NONLINEAR DYNAMICS, MATHEMATICAL BIOLOGY, AND SOCIAL SCIENCE
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NONLINEAR DYN ICS, MATHEMATICAL BIOLOGY, AND SOCIAL SCIENCE Joshua M. Epsteh Senior Fefl~w,Economic SmJies Program, The Brooking$ Instirution, and
Member, External Faculty, Santa Fe Institute
Lecmrrt: Notes Vo S m t a Fe Ilns~trute
Studies in the Sciences of Complexity
Adason-Wesley h b l i s b g C ~ m p m yh, e .
me Adv
B w k &W
Reading, Massachusetts Menlo Park, Callfornia N m York Dan Mills, Ontmo EIzlow, Englmd h s t e r d m Mnn Sydney Singapre T o b o Madrid S m Jum
Paris
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Publisher: Dwid Goehring Editor-in-GhieE: Jeff Rsbbins Production Manage: Pat Jafkrt-Levine
Bireetor sf Publications, Santa Fe Institute: b n d a K. Butler-Villa Produetian Manager, Santa Fe Institute: Dellia L. Ullibarri Publication Assbtant, Santa Fe Institute: Mawllee Thornson
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Abaut the Sanla F@ Inetltute The Santa Fe institz~te(SFI) is a private, independent, multidisciplinary research and education center, founded in 2986. Since its founding;, SF1 has devoted itself to creating a new kind af scientific resear& commtlnidy, pursuing merging syntheses in science. Operating as a small, visiting institution, SFI seeks ta cata1yze new collaboralive, multidisciplinary projec8 that break d m the barriers between the traditional disciptinc3s, to spread its ideas and methadoXogies to other institutions, and to encourage the practicd applieatioas of ilts regutb.
All t i t i from ~ the Santa Fe kstitute 5 t ~ d i e s in the Sciences of Conzplle&tg serieg will carry this imprint which is b s e d on a Mimbres pottery design (circa A.D. 95&1 m),drawn by Beesy Jones, The dmign ww setected because the radiating feathers are evocative of the outreach. of the Santa Fe Institute Program to m a y disciplines and, institutions.
Santa Fe Institute Editorial Board December 1996 b n d a K. Butler-Villa, Chair Director of Publications, Santa Fe Institute Prof. W. Brian Arthrrr Citibank Professor, Santa Fe Institute
Dr. David K. Campbell Chair, Department of Physics, University of Illinois Dr. George A. Cowan Visiting Scientist, Santa Fe Institute and Senior Fellow Emeritus, Los Alarnos National Laboratory Prof. Marcus W. Feldman Director, Institute for Population & Resource Studies, Stadord University Prof. Murray Gell-Mann Division of Physics & Astronomy, California Institute of Technology Dr. Ellen Goldberg President, Santa Fe Institute
Prof, George J. Gumerman Center for Archmlogical Investigations, Southern Illinois University Prof. John H. Holland Department of Psychology, University of Michigan Dr. Erica Jen Vice President for Academic Affa'is, Santa Fe Institute Dr. Stuart A. Kauffhan Professor, Santa Fe Institute
Dr. Edwad A. Knapp Visiting Scientist, Santa Fe Institute Prof. Warold Morowitz Robinson Professor, George Mason University Dr. Alan S. Perelson s Laboratory Theoretical Division, Los A l ~ o National Prof. David Pines Department of Physics, University of Illinois
Dr. Charles F,Stevens Mo3ecuIar Neurobiology, The Salk Institute Prof. Harry L. Swinney Department of Physics, University of Texas
Santa Fe tnstitlrte
Studies in the Seienwiit af Camptexw Proceedings Volumes Mitors L). finess A, S, Peresbon A. S. Perelson G, D. DooXen et d.
Title Emerging Syntkaw in Science, 1987 Theoreticd Immunobg~y,Part One, 1988 Theoreticd h m u n o E o ~ Part , Two, 1988 Lattice Gm Methods far PsurtiizE DiRerentiaJ XV Equations, 1989 V P. W. Andersan, K. Arrow, The Economy as an Evoking Complex System, 1988 & D. Pines Artificid Life: Proeeedirmgs of an Interdiscipiinay Vf C. G. Langton Worhhop on the Synthesis and Simulation of Living Syskms, 1988 Computers and DNA, 1989 V11 G. I. Bell, & T, G. Marr Complexity, Entropy, a d the Physics of V111 W. H. Zurek Information, f 990 Molecular Evolution, on Rugged Lan&ert-pe&;: IX A.S.Peretson&Proteins, RNA and the Immme System, 1990 S. A, KauRman Artifidd Life 11, 1991 C. G*Langtorr et d, X The Evolution of Humm Languagw, 1992 XI J . A, Hawkins &; M. GeU-Mann XII M, Casdsli St 5. Eubank Nonlinear Modeling m$ Forw~tsting,f 992 , Principles of Orgmisation in O r g a ~ s m s 1992 XlIf J, E. Midtentha1 & A. B. Bakin The Double Atrction Market: Institutions, XIV D. fiiednnan & J. Rust Theories, and Evidence, 1993 Time Seria Prediction: Forecasting the f i t u r e XV A. S. Weigend 8d and tfnderstmdr'ng the Past, 1994 N. A. Gershenfeld Tlnberstmding Complexity in f be XVI G. Gumerman & Prehistoric Southwmt, X994 M. Gell-Mann Artifieid Life IIX, 1994 XVYZ C. G. Langton Auditory Dispfiay, 1994 XVXII C. Kramer Complexity: Metaphors, Modelsi, XIX G. Gowan, D. Pin=, and Reality, 199.1: & D, Mettzer The Mathematics of Generalization, 1995 XX D. H, Wolpert SpatieTemporizX Patterns in XXI P. E. Clsurfis & Nonequilibrium Complex Systems, 1995 P, PalEy-M&oray The Mind, The Brain, and XXIT B. I\i"lorowida& Complex Adaptive Systems, 1995 J. L. Singer Madurational Windatvs and Adult XXIlX B, Julesz & Corticd P l a t i c i t ~ 1995 ~, X. Kov&s Economic Uncertainty a d Human Behavior XXIV J. A. Tainter & in the Prehistoric Southwest, 1995 B, B, Tainter XXV J. Rundle, B. n r c o t t e , 8t Reduction and Predictability Ctf Natural Diswters, 1996 W. Klein Adaptive Xndividuah in Evolving Popda-t;iom: XXVZ R. K. Belew 8r, Models and Algorit M. MitcheEl
l X 11 I11
VoI. I I1 III IV V
v1
Lectures Volumes Editar D. L. Stein E. Jen L. N d e I & R, L. Stein L. N d e l & D. L. Stein L. Nadel & D. L, Stein L, N d e l 8t D, L, Stein Lecture Nates Valums Author J. Hertz, A. Kro&, & R, Palmer G.. Weisbuck W* D, Stein lk F, J. Vaela J, M, Epstein
B. F. Wijhout, L- Ndelt & I). L. Stein Vol.
a
Referexsee Volumes Author A. Wuensche gt M, Lmser
Title Lectures in the Scieacw of Complexity, 1989 1989 Lecturm in Complex System, 1990 1990 Lecturw in Complex Systems, 1991 1991 Lrrcturw in Complex Systems, 1992 1992 Lectura in GornpIex Systems, 1393 1993 Lecturm in Complex Systems, 1995
Titf e fntroduction to the Theory af Neural Computation, 1990 Complex Systems Dynamics, 1990 ThinEng About Biology, 1993 Naniinear Rynmics, Mathematical Efiiolo~~ and Social Science, 1997 Pattern Formation in the Physicd and BioIogical Sciences, l997 Title The Global Dynamia of Cellular Automata: Atdration Fields of OneDimensional GeUular Automist;a, 1992
Dedicated in loving memory to my father Joseph Epstein
(1917-1993)
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Contents
LECTURE 1: On The Mathematical Biolog of Arms Rmers, Wms, and Revolutiom LECTURE 2: An Adaptive Dynamic Model of Combat LECTURE 3: Imperfect Collective Security and Arms Race Dynamics: Why a Little Cooperation Can Make a Big Difference
LECTURE 4: RevoEutions, Epidemics, and Ecosystem: Some DynaMlical Analogies
LECTURE 5: A Theoretical Perspective on The Spread of Drugs LECTURE 6: An. Intraducdioa. to N~lnline-it~ Dyn
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ntroduction
This book is based on 131 serim of lwtures I gave at the 1992 Santa Fe Institute Cornpiex Systems Summer School, and on my Princeton University ""Complex Systems, Simple Models" course, ollFered in aademie years 1991-92 through. 1993-94. A god of my texhing, and of this bocsk, is to impart; the mathematical tools and, as important;, the isnpulse to build simple models of eomplm procwses falling outside the artificial confines of the established fields. Many Estscinating and important problems cry out for rigorous interdisciplinary study. And recent advances in seientific cornpuging have m&e the construction and "experimental" "udy of dpamical systems remarbbly emy. The stage, in short, is s t far new synghetic wark, indmd for tz, new dbcipline or, perhaps, tramdiscipline. Thrm themes run through these j e c t u r ~ ,The first is that simple modeh can illuminate essenrtial dynamics of complex, and crucially important, social systems. The second is that mathematical biology oEers a powerhf, and hither2;o underexplaited, perspective on both interstate and intrastate social dynamics. The third theme is the unifying power of xut&bennatics, and specificdIy, of nonlinear dynamical sy~tennsthwry; formal andctgles be-een memirrgly disparate sociai and biofogie d phenomena are highlighted. One overarching aim is to help stimulate some thing of a reconstruction in mathematical social science, relming-in some ernes
2
Nonlinear Dynamics, Mathematical Biology, and Social Science
abandoning-the predominant ~tssumptionof perfectly informed utility maximization, and exploring social dynamics horn such perspectives as epidemiology and ecosptem science.
OVERVIEW OF THE LECTURES There are six lectures. The first is entitled, "On the M&hematica Biolow of A r m b e e s , Wars, and Revolutions." Here, some af the book" recurrent themes are first souabed. It is demonstrated, X believe for the first time, that the most hmous equations in the mathexnatksl thmries of war (Lanchesterk @equations)and a r m races (Mehardson's equalions) are both speeializations of the famous L o t b Volterra ecosystem model. The essay introduces the related idea that explosive mlutions-might be modeled as epidemics, using yet processes of civil violenc other parametrizl2tions of Loth-Volterra. To me, it is surprhing and interesting that the Loth-Volterra ecosystem equations have artgthzng to say about wtzx, a r m races, or revolutions. Of course, to claim that t h e ~ esimple equations say everything on, such complex topics would be foolish. And, in subsequent fectures, I move beyond them, Lecture 2 delves hrthc?.f into the mathematical theory of combat. Drawing on mathematical biology and, of coume, history, this lecture oEers whrzt; f believe to be a fundamental critique of the dominant approach, bmed on the equaekons of F, W. Lanchester. Lanchester Theory, by which P mean the origind equations and their csnt;emporary extensions, produca anomaIous results, mathematically precluding important absemed behaviors, l i b the trizding of spaee for time, Them deep problem arise baause the belligerents, W idealized in the theory, are wmpletely non-adaptive. War is, I argue, precisely a proems of codaptation, though the contestants bear a much closer regemblanee to Ashby" seat than to Homo economicus. My awn Adaptive Dmamic Model tries to capture this as simply as possible, in the process overcoming the anomalies of Lanchester Thmry. The "nonlinear dynamics of hope" iis illustrated in lecture 3. By way of introduction, many applications of nonlineakt. dynamics show that complex systems can be poked at the brink of dis&er; smaEl perturbatism in crucial variables can produce casede @&inetionsin ecosystem, devmtating epidemics in human populatioas, or ozone holes in the atmosphere, d l unhappy events. Sensitive dependence is b d news. Perhaps id is salient that we edl the area ""cat~%rophe thmry'bnd not, for insita~ce,"mirmle thmry." Well, this lecture invites us to consider the Aipside of the nonlinear coin: can we identify cmes in which the right, smdl local perturbzttion, can produce countezin%uitfiveexplosions of happy eve~ts?1think so. As one example, there is an idea cdled '%collective seeuri~'"(65) that is receiving wide attention, Imagine three countries, A, B, iand 6 , Perfect CS would then operatf? as follows: If A attach B, C allocsa2la all force to B; if B a t t w h C,
A aocates all farce to C; and so an. The general rule is simply that *e odd man aut instanfrlg allomtes all J o ~ to e the attackd pady. NW, many aedemic political scientists and &absmen d i s ~ s any s form of collective security because this ps;feet form is implausibly altrubtic. But, as nonlinear dynamichts, we ask: What about a. tiny bit, of mllective ~ c u r i t y a, highly diluted farm of altruism? The bcture shows that in m m race modeh susciently nonlinear to produce really vola-~ilr;dynamics, highly difukd, or imperfect, collective ~ c u r i t y regimes eaa damp the explosive omillatiow atld induce convergence ta stable q u i fibria below inikial wmament leveh. Put differently, the injwkion of tiny degres of altruism can profoundly calm the athemise voXatib dynamics, The benefits of the system are very great and, becaum of the wnlinearity; the level of risk to individual participants is wry low. Here, mnsitive dependence is good news! Whether the next lecture bears good news or bad depends, X suppose, on o ~ e ' s political leanings. It examines the analogy b e t w ~ nepidemics (for which. a welldeveloped mathematical theory exists) and processes of expbsive social chanp, such as revolutiom (for d i e h no comparable body of mathemat;ical theory exists), Are revolutiom "lih" epidemics? If one t h i n k of the revolutianav idea as the infection, the revolutionaricirs as the infee-tives, the public bealth audhoritim ins the pawer elite, m d social indoctrination as inoculation, then an analogy begins to take shape. E t is develop& ixt $he fourth lecture, with, 1 hope, some novel political inhrpretations*The a n d a w to epidemics, which are nonlinear dhreshold procemes, may help explain how small changes in; political conditions-marginal diminutions in central authority-can catalyze explosive sociali tramformations, much to the surprise of eli* and revolutionarim d i k ! The model also sugmts the existence of social bifurcation points at which regression abruptly changm from being stabilizing to being destabilizing and inflaming revolutionary sediment. The &&h lmture combines arms race and epidenriolom perspwtives in building a simple model of the sprearl of drug addiction in an, idealized cornmunit;): revealing basic, and pahaps counterintuitive, relationsfiips bemeen legalization, pricm, and crime. The analysis suggats once again the relevance to social science of seemingly remote fields like mat;l.lema$ical epidemiobg and ecosystem science, and id trirts to illustrate how simple models are built, and explored using methods of nonlinear analysis, These methods are the topic of lecture 6,Entitled ""An Introduction to Nonlinear Dynamical Systems," it consists entirely of mathematics, The ainn is to offer a concentrated course in. the qualitative tlt-imry of nonlirrea autonomous diEerential systems, beginning with linearized stability analysis &and movirrg efficiem_t;ly through Lyapunov functions, limit cycles, the Poinear&Bendixson and Hopf Bifurcation Theorem, Painear4 maps, various negative tests, and on to Index Thwry arrd the celebrated Poinearh-Hapf Theorem from difterent,ial topology. 1 see this as a coherent body of mathematics, much OE which i8 quite beatudifut, m d powerful when applied to social dynamics. I hmten to point out that lecture 6 is not dmigned as a mathematical founclation for the other le&urw, Not all the techniques developed there are ii~pplied
4
Nonlinear Dynamics, Mathematical Biology, and Social Science
in other lectures. Index Theory, for example, is developed for the sheer joy of it, not because I use it elsewhere, though its surprising applications to mathematical ecolom and economics are noted. Xn turn, not all techniques used in other lectures are covered in lecture 6. Lectures 2 and 3, far instance, hvolve digerenc quations, which are not treated in lecture 6. Lecture 6, then, is a frestanding essay oEering a particular dewlogment of nonliaear dpamical systems, from linearized stability analysis through the PoincarkHopf Xndex Theorem via, results of long&anding mathematical interest, such as Hilbert's 16th Problem and Brouwerk s h e d Point Theorem.lxJ One final point regarding the lectures should be made. While I oEten use history or empirical studies to argue for the qualitative plausibility of a model, no new data b m s are assembled or statistical tmts performed. As oRen occurs in science, thmry may ultimately inspire the collection of data and the performance af tests. But these lectures are purely theoretical, the goal being do demanstrate to social, physical, and natural scienti~tsthat simple mathematical models can provide inszghl into a wide range of complex social pracemes and th& mathem&ical biolah~yand nonfinear dynamical systems t hmry in particular offer the social theorist powerful conceptual and analytic tools,
THE LARGER INTELLECTUAL MNDSCAPE These, of course, are not the only methods available for the study of social phemmena. And, in my Princeton course, I gave equal time? to the agent-based modeling techniques employed in Grovring ArtificialSocieties: Social f i e a c e Rom tne Bogom Up, co-authored by myself and Robert k e l l . Both nonlinear dynamical sydems aad agent-bwed models deseme a place in any "comgle~tyc ~ r r i c u ~ u m ~ " But the former techniques are the ones employed here. For agent-bued models of social syslerns, see Epstein and k t e l l (1996) and the g w i n g literature cited there.
