof.
[]
A number of other plausible neutral differential models of population systems are formulated in the exercises. For a discussion of an n-dimensional neutral Lotka - Volterra system we refer to Gopalsamy [1992] (see also Kuang [1991]). Several results related to stability switching and absolute stability of linear neutral differential systems can be found in Datko (1978] and Freedman and Kuang [1991J.
5.5. X(t)
= AX(t) + BX(t -
7') + CX(t - I)
The method of Lyapunov functionals and Lyapunov functions has proved useful in the stability analysis of both ordinary and delay differential equations; however, application of Lyapunov technique to neutral differential equations has not yet reached any level of completeness. Results related to Lyapunov functionals for neutral differential equations of the type
for 0 < I(t) ~ 7'0, i,j = 1,2,"" n can be found in Misnik [1972) and EI'sgol'ts and Norkin [1973). Quadratic Lyapunov functions have been exploited by Khusainov and Yun'kova [1988], Li Senlin [1983, 1987] and Wu [1986) in a comparative study of the stability characteristics of the linear equation
x(t) = Ax(t) + Bx(t - I) + Cx(t - 7')
5.5.1
and the perturbed quasilinear system
x(t) = Ax(t) + Bx(t - 7') + Cx(t - I) + Q(x(t), x(t - 7'), x(t - 7')). If the system (5.5.1) is stable when 7' = 0, can we find an estimate of the delay I for the stability of (5.5.1) to continue to hold when 7' =f O. We consider this aspect briefly and refer to Khusainov and Yun'kova [1988] for other results related to nonlinear perturbations of stable systems.
§5.5. XCt) = AX(t) + BX(t - T)
+ CX(t - T)
437
It is known from the Lyapunov stability of ordinary differential equations, that if the trivial solution of
x(t)
= (1 -
C)-l [A
+ B]x(t)
5.5.2
is asymptotically stable, then for every positive definite matrix P there exists a positive definite matrix H satisfying
One can, for convenience, choose P to be the identity matrix 1 in (5.5.3) and we suppose we have done so. We denote the smallest and largest eigenvalues of H respectively by Amin(H) and Amax(H). We consider a Lyapunov function v defined by 5.5.4 where H denotes a solution of (5.5.3) with P and avO' defined by vO' = {xlv(x)
< a},
= I.
avO' = {xlv(x)
We also consider the sets vO'
= a},
a E (0,00).
The following preliminary lemmas are needed in the proof of Theorem 5.5.4 below. Lemma 5.5.1. If x(t) is a solution of (5.5.1) satisfying Ilx(t)" < 8 for to t :::; to, tben for to .< t :::; to + T, X satisfies
IIx(t)1l
to + T for which x(t*) E 8v a . We have from the above Lemmas 5.5.2 and 5.5.3 that
v(x(t*)) < {[ -
~ + 2(IIH(I VCfH)
+ IIH(I -
T
+ IIBII l-IICII
5.5.11
IIAII + IIBII T ] va 1 -II C II V>'min(H)
C)-l BIl)
+ 811CII [IIH(I - C)-ICIi If
C)-lCII" AIi
(1 + ";I~~~~") + IIH(I - C)BII] 8}X(t*).
< To, then one can choose 8 in (5.5.11) small enough to satisfy v(x(t*)) < o.
Therefore, the vector x(t*) will be directed towards the interior of va. Hence, x(t) does not leave va for t > t* whenever IIx(t)11 < 8 for to - T ::; t ::; to and this
0
c~pk~t~pro~
In the next two results, we derive sufficient conditions for the linear autonomous neutral systems of the type (5.5.1) to be stable independent of the size of delay. Theorem 5.5.5. Consider the linear equation
x(t) = Ax(t) + Bx(t - T)
+ Cx(t - a),
T,a
>0
5.5.12
in which A, B, Care n x n constant matrices. Suppose there exists a differentiable function V : IRn 1-+ III satisfying the following: 5.5.13
1-11 C II> 0,
r_IIA II + II B II \. - 1-11 C II
1\7 V(x)1 = I ( 8V)T 8x I ::; )..Ixl; aV)T ( ax Ax ::;;
_txT x ,
5.5.14
5.5.15
5.5.16
442
§5.5. X(t)
= AX(t) + BX(t -
7) + CX(t -7)
If
5.5.17 then the trivial solution of (5.5.12) is exponentially asymptotically stable for all delays 7 and (J.
