Difference Equations And Discrete Dynamical Systems: Proceedings of the 9th International Conference University of Southern California, Los Angeles, California, USA, 2-7 August 2004

Difference Equations and Discrete Dynamical Systems Editors

Linda S a b e r

J

S Allen E l a y d i

•

Bernd

•

Robert

Aulbach Sacker

Difference Equations and Discrete Dynamical Systems

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Difference Equations and Discrete Dynamical Systems Proceedings of the 9th International Conference University of Southern California, Los Angeles, California, USA

2 - 7 August 2004

Editors

Linda J S Allen Texas Tech University, USA

Bernd Aulbach University of Augsburg, Germany

Saber Elaydi Trinity University, USA

Robert Sacker University of Southern California, Los Angeles, USA

ISPE

1SDE

\fc World Scientific NEW JERSEY • LONDON

• SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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DIFFERENCE EQUATIONS AND DISCRETE DYNAMICAL SYSTEMS Proceedings of the 9th International Conference Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-520-5

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Preface

Difference Equations or Discrete Dynamical Systems is a diverse field which impacts almost every branch of pure and applied mathematics. Not surprisingly, the techniques that are developed vary just as broadly. The annual International Conference on Difference Equations and Applications is organized under the auspices of the International Society of Difference Equations. The Conferences have world wide attendance and cover a wide range of topics. It is through this Proceedings that the mathematical community can (1) obtain an overview of all the presentations at the 9th International Conference on Difference Equations and Applications, (2) see a variety of problems and applications having one ingredient in common, the Discrete Dynamical System and (3) keep abreast of the many new techniques and developments in the area. The emphasis of the meeting was on Mathematical Biology and accordingly about half of the presentations were in that area, Mathematical Ecology and Mathematical Medicine.

v

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Contents Preface

v

A Discrete-Time Beverton-Holt Competition Model Azmy S. Ackleh, Youssef M. Dib and Sophia R.-J. Jang

1

A Dynamic Analysis of the Bush Fiscal Policy Richard H. Day and Chengyu Yang

11

A Hybrid Approximation to Certain Delay Differential Equation with a Constant Delay George Seifert

27

Compulsory Asymptotic Behavior of Solutions of Two-Dimensional Systems of Difference Equations Josef Diblik and Irena Ruzickovd

35

Discrete Models of Differential Equations: The Roles of Dynamic Consistency and Positivity Ronald E. Mickens

51

Enveloping Implies Global Stability Paul Cull

71

Global Asymptotic Stability in the Jia Li Model for Genetically Altered Mosquitoes Robert J. Sacker and Hubertus F. von Bremen

87

Global Behavior of Solutions of a Nonlinear Second-Order Nonautonomous Difference Equation Vlajko L. Kocic

101

How Can Three Species Coexist in a Periodic Chemostat? Mathematical and Numerical Study Shinji Nakaoka and Yasuhiro Takeuchi

121

vn

vm Information-Theoretic Measures of Discrete Orthogonal Polynomials Jesus Sanchez Dehesa, R. Alvarez-Nodarse, Pablo Sanchez-Moreno and R.J. Ydnez

135

Local Approximation of Invariant Fiber Bundles: An Algorithmic Approach Christian Potzsche and Martin Rasmussen

155

Necessary and Sufficient Conditions for Oscillation of Coupled Nonlinear Discrete Systems Serena Matucci and Pavel Rehdk

171

Non-Standard Finite Difference Methods for Dissipative Singular Perturbation Problems Jean M.-S. Lubuma and Kailash C. Patidar

185

On a Class of Generalized Autoregressive Processes Kamal C. Chanda

199

On xn+i = P^+l"*"-1 with Period-Two Coefficients Carol H. Gibbons and Carol B. Overdeep

207

Periodically Forced Nonlinear Difference Equations with Delay Abdul-Aziz Yakubu

217

Regularity of Difference Equations Jarmo Hietarinta

233

Robustness in Difference Equations Jack K. Hale

247

Solvability of the Discrete LQR-Problem under Minimal Assumptions Roman Hilscher and Vera Zeidan

273

Some Discrete Competition Models and the Principle of Competitive Exclusion Jim M. Cushing and Sheree Le Varge

283

IX

Stability under Constantly Acting Perturbations for Difference Equations and Averaging Vladimir Burd

303

Symbolic Dynamics in the Study of Bursting Electrical Activity Jorge Duarte, Jose Sousa Ramos and Luis Silva

313

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A discrete-time Beverton-Holt competition model

Azmy S. Ackleh, Youssef M. Dib, Sophia R.-J. Jang Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010

A b s t r a c t . A model of competition between multiple populations in discrete time is proposed and studied. It is assumed that individuals within a single population are identical and therefore there is no structuring variable within each population. Under the assumption that population growth is modeled by Beverton-Holt functionals, it is shown that the population with maximal fitness will out compete the other population. While global stability results are provided for the case of two populations, only local stability results are obtained for the model with more than two populations.