ACKNOWLEDGMENTS A number of colleagua ~ n institutions d dwewe special thank. For careful reviews of the manuscript or portions thermf*for stimulating discussions, encouragement, or flf%heIstare msumw familiarity with vector csleulus, linear diEerentiaf equations, eigenvalu+ eigeavector methods, phase plane analyak, and c&rain elements of camplm variables, real analysis, and pa&ial differential quations. Partions of the other Imlurm &also m u m e exposure to certain of thme topies. f210ninsight, ars against pre-didion, as a goal of m~deging,see Hirsch (1984).
advim, I thank Robe& Axelrod, Robert &ell, Bruee G. Blair, John Casti, Malcolm DeBevoise, George Dawns, Samuel Bavid Epsdein, Marcus W. Feldmm, Dunean Foley, Murraty Gell-Mann, Attee Jmkson, Jean-Pierre Langlais, Steven McCarrofl, Etaine C . MerJulty, Goafried Mayer-Kress, BenoiL Morel, Lee Segel, Car1 Sinnon, Daniel Stein, Arthur S. Wightman, and H. Pefion Young. I thank the Princehn University Council on Science and Technoloa for funding, and the Woodrow Wilsan School for hosting, my courE, f am grateful to the Brooking@fnstitution for its support and especially to John B. Steinbruner far the climate of unfettered inquiry in which this research W= eorrdueted, I thank Daniel Stein for organizing the Santa Fe Institute Complex Systems Summer School, I offer dwp t h a n k also to my farmer SF1 and Princetan students. For expert wistance in preparing the manuscript, I thank Risha Brandon. I am grateful to Ronda K, Butter-Villa for editing the manuscript, and do Dells L, Ulibarri far production msistranee, Finally; far their love and support, X thank: my wife Melissa, our daughter Axlna M&ilda, my mother Ltrcy, and my brother Sam. The views expressed in this book are those of the author and should not be aseribed to the persons or organizstt.ions aeknowIedged above.
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LECTURE 1 On The Mathematical Biology of Arms Races, Wars, and Revolutions
In this openbg lecture, I will attempt a uni@ing owrview of cerl;ain social phenomena-war, arms raeing, and revolution-from the perspective of mathematical biology, a field which, in my view, must ult;im&ely subsume the social scienees.[31 Uafortunia;Lely, few social scientists we exposed to mathema;tieal bioloa, specifically the dynamied sy~kemspmspective pianer& by Alked, htb, Vita "Vofterra, and athers, In turn, few m&hemil,tical biologhts have considered the application of mathematicd bia.ofowt o problems of human soeiety.f41 Particularly in are= of interstate and intrastate conflict is there a need to explore formal a n a l o @to ~ biological system. On the topic of animal bebavior and human warfare, the anthropologist %chard Wrangham obmrves, 1 3 ) ~ hpempmtive e t&en here, however, is quite distinct from that t&en by Wward 0, WiLon, in his book Soeiobzolog%r(1980). Specifically, X do not disc= the role of gena in the eantrol of h u m n social behavior. h t h e r , the wgument is that mwro social behaviofs such m was, rewlulion, arm races, and the s p r d of clrug nnsy conform well ta muations af mathematical biob f%Y ltnd epidemiolow in p&icutas, h r h a p s ""socioeeolo@" would be a suitable nBme for this bvel of analysis. W or a notable mmption, Gam1Xi-Sforza and Feldman f 1981). See also the innovative and undemtudied -works, b h e v s k y (1947) and h h ~ v s k y(1949).
Nonlinear Dynamics, Matnthemiliti~alBiolagy, and Social Science
"The social organization of thousands of animals is now kxtown in considerable detail. Most animals live in open group with Auid membership. Neverthekss there are hundreds of mammals and birds that form semiclosed groups, and in which long-term intergroup relationships are therefore found, T h e e intergroup re1ett;ionsbips are knows well. In general they v a y fmrn benignly tolerant to intensely competitive at territorial bordexs. The striking and remafkabb discovery of the last decade is that only two species other than hum- have been found in which b r d i n g males exhibit systematic stalking, raiding, woundilng and killing of members of neighbor-. ing groups. They are the chimpanzee (Pan troglodytes) and the gorilla (Pan gom'lla beringei) (Wrangham, 19%). In both sgeeies a grow may have p s riods of extended hastaity witb a particular neighboring group and, ia the only two long-term studies of chimpanzees, attach by daminant against subordinate eommurtiti~appeared rmponsible for the extinction of the latter. "Chimpanzws axrd gorillas are the species most closely rela;t;ed to humans, ~ diverged earliest so close eh& it is still unclear which af the t h r species (Cbchon & Chiarelli, 1983). The fact that these thrw species s h x e a pattern of intergroup aggression that is otherwim unhown speak clearly for the imporl;ance of a biological component; in human warfare'"(VVrangham, 1988, p.78). Although man has engagd in arms rmixlg, waring, and other forms of organized violence for a1E of recorded history, we haw comparatively little in the way of formal thwry. Mathematical b i o l o ~may proGde guidance in developing such a thmry m(arrgham writes, "Given that biology h in the process of developirrg a unified theory of animal behavior, th& Iiurnan behavior in general can be expected to be understood better as a result of biological theories, and that two of our closest evo1uLionary relatives show human patterns of intergroup aggression, there is a strong case for attempeing to bring biolow into the andysk of warfiare. At present, wwld like to sm mare efT~Ipt;,specifically there art: few effort;s in this direction." @"l more mathematical eEort, in this direction and hape to stimulate some interest annong you, To convince you that there might coxreeivably be some '"unified field tftmry" worcth. pursuing, I: w a d to shztre mme observations with you. To set them up, a little backpound is required. The Eundamexltd equatiom in, $he mathematical theory of arms races are the so-call~dRiebardson equations, named for %heBridish applied mathematician and soeiat scientist Lewis ~e Richardsan, who first published them in 1939.161 The fundamental equations in the mathematical theory of combat (warfare itself, as against; peacetime a r m racing) were publishcl in 1916 by Rederick Wililiarn Lanchmter.t71 ~ r a n & a m(1988, p.18). [6]~ichm&an( 1939) and (1960). [']SW ifranchmter (1916). For a cantenrporq diaussion with referencm,
Epstein (1986).
The formal thmry of ixrt;erstate conflict, to the extent there is one, rests on these w i n pillms, if you will. Meanwhile, the clwic equations of mathematical x o l o ~ are the Loth-Volterra equatiom. In fight of the remarks above, I find the following fact i~triguing:The Richadson and Lanehmter mdels of humm conflict are, mathematicaXly, specializatians of the Loth-Valterra ecosystem equations. Before proceeding, X must make one point unmistabbly char. I do not claim that any of these modeh is really ""right" in. a physicist" sense. They are ilfumirrating abstrwdions, f think it w w Picwso who said, 0. So, we have
And we will have Det A 3 Q precisely when alla22 > alzaz~,which is to say that inhibition (allazz)outweighs activation (alzazl),confirming our intuition. One c m demonstrate[lQlthat the eigenvdua of the Jacobian of (I.l) at f have negative real parts (indeed, are negative reds) when the s m e condition is met. An isocline andysis is also revealing. W recall that an isocline is a c u r v e h e r e a finewhere one side" rate of groMh is zero; clearly, an equilibrium is a, point where isoclines int;ersect. Erom (1.2), the baclinw ape given by: all 411(21) = -X1 1312 a2 1 (zl) = -21 a22
bfi2
T1 (the s l - isocline) , -a12 7-2
+(the zz - isocline) . a22
For local stability of the equilibrium Z, we require the configuration of figure 1.1, But, this occurs only if the slope of c;bl exewds the slope of (Bzl which is to say allfa12 > az1la22, or a11a22
> a2xa12 .
Our intuition is agt.zin confirmed: stability requira self-inkbitidion t d exceed reciprocal activation in this sense.
12
Nonlinear Dynamics, Mathematical Biology, and Social Science
The main. point, however, is that the classic Loth-Volterra model of mutualistic species interaction embeds, in i b equilibrium behirlvior, the elmsic Mchardson arms rwe modd.
In the models above, of course, the ""penotypes'Vo not change. in fact, ecosystem dynamics mlect against certain phenotypm. Roughly speakng, phenotmic frequencies and papulailion. 1eveh have interdependent trajectories, This is very clem, for exwple, in irnmunalow, where antigens and antibodies coevofve in a s+calIed "'biologicd arms rwe." But, of course, real a r m races work this way, too. Ballistic m k i l w beget antribdlistie missile defemes, which beget various evasion and defense supprmsion teclrnologiw. The mwhine gun makm cavalry obsalete, giGng r h to the ""ion horsen-the tank-which begets antitaxlk weapom, which beget special armor, and so on, Michwl hbinson's analogy between moth-bat coevolutiorr and the coevolution of World War XZ air war twtics is apposite,
"Moths and their prdators are in an arms raee that started millions of years before the Wright brothers made the Brmden rdds possible. Butterflia exploit the day, but their 'sisters2he moths dominate the insectsbhare of the night skies, Few vertebrates conquered night flying. Only a small frwtion of bird species, mostly owls and goatsuckers, made the transition. Bats, of course, made it their realm. Many species of bats are skilled 'moth-ers': they pursue them at speed &&erdettecting them with their highly attuned echolocatian system. Some maths, however, have developed kars' capable of detecting the bat" ultrasonic cries. When they hear a bat coming, the moths take evmive actian, including dropping below the bat's traek. The pstrallels of the responE of Allied bombers to the r d a r used by the Germans in Wrorld War 11 are interesting. If we visualize the bombers as the moths, and radars on the ground and in the night-fighter aircraft;as bats (a reversal of sizes), the situation is similar. Bombers used rearward-listening radar t o detect enemy night; fighters. When they detected a figheer, they took evwive action. But heavy bombers, heavily l d e n , were not wry maneuverable. They couldn't dodge about quite as well as moths. S m e pilots tried to drop their aircraEt into a precipitous dive. Moths also do this; it is ewy h r tbenn to fold their wings and drop. The next; stage in the nightbattle escalation is predictable. The night fighter's radiar was eventudy tuxled to detect the bomber" fifigter-detector, and thus the bomber itself* Bats have not yet tuned in on mothsbars. ""Bombers also used tecfr;tnologiealdisruption. Night fighters came to be guided to bombers by long-distance radars on the ground. The fighters started winning. But nothing remains static. The ground radars could be jarnmed by various kinds of r d i o noise. The technologicd battle swung the other way. Then the fighters wquired radar. Much like a bat, a fighter e m i t t d and listened to radm signals of its own. Thwe, too, proved to be susceptible to countermeasures, however. The RAF could jam the fighters-adar or 'clutter' it with strips of aluminurn foil. Each bomber in % formation dropped one thousand-strip bundle per minute, so that huge clouds of foil foiled the radar. Amazingly, there may be a similar counterweapon among moths. Some rnoths can produce ultrasonic sounds that fall within the bats' audio frequency The moths' voice boxes are paired, one on each side af the thorm; double voiws must be particularly confusing. Alien sounds in their waveband could confound the bats, exactly in thc? same wrtly the foil confounded the f ghters. ""The next steps in the bat-versus-moth war may simply be awaiting discovesy by some bright r-earcher; &er aft, we did not know a lot about echolocation in bats until &er World War II. My guess would be that the detectar will get more complex to m e t the defenses. Thb may already have happened; bats specidizing in moths with ears may have moved to a higher
I4
Nanfineatr Dynamics, Mat,them&tlticaiBiology, and Ssciat Science
frequency sound outside the mot;hsYhearing range!" (&binson, 1992, pp. 77-79). Quite clewIy, levels of asnnamsnt (in the international system) a ~ tevebs d of papulaeion (in an ecosystem) interact;, W in the ZothVo1t;erra and Ridardson models, but phenotwes themsc3lves are stfw changing. In, biolog, there is a mathe In socid science, there isn't. There probably could matical Lhwry of coevol~tioa.[~l) be, so I simply mention it as a prom&ng &redion. NW, let us shift, p a r s froin the rnutualistic/arrns rme variant of (1.1). Speeifically, instead of wuming that a12 and a21 are positive, sassume that they are nt3gat;ive.