Proof. We have from (5.5.12),
+ Bx(t - 7) + Cx(t - 0")1 IIlx(t)l+ II B IIlx(t - 7)1+ II C
Ix(t)1 = IAx(t)
::;11
A
5.5.18 IIlx(t - 7)1·
We let met)
= sup
net) = sup Ix( s)1 s9
Ix(s)l,
s~t
and note that net) ::; Km(t).
Now d dt V ( x)
= (aV)T ax x (t ) aV)T Ax(t) + (aV)T ax Bx(t - 7)
= ( ax
:::: -exT x +
-.e
::; TV(x)
I~: I[II
+ A II
+ A II C II (
-.e ::; TV(x)
B
II
Ix(t - r) 1+
II GIl
Ix(t - r) I]
0")
5.5.19
II sup Ix(s)12 s~t
sup sE[t-(U+T),t]
+ A II
+ A II
B
+ (aV)T ax Cx(t -
C
B
II
II -1 a
(
sup
5.5.20
Vexes»~ )
sup sup
V (x ( S ) ))
5.5.21
sE[t-(U+T),t]
-.e = TV(x(t) + ;;A ( II B II + II C II K ) V(x(t) where
iT =
IX(S)I)
sE[t-(U+T),t]
sE[t-(U+T),t]
K ( a
IX(S)) (
sup sEt t-(U+T ),t]
Vexes)).
5.5.22
§5.5. X(t) = AX( t)
+ BX(t -
r)
+ GX( t -
r)
443
The result follows from (5.5.22) and (5.5.17) by virtue of Halanay's lemma (see Lemma 3.6.12 in Chapter 3) and this completes the proof. [] It is posssible to conclude that when (5.5.13) - (5.5.17) hold, all the roots of det[A + Be- zr
+ Gze-zO'] =
0
have negative real parts. The reader should try to provide an independent proof of this fact. The next result provides an alternative (and easily verifiable) set of sufficient conditions for the trivial solution of (5.5.12) to be asymptotically stable. Theorem 5.5.6. (Li-Ming Li [1988]) Suppose that the coefficient matrices A, B, G of (5.5.12) satisfy
IIGII < 1,
and
j),
(A)
+ IIBII + IIA"IIIIGII < 0 1-IIGII .
5.5.23
Then the trivial solution of (5.5.12) is asymptotically stable and there exist M ~ 1,0:: > 0 such that
for every solution x(t, -(a+r).
i=1,2;
e-atj
5.5.30
)=1
It is easy. to see that
= 1,2,
i
t E
[-(a
+ r),O).
We want to prove
Pi(t) < wi(i),
i
= 1,2;
t E [0,00).
5.5.31
If (5.5.31) does not hold, then one of the following would occur; there exists a t1 > such that
°
and Pi(t)~Wi(t),
P1(t1) = Wl(t1), P1(t1)2:: Wl(tl)
t~tl,
i=1,2.
We also have
+ II B ll wl(t 1 ) + IICll w2(t1) J.l(A)W1(tt} + IIBllwl(t1 - (r + 0"» + IICllw2(t1 - (r + a»
Pl(td ~ J.l(A)Pl(td =
2
Wl(tJ)
= -kc~la[?=pj(o) + €]e-
at1
)=1
> k (1'( A)al + liB lI a l ea(r+·) + IIClla. e'>(r+')] [
= J.l(A)W1(t 1) + IIBIIWl(tl
- (r
t,
+ 0"» + IICIlW2(tI -
(Pi (0) + f)e -at, ]
(r + a»
=Pl(t 1 ) and this contradicts PI (tt) 2:: WI (tt). The other possibility is that there exists ail>
P2(t 1 )
= W2(tI)
and
Pi(t)
~
Wi(t),
°
such that
i = 1,2;
t
~ tl
.