1

Introduction

Interspecific competition occurs when individuals of one population suffer a reduction in fecundity, growth or survivorship as a result of resource exploitation or interference by individuals of another population. This kind of interaction between populations has been frequently observed in nature and it can affect population dynamics and abundance in many different ways. Among these is the well established competitive exclusion principle, which states that at most one population can survive when several populations compete for the same limiting resource [3, 4]. Many experiments and mathematical models have supported the well known ecological theory of competitive exclusion. For example, the classical continuous-time Lotka-Volterra competition model demonstrates that there is no coexistence between populations under reasonable assumptions on the model parameters [3]. Chemostat models have also validated the principle very successfully [8]. See also [1] for a model with competitive exclusion. More recently, Cushing et al. [5] studied a discrete-time competition model of two populations and showed similar results as those of continuoustime Lotka-Volterra system. We refer the reader to [6] for general theory of difference equations. Our aim of this manuscript is to propose a simple discrete-time competition model of multiple populations and to study its competitive outcome. Specifically, Beverton-Holt functional forms are used to model population growth in the absence of other populations.

1

2

The remainder of this manuscript is organized as follows. The model and mathematical analysis are presented in the next section. The final section provides discussion and concluding remarks.

2

A Beverton-Holt competition model

Let n > 2 be a positive integer denote the number of populations, and Xi(t) be the population size at time t, t = 0,1,2, • • • and i = 1,2, • • • , n. Each isolated population without interaction is modeled by the well-known Beverton-Holt equation [3, 4]. The equation shares the same simple asymptotic dynamics as the continuous-time logistic equation, and it has been used to study many populations such as fish and insects. The BevertonHolt equation is also often referred to as the discrete logistic equation [3]. The interaction between these n populations is given by the following system of first order nonlinear difference equations *v

Xi(t+l)-

;

l + &iE3U*;(*) (21)

xi(0) > 0 i = 1,2,--- ,n, .

where parameters a^ > 0 and 6j > 0 are assumed to satisfy the following conditions at > 1 fori = 1,2,- •• , n (2.2) and a i - l c t i - l . —> —-— for % = 2, • • • , n. 01

.

. (2.3)

bi

Under the assumptions of (2.2) and (2.3), we see that each individual population can persist in the absence of other populations and population i = 1 has the largest fitness. In the following, we shall study system (2.1). It is easy to see for the following first order scalar difference equation

* + » = *

•



all solutions with y(0) > 0 converge to the steady state —-— if a > 1 and b > 0 [3]. Using the asymptotic dynamics of equation (2.4), it can be easily shown that limsupa;t(t) < —^7— for i = 1,2, • • • ,n for any t-»oo

Oi

3

solution of (2.1). On the other hand, letting ao = min{ai,a2, • • • ,an}, b0 = m&x{bi,b2, • • • , bn} and P(t) = £)" = 1 xi{t), we have for t > 0

p

(' + ^TOT'

Consequently, solutions (xi(t),x2(t), isfy liminf P(t) > -\—. t—>oo

• • • ,xn(t))

0 converge to E\ = (—;

, 0). Notice that system (2.1) is compet-

itive and it is known that bounded solutions of planar competitive systems either converge to a steady state or have a 2-cycle as its w-limit set [9, Theorem 4.2]. An explicit calculation was carried out in [7] to eliminate the existence of an interior 2-cycle and one can thus conclude that the interior steady state is globally asymptotically stable. Our analysis performed here is different from that given in [7]. System (2.1) with n — 2 can be written explicitly as *i(t + l ) = W t + 1) =

aiMt)

l + h(xi(t)

+ x2(t))

^ i W l + b2(x1(t) + x2(t)) n(0),ij(0)>0, X2Kt + l)

(2-6)

where 01,02 > 1, 61, b2 > 0 and — - — > —; . Let / : R2 —> E 2 denote 61 62 the map induced by system (2.6), i.e., / (£1,2:2) = (h(xi,X2),f2{xi,X2)), where /1 (0:1,0:2) = , , , ) *, r and f2(x1,x2) = ,2 2 r. The 1 + 01(0:1+12) 1 + 62(0:1 +o; 2 )

5

Jacobian matrix associated with the system is given below.