Rearranging slightly?the equations (1.1) eake the form
where ki E ( r i / a i i ) > O is the carrying capacity of the environment for each species. These equadions were published in 1934 by the gre& Russian mathematical bbiologist G. F. Gause in his boak The Stmggke for Existen=. Indeed, he termed nlz and a21 "LcoeEeients of the struggle for dstenee." im1 Now, exmining (1.6), each species wutd @hibitlogistic gro*h t a its respective carrying capacity but fix these Interaction-truggl erms- XneEuding them, (1.6) gives a picture of uniform mixing of the populations z l and zz, with contacts proportion4 to the product slsz,Now, however, since the interweion coeEcienQ and species 2 at rate asr. Quite are negative, ea& contwt kills spmies 1 at rate clearly; a parallel tto combat is suaested, But more is true. TR fact, unbebownst to Gaum, (li.6) is an exmt form of the famous-and to this day uttiquilous-Lancfrestc3r model of warfaretim"l The tramition fom arms race to war, then, might be seen as a transition from the e w of al2,azl > O ta tbe erzse of alz,a2l < 0. In the fatiter context, the wUknown biological "principal of competitive exclusion" simply maps to the military principle that;, usutxollyt one side wins and the other side loses. Both these competitive
exclusion behaviors reflect the mathematical faGt that the interior (zl, 22 > 0) equilibrium of (1.6) is a saddle. The stable equilibrium in the mutualistic-peacetime me was a node. To the extent thew models are correct, then, we can say (pacem Poincar6) that war is topologically differ& from peace; the outbreak of war is a bihrcation from node to saddle. Thus far we have been exploring a mathematical biolou of interstate relations; what; about intrwtate dynamics? 1s there a Loth-Volterra perspective on rwolution, far instance? And, to what biological process might such socid dynamics correspond?
REVOLUTIONS AND EPfDEAnfCS Consider the fallowing specialization of (1.1):
Then (1.1) bwames kl = -(3125122, 52
= a12z1x2,
which is the simplest conceivable epidemic modet. Now, ratficsr than armament levels, $1 represents the level of susceptibles, and licz the level of infectives, while the parameter a12 is the infection rate, expressing the contagiousness of the infection. Ideal homogenmus mking, once mare, is assumed. If population is constant at PO, then z1 = Pi - 22 and we obtain
our familiar friend the logistic diEerential equation. Here, rcz = O is an unstable equilibrium; the slightwt introduction of infectives, and the disewe whips through the whole of society. A trad&ional t x t i c for combating the spread of a disease is removal of infectives. Sometimes, nature does the removing, as with fatal diseaes; afierr, society removes infmtives from circulation by quarantine. The simpfest possible mumption is that removal is proportional to the size of the infective pool, yielding the bllowing variant of (1.1):
with r2 > 0. This is the famous Kermwk-McKendrick (19-27) threshokl epidemic model,il4J seedled because it exhibits the following behavior, t l 4 l ~ e r m w kand McKendrick (1927). For a contemporary sLaternent, sew Walkman (1974).
16
Nonlinear Dynamics, Mathematical Bioiogy, and Soda! Science By definition, there is an epidemic outbreak only if k2 > 0. But this is to say - r222 > 0, or 1"2 s1>-. (1.11)
alzllslsz
a12
The initial susceptible level ~ ~ ( must 0 ) exced the threshold p z r z / a l z ,soncl* times called the relative removal rate, for an epidemic to break out. The fact that epidemics are threshold phenomena has impofiant implications for public health policy and, X will argue below, for social science. The public health implication, which was very controversial when first diseovered, is that less than universd vacination is required to prevent epidemics, By the thrmhold criterion (l.ll), the fraction immunized need only be big enough that the unirnmunized &action-t he actual susceptible pool-be below the threshold p. "Herd immunity," in short, need not require immunization of the entire herd, Far instance, diphtheria and scarlet Emr require 80-percent immunization to produce herd immunity;(l" lethcote and Yorke argue that "a vwcine could be very effective in cantroning gonorrhert.. .for a vaccine that gives an average immunity of 6 months, the calcufat;ions suggest that random immuniz;l.tion of 112 of the general population each year vvould cause pnorrhett to disappear."i161 Mathematical epidemic models are discussed more fully in lecture 4,With the above as background, let us now consider the analogy between epidemics (for which. a rich mathematical theory exists) and promses af explosive social chmge, such m revolutions (for which no comparable body of mathematical theory exists). Again, a more careful and deliberate development is @ven in lecture 4. Here, we simply oger the main idea. It will facilitate exposition to re-label the variables in (1.10). If S ( t ) and i ( t ) represent the susceptible and infective pools at time t and if r and y are the infection and removal rates, the basic model is:
with epidemic threshold
.%I
The basic mapping from epidemic to revolutionary dynamies is direct, The infection or dimme is, of course, the rewlutianary idea. The iafectives I ( t ) are individuals who are actively engaged in articulating the revolutionary vision and in winning over ('"infecting") the susceptible class S(t), eompxised of those who are receptiw to the revolutionary idea but who are not infective (not aetively engaged in transmitting the disease to others). Removal is most naturally interpreted as the polit;ieal imprisonment of infectiva by the elite ('the public health authority'"). melatein-~msshet(1988, p. 255). i1B1~&h~otc3and York@(1980, p. 41). flSj
Mmy familim t w t i c ~of totalitarian rule can be ~ e asnmrasura to minimhe r (the effective contact rate between infectives and susceptibles) or maximize y (the rate of political remavd). Prms cewor~fiipand other res2;rictions on, free speech. rduce T , while increases in the, rate of domestie spying (to Idc?&i& infeetiva) and of imprisonment without trid increase 7. Smmetricstjtliy, familiar revolutiomary taet iw-ucb W the publication of underground literature, or ""smizdat*%wk to inereme r. Similarly, Mm%directive that revolutianarim must ''swim like fish in the sea," mafxing themselvm indistinguishable (to authorities) &orn the surrounding susceptible population, is intended ts reduce 7,
GORBACHEV, DeTOQUEVILLE, AND THE THRESHOLD Interpreting the threshold &&ion (1.13), if the number of susceptibls is, in fact, quik close to p, then even a sligh2; reduction (voluntary or not) in central authority c m pwti society over the epidemic thrmhold, producing an explosive overthrow of the existing order. To take %heexample of Gorbacfiev, the policy of GIasxrost obviously produced a sharp inerewe in r , while the relmation of political repre-ssian (e.g., the weakening of the KGB, the release of prominent political prisoners, and %he dismantling of Stalin"s Gulag system) constituted a reduction in y.Combined, these meabsures widen;t;ly depressd p to a level below So, and the ikevolutioxls of 1989'' unfolded, Perhaps DeToqueville intuited the tf-treshold relation (1.13), describing this phenomenon, when he remarked that "KberaIizatiort is the most dificult of political arts." As3 a final element in the analogy, systematic social indoctrin&ioxr can produce herd immunity to potentially revolutionary idea. We even s w "booster shots" administered at regular interwIs-May 1 in Moscow; July 4 in Americ occasions the order-sustaining; mfihs ("The USSR is a clwsless ~vorkers$it,rdise"; "Everyane born in America has the same appo&unit.ies in lifen")re ritually celebrated. Now, as 1 said behre, all these analogic are doubtlessly terribly crude, X certainly do nod claim either that any of the models are right or that the dmamical analogies m o n g them are ex Yet, the very faet that a single ecosystem modelould specialize to equations that even caricature, the Lath-Volterra equationt however crudely, such basic and important social processes as arms rwing, warring;, and rebelling is, f believe, very injeerestixlg and serva to reinforce the larger point with which X began: social sciertce is ultimately a subfidd af biology*
18
Nonlinear Dynamics, Matthamatt;ticafBiology, and Social Scisnce
Finally, let me conclude with an, admisssion. X was surprised when I began to no"ciee these colnxsecLions. But why should we be surprised? In certain non-Wedern cultures, where our specia is seen as ""a part; of nature," where gods---like the s p h b can be part man and p a t lion, "all these connections be-mn ecosystems a d socid systems might appear quite unremarbble. But in Wesit;ern cultures shaped by the Old Testament, where God creates only man-not the khes, birds, and bushesin his own image, man is swn as ""apstrt from nature*" And, accordingly1 we are surprised when our modeh of fis worse yet, of viruses-turn out to be i&erest;ing models of man. Perhaps we are true Daminians more in our heads than in our hearts. Creatures af habit, we me captive to a transmitted and slowly evolving cultwe, But, of course, this too is "ody natural*"
LECTURE 2 An Adaptive Dynamic Mode of Combat
In this lecture Z, would like to give m introduction to some simple mathematical models of combat, including my own Adaptive Dynamic Model. Here, we are concerned with the course of war, ritf;her than the arms rww or criss that may precipitate war. Before discussing specifics, it may be we11 to consider the basic question: What are appropriate goals for a mathematical theory of combat at this poinit? First and foremost, we need to be humble. Wxfare is complex, Outcomes may depend, perhaps quite sensitively, on technological, behavioral, environmental, and other factors that are very hard to rneasure before the fact. Exact prediction is really beyond our g a p . But, that" not so terrible. Theoretical biologists concerned with morpbogenesis-the development of pat;tern-are, in some cases, situated similarly. For the particular lmpard, we certainly cannot predict the exact size and distribution of spots. But, certain clwsfj~ of partial differential equaeions-rewtion-diEusiort equationswill generate generic animal coat patterns of the refemnt sort. So, we feel thizt this is the rigfit body of mathematics ta be exploring. The same sort of point holds for epidemiologists. Few would claim to be able to predict the exact onset point or severiw of an epidemic. Theoreticians swk simple models that will generate a reasonable menu of core qualitative behaviors: threshold eruptions, persistence at
20
Nonlinear Dynamics, Mathamaticaf Biology, and So~iaiScience
endemic levels, recurrence in cycles, perhaps chaotic dynamics. The aim is to produce transparent, parsimonious models that will generate the core menu of gmss pralitative sgstem beha~ors.This, id s e m s to me, is the sort of claim one would want to m&@ for a mathematical theory of combat. NW, in elarssical mechanics, the crucial variableu are mass, position, and tirne, In classical economics, they are price and quantity: War, traditionally, is about terrihry and, unfortunately, death, or mutual attrition. A rapectable model, at the very lemt, should oEer a plausible pieture of the rdationship beween the fund* mental processes of attrition and withdrawl (i.e., territoriaf, sacrifice), I will discuss attrition first.
The big pioner in this general area was Reclerid William Lmchester ( 1868-1945), The eclectic English engineer made contributions to diverse fields, including autaMe is best remembered for his motive design a d the thmry of equatioas of war, appmopriately dubbed the Lancfi&er equations. First set h r t h in his 1916 work, Aircrafi in Wa.rfare,these have a mriety of form, the most; renowned of which is called-for reasons that will be given shortly-the Lanchester ""square" m ~ d e l , I With ~ ~ I no air power and no reinforcements, the Lanchester sytlare quations are
of ers of "Blue" and "Red" wmbata EXere, B(t) and R(t) are the which is an idedized fire sourc d 6,r > 0 are their respective firing SS per shot. Qualitatively, these equations say something intuitively very appealing, indeed, seductive: The atf~tlionrate of each belligerent- i s propodional to the size of the adversary. In the phwe plane, the origin is obviously the only equilibrium of (2.1) and the Jaeobian of (2.1) a-t Z is
[l71~mef.lesler (1956). [l81~w Lanchwter (1916). The same model was appwently deveEop& independently by the RUBsian M. Osipov (1915).
The eigenvalues are clearly f G.Hence, the origin is a saddle, though the positive quadrant is all we care about. The system (2.1) is, of course, soluble exactly. With B(O) == B@and R(O) =. &,
with various trajectories for R and B over time. Depending on the parameters (b,r ) and the initid values (h, h), either side can gtast ahead and lost;, or start; behind and win, W Es observd historicdly~l~ The mast celebrated reult of the thwry is the so-cafled L a n e h ~ k rSquare Law, which is obtain& easily. Ram (2.11, we have
Separating variables and integrating from the terminal values (R(t),B(t))to the higher initial values,
we obtain the state equation
Of course, stalemate occurs when B(t) = R(t) = 0, which yields the Lanchester Square Law: b ~= : r@ or
This equation is very important. It says t b t , to stalemate an adversary t b r e t h w as numerous, it does not suffice to be Lhre timm as effwtive; you must be nine times as egwtive! This prwurned heavy dvantage of .numbers is deply embedded in virtually all Pentawn modeb. For d e e d s , it supported the official dire ssments of the canven$ionaf balmce in Central Europe, giving enormous. weight 1291Xndd,the numerically smaller force was the victor in such Auterjitz (1805); of fi~ntiem(1914); Antiet- (1862); R d e r i c h b w g (1862); Chanceflorsville (1M3 ;the battle of Kursk the fall of Ranee (1840); the invmion of R m i a (Operation Bar (1943); the North K o r m &=ion (1950); the Sinai (1M7); the Gotm Heights (1967 and 1973); and the Falklmds (19821, to name a few.