It is found from (5.5.26) that
P2(tI) ~ I/AI/ Wl(td + I/ B llw1(tl - (r
(t,P;(O) + f) < k(tft;(O) + f)e-
+ 0"» + /lC/lW2(t1 - (r + a»
(e- at , (IiAllal
:s; k
J=1
= W2(tl)'
at1
(0"2)
+ IIBlial ea(r+.) + IIC lh ea (r+Q)) ]
446
§5.5. X(t)
= AX(t) + BX(t -
r)
+ CX(t - r)
Thus, P2(t 1 ) < W2(t 1 ) and this is a contradiction; and hence (5.5.31) follows. We also note from (5.5.30) and (5.5.31) that
t> -(r + 0"). This completes the proof.
[]
The following are examples of population model systems subject to feedback (indirect) controls. It is of interest from the viewpoint of modelling population systems to discuss the existence of positive steady states and their stability characteristics. Assume that all the parameters appearing in the following are positive and the kernels of integrals are nonnegative and normalised. We ask the reader to examine the local asymptotic stability of the positive steady states of the following equations and also examine whether delay independent stability is possible. (At this time there exists no technique for the investigation of the persistence of population systems modelled by neutral differential equations; the interested reader can try to develop methods for the investigation of persistence of the following systems).
N(t) = rN(t) u(t)
N(t) = rN(t) ti(t) N(t)
[1 - N(t; r) - o:u(t)] }
= -au(t)[l + u(t -
= -au(t) + bN(t -
r).
= rN(t) [1 _N(t -
r)
+ cN~t - 0") - au(t)] } 1 + N2 (t - 0")
= -au(t) + b1 N(t) + b2 N(t -
n
Ui(t)
+ bN(t).
[1- (N(t - r) ~cN(t - r)) - o:u(t)] } K
ti(t)
r)]
= -f3iiUi(t) + L
f3ijUj(t) j#i i = 1,2, ... , n.
r).
.
n
+ L 'YijN(t ;=1
rij)
= AX( t) + BX( t -
§5.5. X(t)
1.: + 1
N(t)=rN(t)[l-
K,(s)N(t-s)ds-
ti(t) = -au(t)
K3(S)N(t - s)ds.
b
T) + CX( t - T)
447
1.~ K'(S)N(t-S)ds-au(t)])
N(t)=rN(t)[I- (XJ K1 (s)N(t-s)ds- [ooK2 (S)(
J0
J0
N~t-~)
1 + N2 (t - s)
)dS
- a!t(t)]
1
00
u(t)
= -au(t) + b
iti(t) = -aiHi(t) +
t [" j#i
1
00
K 3(s)u(t - s) ds
+
Kg>Cs)Uj(t - s)ds +
K4(S)N(t - s) ds.
t 1.~ K~>Cs)Nj(t j=l
0
0
- s)ds
.
i=1,2, ... ,n.
~:C:~ ~:),)
-{
N,(t)
+ c,N,(t -
Tn) }] )
. [K2 + Q2Nl(t - T12) { N2(t) = N2(t) 1 + N (t _ T12) - N2(t) 1
+ c2 N. 2(t -
r22)
N,(t) = N,(t) [ K; :
1
N(t) = rN(t)(l- [N(t)]8 [N(t - T»)8
2
+
K 8 1+82
it(t)
[N(t - 0)]9
3 _
}]
.
cu(t») }
= -au(t) + bN(t).
x(t) = x(t)[r1 - al1x([tJ) - a12y([t])] yet) = y(t)[r2 - a21 x ([t]) - a22y([t])]
+ px(t)[y(t + py(t)[x(t -
r) - x(t - T)J } T) - yet - T)J.
5.6. Large scale systems
Asymptotic behavior of large scale dynamical systems described by ordinary differential equations have been considered by several authors (Bailey [1966], Michel and Miller [1977], Siljak [1978), Anderson [1979], Amemiya [1981]). Recently large scale neutral systems have been considered by Liao Xiaoxin [1986] and Zhang Yi [1988a, bJ. Besides discussing the large scale dynamics, our purpose
448
§5.6. Large scale systems
here is to introduce the reader to a stability investigation in the metric of space e(l) (for details see El'sgol'ts and Norkin [1973]). In particular, we explore the following aspect: if a nonlinear system has a dominant linear part with certain stability characteristics, then what type of nonlinear perturbations can maintain the stability of the full system. Let us consider a large scale (or composite) system described by
x(t) = F(t, x(t), x(t - T(t)), x(t - T(t)))
5.6.1
whose constituent subsystems are governed by
Xi(t) = Ai(t)Xi(t) + fi(t, x(t - r(t)), x(t - r(t)));
= 1,2, .. , Tj t 2:: to
i
5.6.2
in which the delay T is a continuous nonnegative function, 0 ::; T( t) ::; To, Ai(t), (i = 1,2, .. ,r) is an r x r real continuous matrix,
xi =
I(
(i») ,
i)
CO • X l ' ' ' ' X n'
J
r
L
F(t, 0, 0, 0)
nj = n,
= O.