J =

a\ + a\b\X2 [l + b1(x1+X2)}-2

—a\biX\ \ [l + b1(x1+X2)}'2

—a2b2x2 [l + 6 2 ( x 1 + x 2 ) ] 2

a2 + a2b2X\ [l + b2(x1+x2)}2

)

Define a partial ordering 0, where l + 6i(2i+4) r-r— — < a + biixi+xZ) has a unique aixi aixi aixx 1 + 61 (Si + x2\ = 2/1 and 1 + —— bi(xi +x2— ) >~ 1 + —— fei(ii + — x2) = j/2- Hence x i < x\ < x\. Denote this x\ by A^arijj),^ < x2 < x2. Since

l + bi(X^(x^)+x^)

— 2/1, we have — T V ^ *"•' dx*2

— -:—, > 0- Define 1 + 610:5

/ V L 2 , N 1—rr for x 2 < x*2 < x2. Using -j-±- given above, 1 + 62 (A 1(0:5)+0:5) 0x2* j* +• I J d *2 a2 + a 2 6ix5 + a262X1*(x5) a direct+ computation yields -T-J- = — > 0. Since Y2{xl) =

x2 < x2 < x2 and x2 < x2, we must have 2/2 = ^2 (£2) < 5*2(£2) — 2/2 and obtain a contradiction. Therefore, / is one-to-one and solutions of (2.6) with £i(0) > 0 converge to the steady state Ei as solutions are bounded and Ei is locally asymptotically stable [9, Corollary 4.4]. The asymptotic dynamics of (2.6) is given below. T h e o r e m 2.2 Solutions (xi(t), x2{t)) of system (2.6) with x\{Q) > 0 converge to the steady state E\ = (—? ,0). 61 We conjecture for n > 2 that the equilibrium E\ is globally asymptotically stable provided that (2.3) is satisfied. We shall use a numerical example to demonstrate this conjecture. Specifically, n = 3, a\ = 1.5, a2 =

Figure 1: This is a competition model between three populations. We can see from the graph that population 1 drives the other populations to extinction. 2,a 3 = 1.75, 6i = 0.25, b2 = 0.6 and b3 = 0.8. The system with these parameters is given below. xi(t + l) = x2(t+l) x3(t + l)

=

1.5a:i (t) l + 0.25(xi(t)+x2(t)

+ x3(t))

2s a (t) 1 + 0.6(xi(t) + x2(t) + x3(t)) 1.75x2(t)

(2.7)

l + 0.8{x1{t) + x2(t) + x3(t))

Xl(0),x2(0),x3(0)>0.

Therefore population one has the largest fitness than other two populations. Numerical simulation presented in Figure 1 with initial population size xi(0) = 3,12(0) = 10 and 2:3(0) = 5 shows that only population one survive while others become extinct. Simulations with the same parameter values and different initial conditions also confirm our observation.

7

In fact, one can indeed see that this is the case under the conditions a, < ai and bi> bi, i = 1,2,... ,n (which are stronger than (2.3)). To this end, using Lemma 2.1 we see that for i = 2,• • • ,n Xj(t+1) = tti(t)ffli(l + &lE"=l 3 : j(*)) Oj Xj(t) xx {t + 1) ~ xi (t) ai (1 + h E " = 1 a;,- (*)) ~ 1, so it converges also. It is always eventually increasing, however, so it has the same upperbound qe given by (19). If xo = q% > qe, the argument is identical except the series are eventually decreasing and the inequalities reversed.2 An analogous argument is followed for 7 positive. Consequently, we have proposition:

17 B Suppose exogenous demand (gt+i + xt+i) = (1 + 6 ) t + 1 (g + x) where g, x and b are positive constants for an arbitrary base year. Assume the average tax rate is a constant, r. If (a)

a0-

4>y' + g +

x>0

and (b) 0 < ( l + &)(-)(-»))'w.

These inferences imply the following policy repercussions. C Assume (a) (6)

gt+i+xt+i =(l + b)t+1(g + x) 7 + g + x > 0.