22
Montinear Dynamics, Mathematical Biology, and Social Science
to sheer Soviet numbers and placing a huge premium on western technological supremacy. That, of course, had budgetary imp1icr;rtions. But, the presumption of ovemheinting Soviet conventional superioritzy also shaped the development of sacafled the;zter-nuchar weapons and producecf a widesgreacf assumption that their early ernployrnent w u l d be inevitable, which drove the Soviets to seek prmmptive offensive capabilities, and sa on, in an expensive and dangerous military coevolution (SW the preceding lecture) . The whole dynamic, while driven by myriad politicd and military-industrial intcrats on all sides, was certainly s u p p o ~ e dby Canchester" innocent-looking linear differential equations (2.1). But, the linearity itself implicitly asumes things that; are implausible on refiection and it mathematically precludes phenomena that, in fxt,are observed empirically Moreover, anyone expctsed to mathematical.biology would have f'crund the Laxxchester vaiant (2.1) to be suspect immediately,
DENSIN The equations, once again, are
IDthis framework, increttsing density is a pure benefit, If the Red force R grows, a greater volume of fire is focused on the Blue force B, and in (2.61, the Blue attrition rate grows proportionally. At the =me time, however, no penalty is imposd on Red in (2.7) wher~,in fact, if the battlefield is crowded with Reds, the Bfue target acquisition problem is eased and Red's attrition rate should grow. In warfare, each side is at once both predator and pmy. Inerewing density is a benefit for an. army as predator, but it is a cost for that same army BS prey. The Lanchester square system captures the predation benefit but completeXy ignores the prey cost of densiw, The latter, moreover, is farniliar to us all. For instance, if a hunter fires his gun into a sky black with ducks, he is bound to bring down a few. Yet if a single duck is Aying overhead, it takes extraordinary accurwy to shoot it down. Ebr duck, considered as prey5 demity carries costs, And, as any ecologist would expect, $be eBect is in&ed obmwed. Quoting Herbert Weiss, "the phenomenon of lossw increasing with force committed \;V= observed by Riehard H, Petersan at the Army Ballistic Research laboratories in about 1950, in a. study of tank battles. It was again obsemed by Willard and the
present author [Webs] has noted its appearance in the Battle of Britain data."(20] The work referrd to is D, Willardk statistical study of f 508 !and b;zt;tles.(211 To his credit, Lanchester rtctually agered a swond, nonfinew variant of these equations, which is much more plausible in this ecological light. Here,
In parentheses are the Ganchester square terms reflecting the '"redation benefit1" of ctensity, but they are now multiplied by a term (the prey force 1eveI) reflecting ""prey costs," as it were. The Red atlrition rate in (2.8) slows as the Red population goes to zero, reflecting the fact that, as the prey density falls, the predator" search ("foraingW")requirements for the next kill increase. Equivalently, b d ' s attrition rate grows if, like the d u c h in the a n a l o ~ its ! density growt;, In summary, a density cost is prment tX3 balance the density benefit reflected in the parentheshed tern, If we now form the casualty-exchange ratio
separate variables, and integrate as before, we obtain the state equ&ion
and the stalemate requirement r& = bB@.
W ,as agaiast the Lanehester Square Law,it does suffice to be three (rather than nine) timw W good to staremate an dversary three times as numerous.
AMBUSH AN0 ASYMMETRY h r t h e r , mymmetrica~,variants of the bmic jtanchester equations have been bevised. Far example, the so-called ambush variant imputes the "qqu~relaw" fire concentration c q w i t y to one side (the ambushers) but denies it to the other (the ambushem). Here,
24
Nonlinear Dynamics, Mathematical Bistqy, and Sacial Science
so that
Now wguming a fight tm the finish (R(1).= B(1) =r 0 ) and equal firing effectiveness (r = h), a Blue force of B. can stalemate a Red force numbering B:-a can hold off een $howand. It" TherxnopoEm.
hundred
REINFORCEMENT Thus far the discmsion has concentrat& on the dp~miersof engaged forces, Ofden, bowever, there is some flow of r i j i ~ f ~ r c e r nto e ~the t ~ combat zone proper, But, there are limits t a the number of forces one can pack into a given area-there are "force to space" constraints. One might therefore think of the combat: zone as having a carrying cwacity and, wcordingly, posit logistic reinforcement. Attaching such a term to the Lanchester nonlinear attrition, model prodlitem
where a,p, K, and L are positive constants. As obsewed in the preceding lecture, this is ezac.11~G w e k ((1835) famous model of competition bebeen two sweies, itself a form of the general Loth-VoXterra ecosystem equations. Equ&ions (2.10) admit four basic cases, corresponding to different ""war h i s b rim," Thae are shown in the p h a portrai$s in figure 2.2.
FIGURE 2.1 Phaw p o r t r a ~for hnch-erfiaulse
Msdel
(c)
Source: Bassd on Clark (l996), p. 194).
C w s (a) md (b) sre clear irzstancm of %hebiological ""principle of competitive exclusion,"" or mihtary principle that one or the other side usually wins. Case (c) shows the horrific stable nod the "permanent war" "at neither side wins. Finally, we haw case ( d ) , a saddle equilibrium. Any perturbation (off the stable manifold) sends the trajectory to a Red or Blue triumph. There is, however, the interesting and important region below both lerocli~es.Each side fwls enmuraged in this zone; reirrforeement rat= exced attrition. rat= so the forces are growing. But, for instance, as the trajectory crosses the b = 0 imclioe, matters start to sour for Blue; goes negative while Red forces continue to grow. Expectations of Blue defeat; may set in, Blue morale may collapse, and, as a result, the Blue force can '%reakVh n g before! it; is physically annibifated. Indwd, the general phe~omenooof
B
''bbreakpoints'?~ common.
26
Nonlinear Dynamics, Mathematical Bialogg and Swial Science
BREAKPOINTS Ziterd fights to the finish are aetuafly rare. Normally, there is same lewl of attrition at which one bc3Uigerent "crach." S~upposeBlue brealrs if B(t) = P& and Red breaks if R(t) == p&, with B < p,@ 5 Z and p not necessarily equal to p. Clearly, breakpoints divide phwe space into four zones, m shown in figure 2.2. In Zone 111, each ~ i d excwds e i t s breakpoint, so %bereis comb&. Red wins if a, traject0l.y crosgm from Zone 111 to Zone 11. All" quiet in Zone I, and so forth, F1GURE 2.2 Breakpoints
Substituting the ddernate conditions, EZ(t) = @Baand R(t) = p& illustration, the Lanchesster square state equation (2.4) yiekts
which implies the (wiLh breakpohts) &alemate condition
inLo, for
GENERALIZED EXCHANGE RATIO As discussed in Epstein[221thesevariants are all special cases of the genclral system
The corresponding casualty-exchange ratio is
where c-values are simply reals in the closed interm1 [O,11. Clearly, from (2.10, cl is Blue's preclfalion benefit ham increasing densitJr while, from (2*12),q is Blue's prey ~ o soft increasing density. Hence the exponent c l - q might be thought af as the net predatzon benefit of i n c ~ a s i n gdensity, which is net; fire concentration capacity in Lmchesterk sense. The Red exponent c3 - ez is analogously interpretd.Therefore, let us define Xb
-- Blue's net predation benefit
X, = Red's net predation benefit
= el - q =: c3
-
, .
Then
Again separating vasiables and intt;egrat;ing from terminal to (higher) initial values,
With skatemate defined as B(1) = R(t) = 0, we obtain the stailfematecondition
which specializes to at1 the ernes discussed earlitir (e.g., X b = X, = 1 implies square law), and many mare, stein (2985 and 1990).
28
Nsntinaar Dynamics, Mathematical Biotsgy, and Smial Science
Equation (2.13) is the algebraic form of the exchange ratio p(t), used in my own Adaptive Dynamic &del. On sepanation of variabla and integralion, it also yieids the memure of net military advantage used in the n o d ~ e a ra r m rrzee modds of lecture 3.824 Of course, mere casualty-exclrange ratios do not mcessarily determine m u a l outeamm, Even dehcfers with favorable exchange r&ios in engagennertt;~may run out of room or run out of time @.g., pqular support; may collal>se bc?forcl the attwkerk breakpoint is reached). Duratio~m d territory-spacc3 and tim loom every bit as l a r s as physical attrition in. determi~ngoutcomes. And this brings us to the topic of ~novement.
MOVEMENT Historically, war has been about territory: On a map of the modern world, the j a a e d borders are o&en simply the places where battle lines finally came to rest, X t is imt;ermtingto compare thee with the straight borders arrived at more esntractrrdly, geaceftrlly-say9 the borders between Nebrash and Kanszts or betwwn the U.S. and Canada, This is the remon that mount;i%jtxlranges are such common borders: they w r e natural lines of military defeme. The Alps, Hirnalaya, Pyrenes, and Caucases; are exampfm. The same obviously holds f a major bodies of water, like the English chaanel, and rivers, like the Yalu. In sho&, poll.l;ical borders refiect military technolow. In any event, movement irs a central aspect of war. And, ;zs I argued at the outset, a pl~twiblemodel should capture the basic connection between the Euadamental procams: attrition and movement. Lanehmter himself had nothing to s;zy about this and offered no model of movement, Contemporary extensions of Lanchmter all handle it in ictssentialtlly $he same way: they posit that the velocity of tfi, front-that is, the r&e of defensive withdrawal-is some function of the force ratio. 50, if
these modeh posit a withdrawal rate, a velocity, W ( %with ) W(lf =. 0; W'(%)> O if > l and eventudly W M ( z< ) Q, imglying some w p p t o t e . (The direetion of movement is alwayg ""fomard" for the 1arger force.) One prrbfkbed example1241 is
la31~orfurLher d k w i o n of the X", see Epstein (1N O ) . g2"1 ~ a u f m n n(1983, p. 214).
Lecture 2
29
The basic setup, then, is this: farcm grind each &her up via the attrition equations; the force ratio changes accordingly; and, as a function of that changing fbree ratio, the front's velociity changm, a;s depkted in the ffw diagram of figure 2.3. FIGURE 2.3 Flow Diagram for the Standard Modst
The frsmewrk is very neat indeed, The only problem is that any comb& model wiLh this bnsie structure is fundamenkdly implausible, and for one bwic rewo~: movement of the h n t 4 e f e n s i w withdrawal-is anornalau~!For a glven pair of attacking and defending forces, the Gourse of &trition on the defender" side, ;ls edeulat;ed in this framework, is mactly the same whether ht.? withdraws or not. The course of attrikion on the attmkerk side is also unchmgd whether the defeader withdraws or not. In short, defensive withdrawal neither benefits the defender nor penalizes the attxker. So, why in the world would the defender ever withdraw? Thc: framework i h l f mathematically eliminates m y ralionale, or inmndive, for the purports to represent. Movement is infiuenced by very behavior-withdrawal-it attritian, but not convtersdy The movement of the front (witbdrawaf) is not fed back into the ongoing attrition process, when the entire point of witMrawal w m presumably to affect that; process-in the prototypical crtse, the point is to reduce one" attrition. Surely, it ki eontrdietory to assume some benefit in, withdrawal (otherwise, why would anyone withdraw?) and then to reAect no benefit whatsoever in the ongoing attrition cdculatbm. Yet, a11 the contemporary LanGh&er variants of which I am aware suEer this inconsisteney.~~~] In turn, because defensive withdrawal cannot slow the defender's attrition (or, for that matter, the attacker's), the sacrifice of territory cannot prolong the war. And so, the most hrndamental taetic in military history-the trading of space for is mathematically precluded. But, this tactic saved Russia from Napoleon and, later, from Hitler. A plawible model should ce&ninly permit it. IZ511t is intermting to note that the battle of Iwo Jima-m island, where movement of the kont the only case (to my knowledge) in which there is any statistical corrapondence betwmn evenb as they unfolded and iu; hypothmizd by the Lanehater quations. Even if the 8t;tatisLicd fit were gm&,there muid be RO basis for eMrapolation to cases where by imuacient data. On this issue, ~ubstantiaImovement is pmibfe. And, in fact, the fit is SW EpsLein (1985). was tsl but impwibl-is
30
Nontinear Dynamics, Mathsmatical Biology, and Social Science
So, how do I fix it-how do X build in a feedback from movement to attrition? As simply as possible. The key parameters are the "equilibrium" attrition r a t e , a r d ~and a , ~ .The first, adr, is defined m the ddly attrition rate the defender is willing to suf"flerin order to hold territory. The second, a,T, is defined as the daily attrition rate the a t t w k r is willing to su&r in order to take? territory: X assume 0 < c r d ~ , r x , l . < 1. War, in addition to being a contest of technologies, is a, contest of wills. So it is not outlandish to posit basic levels of pain (attrition rates) that each side comes willing to suger to achieve its aims on the ground. If "ce defender" attrition rate is less than or equal to ad^, he rem%.insin. plwe. Xf his attrition. rate excmds this ""pin threshold,'"e withdraws, irr an egord to rwtore attrition rates to tolerable levels, an eEort that may fail dismally depending on the adaptations of the attacker, a similar creature. If the attacker's attrition rate exceeds tolerable levels, he cuts the pace at which he prosecutes the war; if his attrition rate is below the level be is prepared to sufir, he increases his prosecution rete.f24 X t is the interplay of the two adqtiue systems, meh searching for ifs equilib~um, that pmduees the observed dynamics, t;he actual movement that occzlrs and the achal att&t.ion suflered b$ each side. Indeed, in its most basic Ebrm, withdrawal might be thought of 4ns an attrition-regulatirrg servomechanisnn. The pain thresholds c r d ~and a , play ~ the roles of homeostittic t;trgc?Ls,in other words. The ixlCrodtxetiian of these thresholds struck m w a n d stilt strikes m as the most direct mathematical way do pernit defensive withdrawal to 8ff:ect attrition and, thus, to permit the trading of space for time, Their introduction &so generates the fertile analom between armies and a broad array of goal-oriented, feedback-control (cybernetic) system. Before delving into the mathematics, one passible xn;tsconceplion. about these ""pain" thresholds should be addressed. 1 do not claim, nor does my model imply, that battlefield commanders are necessarily awaw aE the numerical values of a s d ~ and a,r. Humans in the sixteenth century were not '"ware" 'h& they were sweating and shivering depending on the error: "body temperature minus 98.6 degrees, Fahrenheit ." But the homeostatic behavior was there nonetheless.