j=l
The initial conditions associated with (5.6.2) are
twhere i' ¢i are continuous on [to -
IIcpll =
m?-x [ l:::;t:::;r
TO,
TO ~
t ::; to , i = 1,2, .. , r
to]. We define
sup
t- r o99o
(1Ii(t)1I
+ lI¢i(t)ll) ].
The exponential asymptotic stability of the trivial solution of (5.6.2) in defined as follows:
"If there exists a ). > 0 and if for any Q > 0 there exists a K such that,
Ilcpli ::; ===> II Xi(t) II + II Xi(t) II ::; K( Q
Q
)I!CPlle->.(t-t o),
i
e(l)
is
= K (Q) > 0
= 1,2, .. , r, t 2:: to,
then the trivial solution of (5.6.2) is said to be exponentially asymptotically stable in the metric of e(1)".
449
§5.6. Large scale 8ystems
We assume throughout the following, that the fundamental matrix Y associated with 5.6.3 i = 1,2, .. ,; defined by 8Yi( s, t) _ A :(t)Y;( t)
at
-
l
J
S,
i = 1,2, .. ,r
yes,s) = Ei (unit matrix) satisfies
6> 0,
t 2:: s
i
= 1, 2, .. , ;.
5.6.4
The property (5.6.4) will be called (0'1); if instead of (5.6.4), one has
t 2:: s;
i = 1,2, .. , r
5.6.5
where 6i is a nonnegative continuous function, then (5.6.5) will be referred to as (0'2). In the following we denote the spectral radius of a matrix n by pen). Theorem 5.6.1. (Zhang Yi [1988aJ) Assume that the subsystems governed by (5.6.2) satisfy the following; (i) the property (at) holds; i.e. (5.6.4) is satisfied;
II ti(t, x(t -
ret)), Xi(t - ret))) II S;
t,
( ii)
ret»~ II
+ Cijll Xi(t IIAi(t)!I~ai'
(iii)
(biill xj(t -
pen)
O;
§5.6. Large scale sy.'Jtems
450
Then the trivial solution of (5.6.2) is exponentially asymptotically stable in the metric of C(l) and the stability is not conditional on the size of TO.
Proof. It is found from (iii) that n - E (where E = (eij) is the 2r x 2r identity matrix) is a stable Metzler matrix (see for instance Siljak [1973]) or (E - n) is an M -matrix (for details see Chapter 3). It will follow from the properties of M -matrices that, there exist constants ai > 0, i = 1,2, .. , 2r such that 2r
L:: aj(wij -
eij)
< 0,
i
= 1,2, .. ,2rj
5.6.6
j=l
that is
1 2r -~a'w" 0,
5.6.12
451
§5.6. Large scale systems
By the variation of constants formula we have from (5.6.2),
xiit) = 1';( to, t)4>i( to) +
1:
1';( S, t)/i [s, x( s - r( s)), x( s - r( s»] ds.
5.6.13
From conditions (i) and (ii),
II Xi(t) II 5 1I lIe-"('-'o) +
t. 1.:
e-"('-') [ biill Xj(s -
res»~ II 5.6.14
+ cijll Xj(s -
res»~ 1/ ] ds
and
/I Xi(t) lIeA(t-t o )
~
II q.1/
+ e Aro
t l' bij
j=l
I
e -(6, -A)(,-.) x j( s - r( s»
II e A( .-r(.) -'0) ds
to
t + e ATO 2:: Cij Jt. r
j=l
e-Ui";-A)(t-S)
II Xj(S -
res»~ lI eA (s-d s )-t o) ds.
to
5.6.15
Directly from (5.6.2), we derive that
II Xi(t) lIeA('-'o) 5 eA('-'o) [ aill xii t) II +
+ cijll Xi(t -
5 1I II + e.\to
t.
t.