If (c)

j — ^

o < - r < 1+6

a

+ T&,

then a permanent increase (decrease) in government spending increases (decreases) the budget surplus; and a permanent increase (decrease) in the average tax rate decreases (increases) the budget surplus. If (7(1 T)
y'.

6.3

Stability and Tax Implications

The qualitative stability and KRB conditions (7(a)) and (7(b)) must now be applied separately to each Era. Using the new parameter estimates for the piecewise segments, the picture changes drastically as shown in Table 2.

Comparing column (a) with (e) and (f) to determine which of Propositions B or C apply, we get the results summarized in Table 3.

22

Table 2: Key Coefficients (a)

(b) a

(c) r

(1 + *)

(e) (1 + b) • a

.26

.69

.16

1.0186

.32

(f) (1 + 6). (t+1 • q%+1 • (a + ar) - (7 + xt+i + <j>qt).

(25)

F The Jevei of government spending required to achieve a given target employment rate increases if (i) the labor participation rate increases; (ii) the work force/population ratio increases; (Hi) labor productivity

7

increases.

Discussion

The model presented here cannot be taken as a perfect tool for policy repercussion analysis. Important variables have not all been taken into account. However, as far as empirical macroeconomics is concerned, it is evidently a good first approximation. The results would seem to compel serious attention. Of special interest is the KRB tax policy paradox. Evidently, tax reductions should be expected to stimulate the economy and—given the drastic reduction in the economy's per capita marginal propensity to consume—a rise in tax revenues should accompany the economic growth that follows—with approximately a year's delay. The work force and employment data also support the inferences obtained in the preceding section. Increases in the labor participation rate were substantial in much of the post war period, just as with the work force as a percentage of the population. Together with a continued upward trend in productivity, these facts explain why unemployment lags behind economic recovery after a recession and constitute a continuing problem. The lesson of equation (25)—given fixed labor participation rates and the work force/population ratio—is basically that government must expect to grow along with population if a reasonable level of employment is to be sustained. But options also exist: labor participation can be discouraged by delaying entry into the labor force (military or other service, extended education) by reducing the full employment fraction of the year (shorter hours, longer vacations). A reversion to the traditional family structure of one bread winner, one family manager would have a similar effect. Eventually, in the U.S. as well as other developed countries, the need for one or more of these possibilities is being reduced by, and maybe eliminated by,

25

the graying of the population which will lower the work force/population ratio.

Notes 1. An early theoretical analysis of the Kennedy tax policy will be found in Day, Richard H., 1970, "An Elementary Analysis of the Kennedy Tax Program," Chapter 29 in W.L. Johnson and D.S. Kammerschen, Macroeconomics: Selected Readings, Boston: Houghton Mifflin. The present discussion introduces critical nonlinear ities, distinguishes between short and long run growth effects, and presents empirical evidence relevant to the period 1929-2002. 2. An alternative method of proof is by direct iteration. Since qf+1 = n+Mt+i + -B + C 0, we have lim qf = t—»oo

B

< oo. Consequently, the

1 — C7

series {qf } converges to j ^ as £ goes to infinity. D

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A HYBRID APPROXIMATION TO CERTAIN DELAY DIFFERENTIAL EQUATION WITH A CONSTANT DELAY

GEORGE SEIFERT IOWA STATE UNIVERSITY, A M E S , IOWA

50011

ABSTRACT. We study a delay differential equation with piecewise constant delays which could serve as an approximation to a corresponding delay differential equation with a finite constant delay. We consider a number of special cases.

1.

INTRODUCTION

We consider a differential equation of the form (1.1)

x'(t) = Ax(t)+g(x(-l

+ k/N)

on the interval k/N < t < {k + 1)/N, N a fixed positive integer, and k = 0,1,2,...; here x(t) : R —» R™, A is a constant n x n nonsingular matrix, g(x) : Rn —> Rn, and n is a positive integer. The function x(t) : R —> Rn is said to be a solution of (1.1) on [0,oo) if it satisfies (1.1) on each interval k/N < t < (k + l)/N,k

= 0,1,2,..., is

continuous on [0,oo), and satisfies an initial condition of the form x(—1 + k/N) = 4>k where k 6 Rn,

k = 0,1,... ,N — 1. it is easy to see that in a sense such a solution

approximates a solution on [0, co) of the equations (1.2)

x'(t) = Ax{t) + g{x{t - l)),x(t) e Rn

with initial condition x(t) = k,k = 0,1, 2 , . . . , N — 1, and where the approximation become better as N —> co.

27

28 We assume that