OVERVIEW OF THE MODEL Let me now turn to the Adaptive Dynamic Model itself. The fufE atpparatus includes air power as we11 m air and ground reinforcements, faci;ors I will not discuss f 2 " ~ o rearlim versions see Epstein (1985,1990). [271~hw parameters reprment daily rates of attrition, not total or cumulative attrition levels, discuss& above in conneckion with breakpoints.
ilrs
31
~ecture2
ltzere.f281The model is a system of delay equr%lionswhere the unit of tirne is usually inkrpmted as the day. If A(t) and D(t) are the altackr's and defender's ground forces sufvivhg at the sta& of the t t h day and a,(t - 1) is the attacker's attrition rate over the preceding day; we have the wcounting idemtity
The attaderk force on n e s d a y is his force an Monday, minus total losses Monday. Likewke, it must be true that D ( t ) = D(t
- 1) - (Defender's losses on day(t - 1 ) ) .
What are these logs=? Well, if we define the csually-aehange ratio as p(t
-- 1)
Attackers Lost on day t - Z Defender8 Lost; an day t - 1
the defender" losses must be
since the numerattor is the at;txkers b t on (1 - 1). Thus, accounting identity
W
have the second
Obviously, mce we attmh specific functional forms to cr,(t) and p ( t f , we no longer have aecaunting identidies; we have a model. Above tve discuss& p(t) and a r g u d that a plausibie and relatively general functional form is
where X, Ad E [O, I] are parameters. The red action--&l feedback from movement to attrilian-is inside cr,(t). Here is where the interplay of adaptive belligerents tmfalds, As men"F-ioned,this iaterplq is betwwn the attacke~"'~ prosecution rate (reflecting the pwe E& which he chooses to press the izttaek) and the defender's sewomechanisms, in @Recta witfidrauoval rate, both of which are 5tttrition'reguIitti~;tg The defender is, in same respects, simpXer. We discuss him first;.
Nonlinear Dynamics, Mathematical Biology, and Swial Science
ADAPTIVE WITHDRAWAL, AND PROSECUTION The defender" withdrawal rate for day E is wsumed to depend on the difirence between his actual and his equilibrium attrition rate for the preceding day, day (t --- 1). The functional form of that dependence should satis@ sonre basic requirc3menies:
1. As the actual attriLion rate for day ft - 1) wpraaehes 1, the withdrwd rate for day t should approach the mmimum fesible ddly rate, W,,,. 2. If the actual dtrition rate for d a y (1 - 1) is greater than the equilibfium rate ad^, the withdrawal rate for day t should be greater thaxt for day (1 - 1). 3. If the a u a l attrition rate for day (t - 1) is less than or equal to the equilibrium. rai,-t;e ad^, then %hewithdrawal rate for day t is zero. X t may nod be correct;, but the simplest functional form I can think of that satisfies these requirements is
W ( t )=
Q W ( t - 1)
+
if ~ (- 1)t ( a d (t - l ) - a n ) othemise ,
zfS, then zf > zYS for all t > i. Proof,
zLl = ~ s f + b since ~ s >f 0 > ( A - ~ ) s +f b > ( A - B ) Z ? ~+ b since zf > 5y by hypothesis r==
2F5
L+ l
by ( 3 . 5 ) .
a, a proof that zp > sfS for all positive t will be in h a d once we show that zf >'?X But this is simple. Since zg = zfS, call them both zo > 0. Then, z p - s f S = [ A X ~ + ~-] [ ( A - B ) Z ~ +=~ B] Z > ~ 0, [561~a ensure physical rmlhnr (z> 01,vur? must stipulate that a1 -i a2 -3- a3 61 3I- c3 < 2. X thank Jean-Pierre Langlois for this o k w a t k n .
and
< 2, bX f b2 + 63 < 2,
aad we are through. C3 Now, as noted earlier, these rmults obtain. so long ins BJ: > O for z > 0. The specific B matrix above can be altered considerably while leaving the strictly depressive effect of collective seeurit;?r ixldwt. Connwtionist b r m i n o b g wilt prove natural for discussing imperfect collective si~tcuriw
THE CQNNECTIONISM OF COLLECTIVE SECURITY This perspective emerges f m closer scrutiny of the B matrix. The ijth entfy, bij, reprmxttrz the level of altruism that pafty j shows party i. If we let the s p b o l "'re --r y" represeat the altruism z shows y (i.e., it is > Q ) , then, eonceptaally, the B matrk is
a
y--+Z
z--rz
z+y
0
z-+y
4;+2
y-+x
0
Gr;l_phieally;this would correspond to the "(altruism web'hhowrz in figure 3.6, Pairwise, iaff altruism is reciprocated; arrows run in both directions. If z a t t w b z,y aflocaterz force to z and vice versa if s atdwh y, and so on. When this is the c s e , we will say th& the collective securiw system is m&mallg connected* FIGURE 3.6 Maximally Connected Altruism Web
AXE off-diagonal elements of the B-matrh are strictly positive; the sign structure is then
The strength of any c~aneetion(in cont;rwt to the connection pattern) is: the! red number, b,, which can assume values in 10, l]. So, in these terms, perfect collective acurity entdfs m d m u r n connectivity and, maimurn connection stre~@h,In turn, imperfect co11ective security regimes result from: reductions in conneckivilty, rductions in connection strength, or both.
54
Nonlinear Dynamics, MathematicalBiology, and Soeial Science
MAXIMAL CONNEGTIVIW WITH DILUTED STRENGTH It is obvious that, if Bs > O then > 0 for a y real y E (0,I). This is the most transparent cme of imperfat or ""dluted" wllectiw security, Mmimal connectivity-reciprocal altruism-prevaiis~, but, irrsted of sending all of one's forces to the aid of the attackd party, one sends a fraetion y. In fact, the strictly depressive effect is preserved if every oE-diagonal entry in the IS-matrix is a different yij E (0,l). Everyone is better off even if the reciprocal altruism is, in this sense, discriminatory.
MtNlMAL COMMECT1VEW WITH DILUTED STRENGTH More intriguing, however, the altruism need not be; reciprocal to leave all parties strictly bfjtter off. Specifically, altruism matrices far more sparse than B will fulfil1 our drictly depressive requirement, Bz r 0. Indeed, it is necmsary only that each row contain a single positive entry. So, for instance, any matrix with the following s i p strtleture will do,
Graphically, this would correspond to the cyelie LLaltruism web" in fifigure 3.7, FIGURE 3.7 Cyclic Altruism Web
Here, z is unilaterally dtruistic to y; y is unilaterally altruistic to z; and x is unilaterally altruistic to z. Everyone is better off, but there is no mzprocat altmism, Imtead of "mu scratch my bwk and 1'11 scratch yours," the appeal i s " p u scratch my back, and 1'11 scratch Samks,and Sam will scratch yours." 1 c111 this "qyslic ~ l t r u i ~ r nThe . " ~direction ~ ~ ~ of the cycle Is reversed if B has the sign patttern shown below.
[ w ~ ~ b v i o u sthis ~ y , is a form of d i l u t d altruism, with some yid'S equal t o zero. But, as it has a different f i ~ v aand, t beeause the positzon of the zeros mtsttters, E give it a separate name,
It is, in f a , not necessaq thsrt these altrukaz, graphs, or '%eh," h closed. For instance, any B mitttrk with the followi~gsim pattern will satis& our strictly degremive, Bz r 0, requirement.
But, its graph is open, as shown in figure 3.8, FIGURE 3.8 Open Altruism Web
Everyone is strictly better off if z and are reeipracal altruists and g is unilaterally ait;rui&ic t o z , eveat if z is. dtruisdic to no one! In summaryl for the linear models &ove, there are b ~ i e d l ytwo sengm in which collective s w u r i e can be imperfect and stiitl exert a strictly d~j~remive eff:ect on dynamics. A1tru;ISm can be perfmtXy reeiprocd but diiukd in stren@h. It can also be imperfectly reciprocal (as in fipre 3.8), even unreciprocated (M in figure 3.7). As we will see, it may in faet be both highly diluted and imperketly reciprocal a d still have a profoundly depressive effect. It is in precisely the systems that concern us most-the volatile! systems-that such highly imperfect eoflective security regimes can have dramatic stabilizhg e E ~ t sSuch . dynamics, however, really arbe only in nonlinear systems. Let us turn to thwr?varimts,
PART II, NONLINEAR MODELS Nonlinearitties may enter the model in numerous ways. One way is through the balance assmsmexlt, s r e ~ e r n a threat;, l terms, Moder n military e~ta'blishmendsdo m t memure military b a l ~ n c ernal theats-by simple subtractions of the form y - z, EM in the above models. b t h e r , they o&en usc; methods that;, at some level or other, embed mutual attrition models implfing that net military advantqe i s a diflerence of levels raised to powers. Where does this come &am? BasicaUy, konn the attrition stalemate conditions of generalized Lanchester equations which were discussed in the preceding lwture. Allow me to derive this quickly.
56
Nonfinear Dynamics, Mathematical Biology, and Swiat Science
GENERALIZED AmRITiION STALEMATE Let R($) and B(t) be Red and Blue forces at time t , and let r and b (real numbers between zero and one) represent their effectiveness per unit. With constants cl throu& c4(0 c < l), the most general Canchester attrition system is
0, I(Q) == l. > O and R(O) = 0. The comtmts r m d 7 are called the infection rate and the removal rate, and p = ylr is termed the relative mmovd rate.
THE THRESHOLD CONDITION Now, under what con&tions will m epidemic occur in this mod47 To say that an epidemic occurs is to say that the infectious class grows or, equivalently; that > 0, which h m (4.1) implia that rSI - y l > O or, simply, tbzvt.
[ 7 " 5 1 ~ d t(1914, m ~ p, 2). [76]~mam the A m is from susceplible (S)to infmdive (If to r e m w d ( R ) ,this is term4 an SIlR rmodk?l.ff the inf&iuw p h w is fo'otlawd,not by remod (e.g., irnmuniky), but by rentry into the sweptible pool, tm SIS model would be call& for. "Is general, SIR madeh axe appropdatrt for such ~ 3 8m ~ l wmumps, , md smdlpm, while SIS m&eh itre appropriate for such as meniagitis, phgue, and venered d i some bmteriaf agent d such W d w i a and sleping siekn-." Hdhcot;e (1976, g. 336). See &m Hetheode (1989). The wrnerstone af the nnrrlhemalical epidamiolu~literature mmains Bailey (1957). See h Baile3y (1975). A cormprehemive mntennporw t a t k Anderson and May (1991).
74
O\lanlintsar Dynamics, Mathernatk-calBiology, and Social Seienee
This is a basic result.iT7jFor an epidemic to occur, the number of susceptibles must e x c ~ dthe threshold level -the relative removal rate defined above,
POLITICAL INTERPRETATION The bwie a~fS\lfom to revolutimi~fydynamics is dirwt. The infection, nr diwase, is af course the revolutionary idea. The infwtives Ift) are individuds who are xletively engaged in articulating the revolutionary vision and winning over ("infecting") the susceptible class S(t), comprised of those who are receptive to the revolutionary idea, but who are not infective (not netively engaged in transmittiag the diseme to others). Rernwal is mast naturally interpreted ijls the political imprisonment of infectives-----R(t)is the ""Gulag" "population, the set of unfortunate revolutionaries who have been captured and isolated frorn the susceptible populatim.f781 Many familiar tactics of todditarian rule cttn be seen EM measures to minimize r (the eEective canLaet rate b e w e n inkctives and susceptibles) or m&mize 7 (the rate of political rcjmoval). Press cemarship and the systematic inculcation of counl;errevolutionafy beliefs reduce r , while increases in the rate of domestic spying (to idedify infcxtives) and of imprisonment w i t h u t trial increase y. Spmetrically, familiizf revolutionary tactics-such as the publication of underground literature, or " s ~ m i z d a t " ~ e etok increae r. Similarly, Mm% directive that revolutionaries must ""sixn like fish in the sea," making themselves iadistinguishable (to authorities) from the surrounding susceptible population, is: inteded to reduce y.
GORBACHEV, DeTOQUEVILLE, AND SENSITIVITY TO INITIAL CONDCTIQNS Interpmting relation (4.2) =mewhat differently, if the number of suseeptibltss, &, is in fact quite close to p, then even modest reduct'rans (voluntary or not) in central authority can push society over the epidemic threshold, producing an explosive overthrow of the edsting social order. To take the example of Gorbwhev, the policy of Glasnost obviously produced a sharp increadie in. r , while the relaation of potitieal repression (e.g., $he weakening of the KGB, the releae aE prominent political prisoners, the dismantling of Stalin's Gulw @@ern) constituted a reduetion In y. Combined, these mewures evidently depresmd p to a level below So, and the ""rvolutions of 1989" unfolded. Perhaps DeToqueville int;uited reldion (4.21, describing f771~bviously, the system (4.1) has a gre&t many further mathematical properlim sf interwt. For a d i ~ m i o now , Braun f 1983, pp. 456-73). IT81ln tfib disewion, we ignore exmutiona
Lecture 4
75
tbis sensidivi/t;y to istial conditions, when he remark4 that '"iberalization is the most diEeuEt of pafjiLieal arts."