(b ij II x j( t - r(t»
II
r(t))!]) ]
5.6.16
hij '-;01.9
(II x j( s) lIeA('-'o»)
+eATotcij sup (lIxj(s)lIeA(S-tO»). j=l t-To::;s::;t Define
Si(t) =
sup to-ro<s II
+ eAro
j=l
Sj(t) ::; II 0 such that
then the trivial solution of (5.6.28) is said to be asymptotically stable. Theorem 5.6.2. (Zhang Yi [1988a)) Suppose tbe following conditions bold:
(i) Tbe fundamental solutions Yi( s, t) of tbe isolated subsystems i
= 1,2, .. , r
i
= 1,2, .. , r
,t ~
to
satisfy
\I Yi( s, t) II ~
e-
f
o;(u) du ,
t >s
for
and
t
as
II fi(t,X(t -
r(t)),i(t - ret)) II S
i = 1,2, .. ,r.
-* 00,
t
[bij(t)1I Xj(t - r(t)) II
+ Cij(t) II ij(t -
(ii) sup bij(t) = bij
,
i,j = 1,2, .. , r,
exp ( -
0i(U)dU)C;j(S)ds
s wl]>,
i,j = 1,2, .. , r,
p(f!) < 1 wbere
f! 1 -
(W(l») ij rxr'
n2 =
(w~:») I) rXr
- .. _ { bii
bl )
C
= (c··) rXr . 1)
-
b
+ ai,
ij,
. -I-
ZT
Z=J .
J
§5.6. Large scale systems
455
Then the trivial solution of (5.6.28) is asymptotically stable in the metric of e(l).
Proof. A consequence of condition (iv) is that there exist numbers 1,2, .. , 2r such that 1
I 2r } max { - ""' a ·w . . = h 0,
< 1.
i =
5.6.29
j=1
By the variation of constants formula, one derives from (5.6.28),
II Xi(t) II
~ 114> II +
t, l:'xp (-[6 u) i(
dU) [ bi;(s)1I x ;(s - r(s) II 5.6.30
+ cij(s)1I :tj(s - res) II] ds; directly from (5.6.28),
II Xi(t) II
~ 114> II + adl Xi(t) II +
t,
[bi;1I Xj(t - r(t)) II 5.6.31
+ Cijll :ti(t -
ret)) ,,],
Define
Si(t)
= {SUP -
<S9 sUP-oo<s9 OO
(II ~i(S) II), (II Xi-r( s) II),
l::;i::;r r + 1 ::; i ::; 2r.
. - 00
< t < 00 5.6.32
It follows from: (5.6.30) - (5.6.32) that
II Xi(t) II
~ 114> II +
t. U>xp {-
[Oi(U) du }bi;(s) ds )S;(t)
+f (1' exp {- J.' }=r+l
to
Oi(U)du}ci,j_r(S)ds)Sj(t)
s
2r
::; II ~ II + LwijSj{t); i = 1,2, .. , rand
5.6.33
j=l
r
II :ti(t) II ::; II ~ II + L j=1
2r
bijSj(t) +
L
Ci,j-rSj(t)
j=r+l
2r
::; " ~" + L Wi+r,jSj(t) , i j=l
= 1,2, .. , r.
5.6.34
§5.6. Large scale systems
456 From (5.6.33) and (5.6.34),
2r
II
0; odd positive integer and derive that
+ py(t (j
> 0;
0") T
~
=0 O. Assume that n is an
is a sufficient condition for all solutions of the above equation to be oscillatory. Can you derive a sufficient condition for the existence of a nonoscillatory solution? 25. Establish necessary and sufficient conditions for all solutions of
to have zeros on (-00,00) where c, a are positive constants and the kernels K11 K2 are nonnegative and integrable on [0,00). 26. Obtain a set of sufficient conditions for the neutral system of equations
i=1,2,···,n to be stable independent of the delays where stants.