TRAVELING WAVES In the discussion thus far, the sqtatial dimension has only been implicit. In fact, epidemics spread wross geographical a r e a over time. And one generally t h i n k of revolutions spreading as well. Specifically, we o&en invokcl, the t e r m i n o f o ~of waves, Recently, we saw "a wave of democratic revolutions" sweep acmss Eastern Europe, Perhsbps this sort of language swnrs natural for a reaan: if one generalizm model (4.1) to explicitly include the spatial dipusion of infectiws, traveling w m s do indmd emerge. And this process, of course, has a political interpretation. The one-dimensional spakio-temporal gen~ralizationof (4.1) is:
An infective spatial diffusion term, Da21/@x2,has been introduced into the second equation, which bears some resemblance to the clmsicaf heat equation, It == BI,,, where D is %betherma r, in this cwe, the political-""dEusivityB of the medium. The presence of the paredhesized term makes the equation a so-called reactiondiEusion rdation, Now, as set h r t h in lecture 6, one posits traveling wave solutions to (4.3) of the form S(z, t ) == S(z), I ( x , tj = I ( x ) , z = z.- - &, (4.4) where c is the wave speed. The boundary conditions S ( m ) = 1,S(-m) = O,I ( m ) = I f - m ) = O must also be met. B y p ~ s i n gmat;hema%iedspecifies that are well presented elsewfiere,[7" the basic conclusions are, first, that no epidemic wave prop& < y/r. This, of course, is the basic threshold condition from model (4.1). gates if What is new, however, is that if that threshold level of susceptibility is exceded, an epidenaiclrevolutionary wavefr;onl propagates. And its speed of propagation, c, is given by c = 2[D(rSB- y)]'/2 . (4.5) Basic cortnterrevofulionary tactics aim not o n b to minimize r (the rizte at Ftakich contact produces a transmission) and metximize 7 (the removal rate), but to minimize D as well. Physical curfews, ratrietiom on free w~embly,internal lT9f~e@ Murray (1989, pp. 66143), BritLon (1986, pp. 51-71), and the discurnion in lecture 6 of dbk volume.
76
Nontinear Dynamics, Mathematical Biology; and Social Sciems
pmsport requiremexrt;~,a g a ~ h e i din all its forms, are m e w to '"atchifyt" and limit the ""politic& diEusivityB of, the sacid medium. A high density of ipternal police reduces the diffixsion af revolutionaries just as a high demity of wet insmts or Dr&ards the spread of a brush fire.@qfndwd, the mosti obvious ""fiebre*," minimizers, are the borders bewwn e o u n t r i ~which , physicdly enforce certaLin. of the ideological '"patches" "viding h~mpanity*[~~J
VITAL, DYNAMICS AND THE EWLUTION OF IC)ISCONTENT Thus far in the dbcussbn, the total popuiatbn has bmn assumed eanstant. Proeases of profound social change may unfold owr periods in which birth md deathwhat epidemiologhts cd1 ''vital dynamicsn-play a rate. The introduction of thwe factors expands the range af gomible revoIutionary/epidemic trajectories?. Indwd, the ixlt;roduction of some rudimentary vital dynamics connects our disetrssion, perbaps surprisingly, to the field of mathematical e c o l ~ o ~ Fbr expasitmy ewe, let us recall the basic epidemic model (4.1). Keeping matters simpk, now introduce a MaXthusian birth rate i&o the susceptible papulation, If p > Q is the birth rate, we obtain the sys;t;em
Students of ma;t;hematicaI,ecology will recognize this as prmimly the Lath VolLerra predator-prey modc31. The prey (susceptibtes) would increm exponentidly if not for the pred&ors (infeetives), who m u l d die off exponent;ially *bout their ""foodsource,'"he prey, Qnee a birth term is introducd, the infwtives and suscep tiblm may be w n as forming an ecosystem* As shown in figure 4.1, the orbits of (4.6) are closed cuwes in the SI p h a e plane; soluti<ms are undarnped 184sm Murray (1989, pp. 375-76)f 8 Xbait i~ strate~ to collf;r~li the s p r d af rabim in foxr?s,for =ample, is to thin the susceptible fox population to a level belaw the eriticd thrmhold demity acrm some swath. lying in &ant of the d ~ ~ e i epidemic n g wave; to create a 6re:c?bre&,as it were. Far an andflic approxinnstion for the minimum width of a rabies control break, as sueh b m i m are knwn, see Mumay (1989,pp. 681-89). fS2fftdamp& mriants rmulb if pS is rmh& simply by p. See Elailti?y (1975,pp, 135-37).
Predators
Prey
The political interpretation is straightforward: extending model (4.1) to include rudimentary vital dynamics ('het births"), as in (4.61, society may experience periodic cycles of rewlutionary discontent (the infection) analogous to cycles GharwLeristie of basic model ecaystem, and rt3current epidedcs.[=l InteratinglLy; Gaodwin's madet of tht? clitss strug1.e b e w e n workers and capitalist;^ is of e x ~ t l y the s m e f"orm.f8"1 In the physical sciencw, it is often of great interest to identify entities that are ceased by the dpamical system-funetions that are comtant along sy-m trajectories. As discussed in lecture 6, such functions are termed Hamiltonians of fsqOn cyclical behavior in - e g i d e ~ cand waI~&calmdeb [sspwificisily,the inhrw%eclr d a r might (1989, pp. 192-211) and H u a g sfld MerriII (1989). [&4j~oadwin. (1967). The mdei is3 e!ewky d k u d in brenz (1989).
78
Nonfinear Dynamics, Mathematical Biology, and Social Science
the system. Interestingly, this model admits a Hamiltonian formulation. Under the transformation p = In S, q = In l,system (4.6) becornesI85J
Separating variables and integrating the equation
dp
p-reQ
dq
rep
-5-
-7'
we obtain reP-7p+reQ-pq=c,
where c is a constant determined by initial conditions. The left-hand side, constant along trajectories, defines a Hamiltonian:
It is easy to verify that (4.7) is, equivalently, Hamilton's equations:
A socid interpretation of this energy-like Bamiltonian-a
constant of r~lutio~asy/counte~~revolutionitry motion-wouldb-intrigdg. This purely o s c i l l a - n e u t r a l stable-behavior of system (4.6) breaks down when the assumption of Malthusian growth is abandoned. Indeed, the model is structurally unstable in the language af lecture 6. Assuming some carrying capacity for the susceptibles, a more plausible system would be
where K is the carrying capacity.. [ 8 s l ~Verhulst (1990,pp. 21-22). For further generalization, and an elegant discussion of HadE. tonians in this cantext, see Samuelson (1971).
Once more, the int~oduGtionof' spalial diffusion terms renders (4.10) into a
reatet ion-diEwian system. f n one spatial dimension,
And, as in the e m of the generalized epidemic model (4.3), traveling wavefront soltrdions mist. Assuming for consistencyb sake that it is again the infeetives th& diffuse, let D1/Dz = 0. Then, under the usual boundary ~onditions,[~~1 a minimum wavefront ~ e l a c i ht i~n~t @ven by
An inl;er&ing digerenee betwwn (4*11)and (4.31, however, is that this wavefront solution can be approach4 in. an. oscillatory or monotoxlic fwhion depending on parmeter vaalues.fB7J
THE REMOVERS The modcsls thus far prmented plrtee no upper bound eta R(C);thme is an unlimited removal czlpaeily. One refinement is the imposition of same upper bound on Rlt). Second, the counter-revolutionary elms (the elite and its gendarmerie), or "public health authority," b not explicitly represented. Clearly, since the power elike is doing the removing, it desemes a place in. the model. Let us denote this group by &(C) and, for simplicity, assume it ta be constant at some initial level Eo. Then, i f G denotes thq upper bound an removals, a slightly refined version of our hrst (Kermaek-McKendrick) modet might take the hUowing form:
Nonlinear Dynamics, Mathematical B i o l ~ g and ~ Sscial Science
The basic m m o d (repression) rate parmeter y has 'bwn supplanted by aEo, where cu > O if R(t) < G and a = 0 o t h e m h . In other words, a ctquals a positiw equals zero once the Gut% constant so long as there is ""room in the Gul;sg,'"ut is full (or, more generally, once the removal capacity of the State is reached). We saw that, under the model f4.1), with no upper bound on removals, the infection ultimately dies out, At the other extreme, if G = O (no removal capacity), the tramfer of susceptibles into the infective c l ~ ~ - t h &is, the sprt3axli of the idectian-is in c;@ectgavernd by the sysgexn
dt
Since tfie total papulaeian P and the elite E are constants, we have S(2) -t I ( t ) + E. = PO,or S ( t ) = [Po- Eo]- I ( t ) , and h n w e m wrik
a Bernoutli equation whose solution,
is precisely the Lo@stichnetion, This same hnetion, interestingly, has been found to govera the digusion of ce1.lain befinoliogicd innovatiom-'%whnoJaw epide~es."@88f The clmfy related discrete Logktic map b,af course, the prototypical chmtie dyamical system. h d , in, fact, the question whether epidemics ehiibit chastic behavior is under actiw st~dy.l8~1 Such canneetians, it seems to me, are potentially quite intermling.
IMPERMANENT REMOVAL Thus far we have wsurxrd any removals to be permanent; the Kermmk-McKendriek flow is from S to I to R. In fact, even when the state" removal capwity k unbounded (a reasonable assumption in most practical cmtts), removal need no$ he permanent. Removed individuals may eventually reater the susceptible pool-= when we rwmerge from a stay in the haspitd, or the prison, a the case may be. Extending Models capturing this are, for abvious remons, termed SIRS modets,@Ql ls8f see Mmsfield (1C)Cil).See also Cam11i-Sfon;a and Feldman (1981). f891~ee, for example, Olmn and SchaEer (1990). For ai rigorous mthernatical definition of the much-abud tern "chws," m, for sample, D m n q (1989). [ ~ X principlplr?, R one could return Erom prison to the infative, rather than suseplible, pool, producing an Sf Rl wdel.
Kermwk-McKendriek, we obtain. the system
where y = aEo from the previous (political) interpretation.lQ1lWe are most interested in the positive equilibrium. Clearly, from (4.18), $ = 7/Pf and we can write
Since, in this model, population is m & m t at N,we may witc? R = N so that the system, in $1-space, becomes
-S -I ,
Denoting by F the vector field (fi,f2), the Jaeobian at any equilibrium Z = ( S ,f)
At the positive (or interior) equilibrium of interest, S = ?/P, so that P$ - y = O
Clearly; the traee is negative and the determinmt is positive; so, as reviewed in lecture 6 , this quitibriunn is stable. With T the trace and D the determinanc of D F ( f ) , its eigendum are:
Tbe resal p;art;s are negative, but we m y fitwe nonzero imcxg~h~wry pms, and spiral convergence to the equilibrium, as shown in figure 4.2. ~ b i analysis s pwdiek MeMein-Kabet (1988,pp. 2 4 H 9 )
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Nonlinear Dynamks, Mathematical Biology, and Swiai Scieme
FIGURE 4.2 Spiral Sink,
Source: Based on Edetstein-Kashctt (1988, p. 248).
The political interpretatbn of spiral eonvergrtnee w u l d be periodic outbursts that atre reprmsed to it steady level of endemic positive discontent,. Now s u p p o s e m a g e d a d e n experiment-that you are a Mwhiavellian a c i d enginwr and want to ensure that revolutionslry ideologiw pose ~o thre& to the establishd order. One ins"trument is indoctrination. The parallel. $0 inocula;tion may be instructive.
HERD IMMUNIW AND DECENTRALIZED TOTALIITARIANISM Staying with the same model, we see from (4.20) that for f----the endemic levelto be positive we must have N - $ > O which is to say N p l y > 1. With May, we define & =. N p / r ta be tht? intrinsic reproduct;ive rate crf the disem.fg" represents the averii?ge number of secondary infections caumd by irrtrodueing a single infeeted individual inta a host, population of suseept;ibles.'"g31 The epidemic thrmhold condition is thea simply .& > 1. Ig21~aS" (1983). &J is dsa termed the ""reproductive number." 1 9 3 f ~ d e i s t e i n -(2988, ~ a ~ p. 241).
Suppose, however, that we can vaccinate some frxtion, P, of the population (still eanstaa at N),The fraction immunized should be big enough that the unimmunized fraction (1- PIN is belaw the3 threshold*We will then have achieved "herd immunity." Procwding, we require
from which the required inoculation level is given by
For variaus d i s e ~ e swe , have table 4.1, TABLE 4.1 Immunization Levels P Required far Herd Immunity: Various Diseasea
Inlwtion
Whooping cough German memltzls Chicken pox Diphtheria Scarlet fever Mumps
Location and Time
-%l
Approximate Value of P(%)
Developing countries, before global campaim England and Wales, 195+68; U.S., various p I a e ~ 1910-30 , England and Wales 1942-50; Maryland, U.S., l908-lY England and WaXes, 1919; Wmt Germany "CT.S., various places, 1913-21 and. 1943 U.S., various plwes, 19Tte-47 U. S., variaus places, 1910-20 U.S., various plwes, 1912-16 and 1943 Holland, 1960: U.S,, 1955
If we now recover our politic& analogues, we have the required level of social indoctrination given by
An in~rea~se in domestic police (&) or in their eEectiveness (a)will allow the elite to do b s indoctrination, while an inerem in j9 (the infectivenws of revoludionary
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Nonlinear Dynamb, Mathem&ical Biology, and Smial Science
i d e ~ will ) lequire increaed prapagandktic effort, all of which maka a wrtak amount of sense. A somewhat darker reading is invited by the rearrangement
If indoctrination is very high, even the most compelling revolutionary critique will pose no threa;t to the mtablbfind order because too few will li&en. Once herd immunity is whieved, the system is really on autopilol; a type of deeentoalized to talita~anismis possible.
Regarding another crucial variable, reprwsion, there we many subt1&ia that one might wkh to capture, For ple, in some situations, reprmsion has a deterrent, or "chitling," eEect, whib in other sliluations, it simply infjtmeshostiliw toward the regim. Indeed, revolutionaries have been knawn to provoke reprwsion with precimly this infiammahry aim. Morwver, a @ven population may gip &am one &'response made" to the other in the wake 0f pa&iculm incidents, With these thoughb in mind, consider the faElwing extension, which was pleopomd by Jean-Pierre Langlak.
The extensian consists in adding a tern, &RI,to the preceding; model. hpression (removal) has a deterrent effect if S > 0. In that ewe there is a Aow out of the idective, and into the susceptible, pool, The idea is that the awareness of removah (vmishinl;s) has the @Be&of drMng some idectivm out of the revolutianq movement. If 6 < Q, repremion has precisely the opposite eEmt. Histx>ricaUy3both rmpome mod= are obsemed. Ig4I~ccordingto Hoover and K w ~ l m k i reprmion , r d u c d dimnt by the anti-Nai movemnt in Germny (emIy 29403), the human rights mwement in the former USSR (19?&198Cls), a d the demoeray movement in Burma (late 198k). Here, 6 W= positive. b p r m i o n had the revem eEwt of increwkg d m n t by the mti-Vichy movemen6 in France ( w l y 294Os) and the anti-ap&heid movement in South Africa (late 11$809). In them E G W= negative, 1x1Hower ancl KmaImki f 1W2) is a revim of the literature and mferenm.
hgarding thresholds, we see from (4.27) that the revolution spreads when the follawing con&tiaa is met
(i > 0 )
With 6 = 0, are reeover the clmic threshold condition (4.21, w&h aEo rts .g., just a in (4.13). Xf 6 r 0, however, more susceptiblm are nwded; repressian deters so it is harder to initiate the revolution. If 6 < 0, fewer susceptibles are required; repression simply fam the flames- Xn sucb case^;., a ruling elite is best advised to refrain horn all action. Equivalently, revolutionari~in these situations should bend every ego& to provoke brutality. The parameter, 6, is what; dbtinguisbes "a revolutionay situatbd"am others. We can study the d m a m i c ~of this model iix $1 spwe. With a bed total population of IV and k e d elite of Eo, let k = N - Eo. Then R = k - 5 1. Substituting this into (4.26) and (4.271, we obtain ..-
In the S1 plane, there are two equiIibriags51: (k,O ) , and
Far illustrative parameter vdues, the first is a saddle, the second (4.33) is a spiral This, and sink, and the Wa are connected by a heterodinic orbit (sm le&ure 6).rg6J other LrSt;jmfrQria,are shorn in figure 4.3. 1nterpret;ingthe heteraclinic orbit politically, the equilibrium (k,0) represents a world of idwlo@cal purity-there are no r~valutionarim,But, that society is "ripe for revolution;'' (k,0) is a sa-ddle, and heace unstabfe. The slightest subversive agitaLion (I@> 0) and the system will run to the spiral a.t;traetorof endemic diseonterrt, , our param&er 6. whose e x u t location depends, eetelris p a ~ b v s cm fQ51~he s~ond of t k m was obtained symbaliedly by IWathemtica. See Wolhrn (1991). isCil~he wlues emplay& are: r = 0.0.1,ib: =.. 100, v -- 0.45, & =. 0.02, a -- 0.04,sad & ;= 20,
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Mantinear Dynamics, Mathematical Biology, and Sorziai Science
FIGURE 4.3 Hetaroclinic Social Orbit
EXTENSIONS In point of faet, oT course, the class of" in&vidu& subscribing to the elite ideology is not constant in size (at Eo).There is recruitment of susceptibles into the elite in addition to recruitment of susceptibles into the revolutionary class-a "struggle for the hearts and minds" of the susceptibles. And, equally impoftant;, there is ofien d i r s t eonfiicit between. the power elite and the infwtives; there h o\llright civil war, A more fully elaborated syStem might take the form of an ecology in which taro warring (see lecture 2) and diffusing predators feed on a prey species. The resulting reaction-diffusion sy&em muId be complex mathematically, and well worth study It wodd also be interesting to attempt a formulation of such a society usiw cellular automata (perhaps as a generalized Greenberg-Hastings modeliQ7l)or agents.t981
CONCLUDING THOUGHTS Clearly, social dynamics of Fundamental interest can be generated by simple models af the sort I have advanced (wry tentatively) above. Explosive upheavals; revalu-
< p), or that begin to tio-ons that fizzle for faek of a remptive population (e.g., spread but are reversed and crushed by an elite; longer-term cycles (undamped or
Lecture 4 damped) of revolutionary action; even endemic levels of weid diseontenl, and trweling wavm of revolution are aH ewily produced in nonfineizr models. T h ~ modeh e have the additional attractions of reBecting sensitivity to initial conditions recently in Emkrn Europ and of unifying in a few variables diverse totalitaria rule and rewlutianary aetion. Findly, wen rather subtle social ""bhreation poixrds" emerge, where repression abruptly changes from being stabuizing (S > 0) to being inAammatary (6 < Q). Dynamical, andogies are of theoretical value precisely in their pawer to illuminate such nonlinearities, to parsimoniously suggest generd conditions under which explosim, dissipative, cyclical, even cbmtie social dynamics are likely, and in so doing, to foeus empirical attention on the parameters and relatiomhips that, in fact, matter most. Indeed, without some theoretical framework, or model, it is often. quikite unclear wh& we should try to memure!
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LECTURE 5 A Theoretics Perspective on The Spread o Drugs
This lecture explores another social process of considerable interat, the spread of drugs, and is divided into three parts, In Part 1, a simple dynamic mod& of a drug epidemic in an ideaEzed community is built up ham bmie wurrtptiom concerning the interaction of subpopufations-pttshers, police, and no$-yet-ddicted residents of the communiz;y.[g@~ The model combines demerits oT the epidemic, ecosystem, combat, md tzrms rwe modeh dkcusmd above. Quilibria of the rmalting dynamical 8yskem are located md c1wified using took of f i x l e ~ i z dstability analysis, Dajwtories are pfottd for a set of initial conditions. In Part II, a spatid-reactiondiEusion-vwiant is prewnted, Then in Part 111, supply, demand, md.price considera;tiom we intrrrducd; a;sen%iaf,and perlralps caunterhtuitive, relationships 19910b~0wgytpmieuIar dynsnnies dt?pend on: p&ieular drugs. No part;ieular drug is mentiand here. We imgine em idealized drug thag is totdly md imevemibly ddictive after some smI1, but hard to prdiet, number of us=.
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hlsnlinear Dynamics, Ma;thern;lticaf Biola~y,and Social Science
betwen legalization, price, and crime are revealed. And in this light, the role of education is discumd.
PART I, A DRUG EPIDEMIC MODEL, We begin with definitions and a brief discussion of variables and parameters. At any time, the populiltion is msumed to be divided into four disjoint groups,
S(t): The aonaddicted and susceptible population. I ( t ) : The population of addicts, all of whom are assumed, in this simple model, to be pushers, The variable f is used because this group plays a role thak is mathematically analogow to the infective group in epidemiologyf a pilraflel we shdf expbit. L(1.1: The law enbcernent, or policc;, force, whose sole function is msumed to be the arrest and removal of pwhers. R(t): The arrest& and removed, or imprisoned, population. For this simple model, removal is msurned to be permanent,
In ddition to these variables, a number of parameters are involved.
p: p:
y: a: b:
The rate at which a corrtwt beween a pusher a ~ da, susceptible produces a new addict/pusber (price dqendence is discussed in Part XI1 below), The natural grodhi rate in the susceptible pool, ss youths come of age, sa,y; The r&e at whi& a contxt between a pusher and a cop raults in removal of the former. The rate at which an inereme in pushers incream the growth, rate in police. This variable reflects social a1arrn.['~~1 The economic damping to which the police g r a e h rate is subject.
All parameters are nonnegative real numberg. Let us SW if we cannot arrive at a plausible model by remning from first principles, noting conxleetions to related phenomena as we go. Pedagogically, the exerche may illuminate the type of resaning thi3Lt o&en goes i a h the csnstruetion of modelis in mathennatieal b i o f o ~ a, field which, ukinnatefy, subsumes the social seiencw, flooi~wicatllytproblem get mare attention when they impinp on the elite than when they are confind to the ghetto. In a more realistic modd, therefore, a would depend on the meio-economic inLa which drug abum, andlor the trim w o e i a t d with it, had spred. Were a is a, comtant.
As the simplest conceivable model, then, let us imagine t h ~ there t is no population $ro&h and no police force. At every time t , the populztlion is constant at N and is the sum of suseeptibtes S($)and pushers I(t). That is,
How do 5 and I evolve? Well, for a susceptible to become an addict/pusher, he or she rnust first come ilnto contact with a gusher, Reeognbing that real societies are hekrogenwtxs and patchy? let us nonetheless follow the practice of theoretied epidemiology and ecology and, as a first cut, msurne hornogenmus mixing of pushers md susceptibtes. The number of coxlt;ats is then taken to be S I . Of course, only some fraction p of contwts produces new addicts. One may think of ,B as the "ust say DO" "rameter. If P = 6, every susceptible says n0 and there is no growth in the ddicted, or "infected," pogool. Xf = 1, the^ every cant;aet produces a nevv addiet/pusher, On these very primitive assumptions, then, the A m out of the susceptible pool and into the &dieted pool is h l y described by the equations
This system ki none other than the most bmic epidemic model, termed an "SP' model since the fiaw is strictfy fmrn susceptible to infective, Now, by virtue of (5.1), we ma;y write S = N - I, axld (5-3)becomes
whose solution is the well-known equation of logistic gro+h. The acfdicted popula, tion inerewes until it equals the entire population; the "epidemic'bhips through the whole of mciety. In faet, there are some brakes on this process. Hewing to our assumption that the drug is iHegal, there is some rate at which puskerladdicts are removed from general cireulation. As a first refinement on our model, let us imagine a fixed police force of size Lo. AS in the pusha-susceptibfe sphere, "Eaw of nnas action" "dynamics are nssumed. There is homogenaus mMng of pushers and police, so that contacts proceed sts Lo1. And, per contact, the removal rate is y.The idea, then, is that, as before, susceptibles flow into the addictedlpushing pool at rate PSI. But, pushers Aow out of circulation and into the '%removed" dclrass at rate yLoI, Since ?Lo is just a constant, call it Q. Then we have the model:
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Nonlinear Dynamics, Mathematical Blolqy, and Saciai Science
Students of mathematical epidedolaw will recognize "c W the ciwie Kermack-MeKerrdrick SIR e p i d e ~ cmodd. It is a threshold model in that suse e p t i b l ~must exceed some minbum level in order for the infected, or addicted, class to grow, Thb is strdghtfomard. To say the d d c t e d clws grows h to say that
which is to say that PSI
- 01 > 6, or: that
The ratio is o&en termed the relative remaml rate of tbc3 infxtian. It fs the werything evenepidemic thrmhofd. m i l e the infwtion u l t h a k l y dies out-ince tually flaws into the removed compartment-it deerewes manotonically oltlly if S < o/P. Otherwise, it enjoys a period of growth-the epidemic ph dyjing out, as shown in &@re 5.X, in which p = B/@, FIGURE 5.1 An SIR Epidemic Model
Now, from ($.S), f5.6), md (5,7)?it is evident that papul&ion b still consea&, ~ i n m
Of courR, population is not generally corntat. There are sa-cdld vitd dmamics, bi&h and death.
As the n e obvious ~ refi~eme&on our efementary model, then, let us assume n& "birdhs'hr entrmts of pS into the swceptibh cohort, where p if the per capita g r a d h rate. N e d l m t o sayf logbtic rather t h m Maltbusim g r o d h k mather pamibility. But, kmping mat@rs m s h p l e M possible, we then obtain the model:
Notice that (5.9) and (5.10) are the classic Loth-Volterra predator-prey model, wiLh pushers as prdatars and not-yet-addicled susceptibles as p r q Predaf;ors would die out (at rate -cl) were there no prey to feed on (at rate PSI); and prey would flourish (at rate PS) were they not consumed (at rate PSI) by p r d a tors. Aside from the origin, this system has as its equilibrium the point (S,i ) = (@/P,p/@),whieh is a ~ e n t e r . fAs ~ ~shown ~ l in figure 5.2, the populations oscillate; the arbits are closed cumes in ehe Sir phast? pIaae.
Popufations
Predators
0.5
1
1.5
2
Prey
Above, we saw thaut in the SIR e p i d e ~ cmod&, the infected ar addi&d elms ultimately go= to zero, as aIf addicts are remwedt. Here-, we have Loth-Volterfa d y a a ~ e sin whieh predators and prey cycle around an equilibrium. C d d it be that the complete modd wit1 combine t amp& md oseiXtat~ry-tendencies in same way2 We will return to this que&ion. llollksdeveloped in lecture 6 , at the equilibrium, the eigenvalues of the Jaeobian of (5.9)-(5.10) we i m a g h q . The quitibrium i;s nonhyperbolic and Iiaewizd stabifiv andysb dcm not apply. %hequilibsiurn its a cent= or a hew; and b w u e the But b e c a w the eigendua are ima& system admits a Bmilts~lianformulation (mf&ure 41, Lhe wuilibrum. is B center or a saddle, Hence, It h a mntm.
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Nanlinsair Dynamics, Mathematical Biology, and Social Science
THE ARMS RACE COMPONENT Above, we defined as the product ?.Lo where L. was some f i e d level of law enforcement or police. But, the police force is not necessarily constant. So, let us relax this assumption. What, to a first order, would determine the size of the police force7 Well, if no one cares about the level of addiction in society, i ( t ) ,there will not be any groMh. Thinking of the p a a m a e r a izs a coefieient of societitl &arm, we might posit that, without any economic damping, the police force should grow as a I , But, as in arms race modeling, id is remanable to wsume some eeanonnie fatigue or damping, under which mtes of growth decf ine the Iarger is the military establishment. ff the damping coeBeient is Et, then the police growth rate is given by h = u l - bL, just as in the Richardson arms race model of lecture 3, and the complete model is ais follaw~:
df psr dt ==:
-T ~ L ,
Notice that the term -?]L, in (5.13) is a pusha attrition rate reminiscent of a Lanchwter combat model presented in lecture 2, so this dynamied system combines elements of the epidemic, ecosystem, arms race, and combat models developed in precediw ieeturw, Before engaging in a linearized stability maty.sis of this dynamicd system, let us briefly trme through the eEect if, horn some time, everyone says no; t h d is, if p = 0. C/early, sin= P --- 0, the growth rirte in the ELd&ct;ed/gusher pool in (5-13) is strictXy neg;ative; this entire group is eventuatly rmoved. That being the ease, (5.15) reduees to
and the police force, too, "ithers away9"a rexonable qualitative result since the apprehension of pushers is their sole ELxnction in this model. Xn the end, we have a poliwless society of nonaddicts and a removed populat;ion of former pushrs. This little thought experiment completed, let us bring to bear some more powerhl toots.