Cij, aij, Tj,
0"j are all real con-
27. Derive a set of sufficient conditions for all solutions of the neutral integrodifferential system (i = 1, 2, ... , n)
Exercises V
471
to have the property of "equilibrium level crossing". State your own assumptions on the kernels H 1, H 2. 28. Can you derive sufficient conditions for all solutions of
d dt [x(t) - cx(t - r)]
+ ax3(t -
a)
=
°
to be oscillatory? Assume C E (0,1), a E [0,00), r E [0,00), obtain sufficient conditions for all solutions to satisfy lim x(t)
t-oo
29. Assume that aii, bij(i =f j), cij(i,j O(i = 1,2,···,n). Prove that if n
= 0.
= 1,2,···,n)
n
+ Llbijl + L ICijl
1.
Discuss stability independent of delay in the neutral system d
-d [x(t) - Bx(t - h)] = Aox(t) + AIX(t - h), t .'
B
0
where
1]
= [ _~ ~ .
32. Discuss the possibility of stability switching in the two species competition model; consider the cases kl > 0 and kl = 0 : see (Kuang [1991]);
x(t) = rlx(t)[l - klX(t) - ax(t - ..Td - (3x(t - To) - cly(t - 12)] yet) = r2y(t)[1 - C2x(t - T3) - k2y(t - 14)].
Exercius V
473
33. Derive sufficient conditions for the stability of the equilibria in the following logarithmic population systems:
dx(t) = ()(1- IOg[x(t. - T)J :... . ~ log[x(t - T)]) dt rx t K dt K .
i = 1,2"", n.
dx(t) = () [ _ (log[X(At») _ ~ (log[X(At»))] dt x t 1 K dt K '
d~~t) =x(t)[a-b].= K,(s)[logx(t-:-s)]ds+
O O. J. Math. Anal. Appl. 55, 794-806. Li-Ming Li [1988]. Stability of linear neutral delay-differential systems. Austral. Math. Soc. 38, 339-344.
Bull.
Li Senlin [1983]. The stability of superneutral functional differential equations. Scientia Sinica, Series A, 26, 1-10. Li Senlin [1987]. A V-function method for stability offunctional differential equations and its applications. Scientia Sinica 30, 449-461. MacCamy, R.C. and 'Nong, J.S.W. [1972]. Exponential stability for a nonlinear functional differential equation. J. Math. Anal. Appl. 39, 699-705. MacDonald, N. (1978). Time lags in biological models. Lecture Notes in Biomath. 27, Springer-Verlag, New York. Maeda, H. S., Kodama, S. and Ohata, Y. [1978]. Asymptotic behaviour of nonlinear compartmental systems; nonoscillation and stability. IEEE Trans. Circuits Syst. CAS 25, 372-378. Marcus, C.M., Waugh, F.R. and Westervelt, R.M. [1991]. Nonlinear dynamics and stability of analog neural networks. Physica D 51, 234-247. Marsden, J.E. and McCracken, M. [1976]. The Hopf bifurcation and its applications. Springer-Verlag, Berlin.
Reference3
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===
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Index
~braharnson D.L.
264
absolute stability 193, 194, 217 Aftabizadeh A.R. 79 Akhmerov R.R. 393 Alexander J.C. 148 Amemiya T. 447 . almost periodic 37 an der Heiden 208, 307 Anderson B.D.O. 447 Anvarinov R. 386 Araki M. 228,230,337,363 Arino O. 37,256,468 Arzela-Ascoli 45,78 Ashkenazi M. 148 Atkins G.L. 355 Ayala F.J. 195 Bailey H.R. 447 Bainov D.D. 90 Banks H.T. 255, 298, 318, 375 BarbaJat 1. 4, 5, 30, 31, 264, 325, 396, 426,436 Barbashin E.A. 90, 215, 339 Barbu, V. 27 Barker G.P. 339 Bellman R. i, 9, 126, 188, 206, 207, 211, 310 Berman A. 295 Borisenko S.D. 90 Borsellino A. 124 Boucher D.H. 191 Braddock R.D. 109
Brayton R.K. 466 Brelot M. 200 Brian M. V. 182, 298 Bromwich T.J. 36 Brouwer fixed point theorem 360 Brumley W.E. 393 Burton T.A. 14, 31, 217, 271 Busenberg S.N. '263,373,374 eai Sui Lin 239 Carvalho L.A. V. 264 Castelan W.B. 264 Chandra J. 148 Chang Hsueh Ming 239 chaotic behavior 79, 307, 311 characteristic exponent 139, 142 Chew K.H. 222 Chin Yuan Shun 239 Chow S.N. 131 Coddington E. 127 coexistence 347 Cohen D.S. 124, 148 comparison 48,54,222,225 compartments 355,361,363,366,368, 460 competition 168, 182, 195 competitive exclusion 306 contraction 429, 430 Cooke K.L. 14,79,239 cooperation 168, 172, 182, 191, 194, 318, 326, 340 Coppel W.A. 192, 321
498
Index
Gantmacher F.R. 219, 230 Corduneanu C. 26,91 Gard T.C. 383,384 coupled oscillators 148 Gersbgorin's theorem 259,308,364 Crandall M. 138 Cushing J.M. ii, 124, 125, _131, Goel N.S. ii, 210 global attractivity 87, t10, 292, 367, 173,200,201,327 375 Datko R. 264, 393, 436, 472 global stability 55 Dean A.M. 191 Gopalsamy K. 79, 90,95, 100, 107, delay independent 60, 180, 217, 472 148, 149, 186, 196, 208, 222, 253, delay logistic 2, 55, 71, 87, 95, 116, 277,296,298,306,399405,408,418 123, 162, 173, 201, 314 Gosiewski A. 212 density dependent 1, 172, 182, 183 Gromova P.S. 217,393 difference equation 87, 88 Gurgula S.I. 90 differential inequality 32, 41, 43, 73, Gyori 1. 79, 81, 361, 366, 460 227,229,300 Driver R.D. i, 12, 18, 103 Haddock J.R. 14 Edelstein - Keshet L. ii Eisenfeld J. 388 El'sgol'ts L.E. i, 145, 448 equations with impulses 90 exploitation 196, 298 Fargue D.M. 2,336 feedback control 95, 121, 446 Field R.J. 148 Fisher M.E. 87,88 Floquet exponents 138, 156, 157 Floquet technique 138 food limi ted 107 Fox L. 36, 236 Franklin J. 45 Franklin J.N. 259,308 Fredbolm alternative 133, 141 Freedman H.I. ii, 253, 436 Fukagai N. 68,403
Halanay A. 64, 126, 133,·227 Hale J.K. i, 37, 126, 131, ,179, 187, 255, 419, 422, 466 baematopoiesis 107 Harrison G. W. 338 Hassard M. W. 127,147 Henry D. 37, 393 Hirscb M. W. 307,318,326 Hofbauer J. 347, 351, 372 Hopf E. 126, 128, 130, 131 Hopf-bifurcation 124, 125, 126, 130, 151 Howard L.N. 148 Hsu S.B. 283, 298, 384 Huang Z.X. 219 Hunt B.R. 14 Hutchinson G.E. 1, 173, 196, 201 bypercooperation 318, 324, 326 hyperlogistic 60
Index
Implicit function theorem 132, 136, 142, 157 impulsive 116, 117, 121, 356 Infrulte E.F. 264 infinite product 36 in-phase 151, 154, 155, 159 integral representation 11 interference 182, 195, 196, 298, 299 interspecific competition 183, 189, 298 intraspecific 183, 186, 298, 299 invariance principle 89
Jacquez J.A. 355 Jansen W. 372 Jiong R. 352, Jones C.S. 1
Kakutani S. 1 Kato J. 34 Kawata M. 148 Kaykobad M. 294 Khusainov D. Ya. 214, 436 Kirlinger G. 351, 353, 372 ,j(olmanovskii V.B. i,394
499
Ladde C.S. 54, 95 Landman K.A. 124, 125, 151 large scale systems 447, 448, 453 LaSalle J.P. 88, 339 Lebesgue convergence theorem 46, 413 Lefever R. 148 Lenhart S.M. 62 Levin J.J. 6 Lewis R.M. 355, 361 Liao Xiaoxin 447 Lim E.B. 235 Li Ming: Li 291, 443 limit cycle 127, 347 linear analysis 172 linear stability criteria 3 linear oscillators 37 Li Senlin 436
MacCamy R. C. 112 MacDonald N. 2, 336 Maeda H.S. 355 Marcus C.M. 369 Marsden J.E. 127, 128, 142 Koplatadze R.C. 73 Martin Jr. R.H. 318, 319, 321 Kozakiewicz E. 48 Martynyuk A.A. 387 Krasnoselskii M.A. 318 Matano H. 326 Krasovskii N.N. i, 126 matrix measure 257, 356, 443 Krikorian N. 339,385 May R.M. ii, 79,307 Krisztin T. 361, 367 Maynard Smith J. ii Kuang Y. 430,436,472 Mazanov A. 355 Kulenovic M.R.S. 67, 399, 400 mean diagonal dominance 330 Kuramoto Y. 148 Michel A.N. 447 ~adas G. 18,20,38,68,401,403,407 Mikhailova M.P. 321 Miller R.S. 298 469
500
Misnik A.F. 436 M-matrix 227-230, 232-235, 317, 337 Mori T. 222 Morita Y. 148 Mulholland R.J. 355 Murdoch VV.VV. 196 Murray J.D. 11 mutualism 172, 191, 318 Mysbkis A.D. 54
Index
positive definite 61, 219, 333, 336 positive feedback 179, 186 positivity condition 26 Post W.M. 336, 337 Qin Yuan Xun
239, 285
Razumikhin B.S. 218 respiratory model 107 Ricklefs R.E. 2 robust stability 460 414 Nashed M. negative feedback 9, 60, 174, 180, 186 Rose M.R. ii Rouche's Theorem 11,365,423 neural networks 369, 473 Routh-Hurwitz 150, 204, 248, 393 neutral equations 393 Royden H.L. 46 neutral logistic 418 Rozhkov V.I. 212, 237 neutral Lotka Volterra system 430, 467 Sandberg I. W. 321,355 No on berg V. W. 348 Sattinger D.H. 136, 138, 154 N unney 1. 203 Schauder-Tychonoff 45, 76, 78, 414 Obolenskii A. Yu. 374 Ohta Y. 318, 321 Oliveira-Pinto ii Pandit S.G. 90 Pavlidis T. 148 Perestyuk O.S. 90 Perron-Frobenius 230, 295 persistence 347, 348, 351, 352, 353, 460 Peschel M. 318 Philos Ch.G. 469 piecewise constant 78 Pielou E.C. 418 Pirabakaran R. 122 Plemmons R.V. 61,337
Schoener T. W. 196 Schuster P. 383 Scudo F.M. ii Seifert G. 32 Selgrade J.F. 318 Shibata A. 307, 311 Siljak D.D. 447,450 simple stability criteria 263 Simpson H.C. 125 Sinha A.S. C.· 32 Slobodkin L.B. ii Smale S. 148, 307 Smith H.L. 318, 321 Smith F.E. 418 Snow W. 393 spectral radius 230, 295, 449
Index
stability switches 193, 208, 239 Staffans O. 112 Stech H. W. 131 Stokes A. 137, 138, 139 strongly positive 26, 27, 29, 62 synchronous 154, 155 Tokumaru H. 225,227,229,317 Torre V. 166 transport delays 365 Tsalyuk V.Z. 212 Turner Jr.M.E. 318 Unbounded delay
30,34, 453
Vandermeer, J.H. 191 variation of constants 19, 451, 455 Vescicik M. 75 Vidyasagar M. 257 Volterra V. 124 Waltman P. ii Wang Lian 239 Wangersky P.J.
201
"Vendi W.
501
352
Wheldon T.E. 279 Winfree A. T. 148 Winsor c.P. 196 Winston E. 51 Wolin C.L. 191 Worz Busekros A. 148, 327, 336, 378 Wright E.M. 1 "Vu J. 436
XU D.Y. 274 Van J. 45 Yamada Y. 62 Yodzis P. ii Yoneyama T. 14,20 Yorke J.A. 14
Zhang B.G. 19,76,407 Zhang Yi. 447, 449, 454, 457 Zbivotovskii L.A. 284 Zverkin A.M. 393, 394