LfMEARlZED STABILITY ANALVStS Assuming all parameters to be pasitive in (5.12)-(5.15), what are the nontrivial equitibria of the system, the nonzero population levels where all derivatives are zero? We really care only about S, I, and L, and a bit of algebra quickly leads to the unique pasitive equilibrium:
Evaluated at this equilibrium (call it $1, the Jacubian matrix of (5.12)-(5.141, which ecologists term the community matrix, is given by
The eigendues are solutions to the third-order ch~rl"acteristic equation Det(J@)
- Aid) = 0,
(5.18)
where id is the identity matrix. Expanding, this characteristic equation is
Equilibrium is stable if and only if the roots Xi of this equation have Re(Ai) < 0. For a third-degree equation,
the RouthHurwitz necessary and sufficient eondition~l~~~l for %(X) < 0 are a1
>Ota3 > 0, and alaz -aa
>0.
(5.20)
The first two of these are obviously satisfied by (5.19)) and so is the third, since
Therefore, the positive equilibrium in (5.16) is stable. There is an endemic level of addiction. Earlier, I raised the question whether our simple made1 might somehow manifest both the damped behaviar of the Kermwk-McKendrick SIR epidemic model and fla2lSee,for example, M m y (1989,pp. 702-04), or May (1974,p. 196).
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Nonlinear Dynamics, Mathsmllltkal Bislwy; and Soeial Seieme
the cyclical behavior of the LotkaVolterra predator-prey model, each of which is a special case of (5.12)-(5.15). A canonical bebavior combining these would be a spiral approach to our positive equilibrium. And this is precisely the behavior we have, as shown in figure 5.3, which - - offers a small gallery of phase portraits. Here, the equilibrium happens to be (S,I , L ) = (108,2, 12).11"l FIGURE 5.3 Drug Modet Orbe and Sotutions.
Mote: Here z is our S, g is our X, and x is our .L.
A natural extension is to add spaee. As demonstrated earlier, t h k can be Wcomplished by appendinef;vmious dzusion terms to an underlying dynamic model, yieldinpf a sa-called remtion-diffwion system. One such generalization is offered next.
PART !I. DRUG WAR ON MAN STREET": A NONLfNEAR REACTION-DIFFUSfON MODEL The population is cornprig& of three subgroups, who~enumbers and spatial digtributioxls evoke over time. We ima@ne th& event8 unfold an on+dimensional inkrval-it "'street." h t us define S(;e,tf, l(%,t ) , and L(z,t ) as the susceptible, infecdive, and law enfurcennent levels at; &rwt posilion ;e sbt time t.g1WJ Denotinf: these functiom (of z and t ) simply as S, 6, and L, the generalized quations are as follows: bS d25 == -PSI 4- p S 3- &ss-,
at
O%Z
Ig~oringd l diffusion and cross-dift"usion terns, the first two equat;ions are exwtly as before, The third equation has been refind slghtty. The Ricfiardsoxlian damping term (-bL) is retained, but the first expression is now ESIL rather than the previous aI. The idea, recall, is that the poliee force grows with the level of mcietal alarm at the drug problem itselE This level of dmm is assumed to be a function of arrests of which the (tax-payiag and police-buying) susceptibies are awue. Under our normal msumpti;an, the arrmts are proportional to PI;, and suxeptible awareness grows with exposuxe to these arrests, hence further multiplication by S, yielding the overall term [SPL,[1051These, then, are the reaction k-i~eticsin the reation-diEusian system (5.21). n m i n g to the diEwiw process@, the simplest is the susceptible c m . Here, the term bss(a2S/ax2)is added, as in the models of the previous chapter, indicating that the susceptibles-while int;ersctim with other groups-diBust3, Analowus and police diffusion terms appear in the equations for infectives (611(a21/a22)) (6LL(a2L / a x 2 ) ) .All t h e ~ ediffwivities (6,) are positive. However, the infective and police equations are more complex than this, In the paliee equation, there is dso Pa4f ere, we will not drmk the remavd
(i.e., ammted) group c?xplicitly. [ l D S 1 ~mume e again that the exposurw occur through homogenmus mixing, or mkinetics.
atian
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Nonlinear Dynamics, Mathematical Biology, and Social Science
a cross-diffusion term (-61L(a21/a22)), indicating that police diffuse toward infective eanwntrisrtioions; they @&gage in " " c i m + t ~ s , "if you will. In turn, infectiws (i.e., pushers) cross-diffuse in the direction of susceptibles (-6sr(82S/a~Z)), and cross-diffuse away from police (6Lr(a2L/@z2)).I further m u m e that fiLJ > bsl: a pusher would rathm amid arrest than convert a susceptible to a new drug user. With all constants the assignment; of initial spatial distributions for the subpopulations is all that remains to specilFy the model. Imagine, then, that everything transpires on a. &rwt 12 blocks long. At time zero, the susceptibles occupy the middle four blocb, and we 1000 strong at every point, Up at blacks 8-12 are the infectives, initidly numbering but 100 ,zt; each point. And wagt down at block 1-3 are the cops, initially a-t t o k n levels of 25 per poht, We trwk the spatial. evolurtion of each a o u p over 50 time intervals in fiwre 5.4. FIGURE 5.4 Drug War Reaction-Diffusion Model
Tbe susceptible, infective, and police evolut;io~sare shovvn in the left, middle, and right graphs, respectively.[l071 At to the levels and positions are as noted above. Haw do things evolve?
A SPATIO-TEMPORAL STORY Seeing that there is a large concentralion of susceptibles and few cops d a m the s t r e t frarn them, the infectives cross-di&se to the center, Many susceptible^ we conve&ed into ixzfectives, so the smcept'xble populat;ion falk and the infective one rises, now swelliq nrith "converts" "into the micldle blocks. This buleng problem,
["l~)~ere,theva-tucr?s@re:p = 0.005,p=0.5,7=0.03,t = 0.0001, b = l.O,bss =0.03,61f =0.01, = 0.02, liLz= 0.006, Ss1 = 0.001, lilt = 0.006. ifa7)~hese were genaated in lGiathematim (Wolfram, 1991) using the Numerical Method of Lincs. X thank &be& Axtell for his misLance.
Lecture S
FIGURE 5.5 Drug War Reaction-DMusian Model: Overhead View
however, inspires a reaction, in the h r m sf dramatic inerernes in, police, who crossdiffuse from their initial barrwk at the end of the strmt into the hem%of the problem in the center, This surge in poEic vident; in the peak of the rightmast graph-literally scoops away the ideetive mamd. By t = 40, there is hardly a problem. Hence, m before, the polke "ither awaySb&er%h& p a i ~ t lewing , the suseeptiblets to coxlt;inue in their untroubled dausion, as shown. An owerhead view of the same pr~cessis oEered in figure 5.5. Here, the higher the numbers at a point, Ghe lighter the s h d e . We c m clearZy see the surge of pushers, followed by the police response, the hollowing-out af the pusher mound in the center, a ~ the d withering lzsva~yof the palice. A nonlinear reaedion-digusion model allows us to generate a plausible spadiotemporal story of basic interest;.i1081 i l o s l ~ y point here is tZI& there exist parameter wXuw and initial conditions under which the spadio-temporal stow emerges, A aperate study would marnine the robustnw of this result under a wide range of pasameter and initid valua.
3 oo
Nanlinaar Dynamics, MathematicalBiology, and Social Science
PART If!. EMSTtCXTIES AND THE DEMAND
COMPONENTS OF
Qonomkts will have recognizd one glaring oversight in the models thus far developd-the price of the dug is not repraentd* Clearly, P-the rate at which a contmt b e w e n a pusher and ai. not-yet-addieted susceptible raults in a use-z~ likelihood s , that som~fonewho is not yet a& w e n & on price. GetePis w ~ ~the d i c k d wiU ""just say no'bhould rie;e (P should fall) with price. So, for that group we -urn@ @ ' ( P )< 0 . (5.22) I f we hfdbw w u m e price to be determined by supply and. demand, then, in prineipie, the effect of drug interdiction operations on h s t uses of the drug can be gauged, as shown in figure 5.6. Interdiction shifts supply in (further discussion below), inc r e a a the equilibrium prim, reducm @, and in principb changes the q u i l i b r h level of drug consumption (from Q0 to Q'). By contrwt, wiLhin the addicted population, the quantity deatnnded k not sensitive to price; the demmd came i~ a verticd fine, as shown in figure 5.7. FIGURE 5.6 Interdiction Increasing Price
Lecture 5 FIGURE 5,"irddkted Demand
price effect on the Zrt sueh a market, supply interdiction merely incm equilibrium drug quantity. The result is simply more street crime, as addicts do whatever is necessary (e.g., rob and loot) to come up with the price. Now, figure 5.6 assumes the market to be eomprM only of not-yebaddicted susceptibles, while figure 5.7 arsumes it to be comprised only of addicts. In fact, both groups are present. These two components of total demand are represented in figure 5.8, FIGURE 5.8 A Drw Wlai&et
*I Q2
Nonlinear Dynamiw, Mathematicat Biology, and Swial Science
The diagram encompnsses an addicted pool, with its dislinmislhing feature, a vertical demand curve (%)oddiets), represeneing complete inelasticity of demand with respect to price. The addicted group h= ""f~tto have" QQ,and that" that;. The Da,, cuwe represeds the nonaddieted population, with its downward-sloping demand curve.. Here, price does aEect the demand by potential first users of our drug, Total demand (&,tal) is the horizontal sum of these two components. Finally, we imagine SO to represent the current supply curve in this economy. Then the equilibrium price and quantity in the market is (p, QO). Let; us use this fiamwork to campare the short-term eEwt of two supplyoriented policies, returning later to the longer-term hsue of demand adjustment, The twa policies are, of course, interdictionlprohibition and legaXization,t"~l Assuming (perhaps generously) that interdiction is techniedfy femible, it s h a s s u p ply inward to S" Legalization---whose aim is to drive, underbid, the cartels out of business-would shift supply outward to S&.
First, under interdiction the equilibrium price rises to P< Since addicts have "got to have it," they will do wh&ever is xlecasary to raise the money required to support, their habits, and we can expect an inereme in strwt crime. Xn the normaddicted group, the quantity demanded falls as price increases. So, interdiction produces fewer first users but mare c.rime.
Legalization shifts supply out to SL. The equilibrium price falls to P&,so addicts need less money than they did before (at So) to satisfy their habits; not all of the erirnc; in which they had been engaged would now be necessary, and we would expect to SW street crinze deerese, New users, however, may be expected to increm (and to increase their consumplion) as f?xperirnentationbecomes cheaper (and, ohiously less risky legally) under legalization. So, this policy produces less cT.i;nze, but more new users. All of this is enicapsulated in the purely heuristic diapam in figure 5.9. The debate is, predictably; wide open becaua no oxrc3 real& h a w s much about this cuwe, about the factors underlying it-price elwticities, propensities to engage in crime, eEectiveness of interdiction, and so o r about the social welfare do be associatd with diRerent points on the curve. As a result, the most fundamental issue is open: Do we want d m g prices to .rise dramaticall@or fall dramaticallg? But, at least the analysis reveals those fators about which advocates of the diEerent policies are, in fact, disapeeing.i1lo"l modest claim, to be sure. f"@l~ora thorough historicat dimumian and policy analysb, see Star= (1996). i1"1 ""Just swing no" rduem crime indirwtly bsause DLOLal shi&li left, rdueing the ~?quiIibrium price, and hence the incentive to engage in ~trmustcrime,
lecture 5
FIGURE 5.9 The tsgatization-Interdiction Trade-off
EDUCATION AND P FOR THE NONADDICTED POPUUTIQN further point concerns demand among the nonddicted susceptibles, where the likelihood that; somwne will "ust say no" probably depends on prices. While the incorporation of supply; demand, and price into the model dlows us to connect, through P, the "epidemic dynmics" "veloped earlier to the alternative policies, maMng P a function salelg of price is not completely satis$ing, for the following reascln: Imagine an a11 out, %wefront"war on drugs" that simultaneously cut supply through interdiction and cut demand (here in the new-use senm) through education. The equilibrium price could be left unaffected. If P were a ftmetion solely of price, then it, too, would be unaffected, and the drug epidemic dpamics would be exactly M they were before, which swrns implausible. One would therefore want ,& to depend on price and education ( E ) in some (perhaps pasametric) way such that
A,
Many "by" "candidates for @(P,E), could be constructed h r sinnulation purposes. But, basic empirical, work is essential here, W elsewhere. Indmd, it would =em that educatLian Is the prime way to avoid a very disturbing long-run scenario under legalization.
IQ4
f\loniinear Dynamics, Ma%i~?matical ~ioiogy,ancl Sociat Science
LEGALIUTfOPI, EDUCATION, AND THE LONG RUM Speescally; tz potentially mrious problem is that with repeated use over t h e , the price elasticity of demand for individuals initially in the nonaddicted pool will go toward zero: they may bwome ddicds. FIGURE 5.10 Legalization and Lang-Run Price
In turn, we c m expat inerewing levels of ddictim (a) to inerewe the total qumtiey demanded at any price and (b) to reduce the price elastic@ of aggregate demand. The h$ddemand crxwe, thus, would be expected to shift righward and bemme snom vedical, W shown in figure 5.10. Counterintuitivelty, in, this case the equitibrium price could rise despite increased supply. That is, we could have (PL> PO),as shown. Street crime, driven by rising drug prices, could in principle actually inereme under legalizadian! The magnitude and direetion of price changw, it mmt be ernphwhed, would b e p a d on. the s u p p l p r ~ p a mdo iegalization, the rate at which new users become addicts, and sa far(;h, But, the obviow way do undercut %histwt: of watution-whose liklihaod X do no$ cldm to &iurr&t is dueation- The aim of education, in a dwhnieal eeonorxzic seme, muXd be to "flatten" the individuaik iindiEerence cuwes ( b e ~ w n drug wn~umptionand engagement in: other forms of rtrj~reation)such that there is no pokt of tangency betwen the indigerenee curve and the budget comtraint,
'Leaving only the corner solution, "just say no," as fewiibe.tlflf The production of horizontal indifference curve8 for fur coats is the d m of animal rights wtivists, for exmgle; there is then na pair of positive prices-for real arrd imitation hr-at which the indigerence curve is tangent to the individud" budget constraint, and he or she "use SW 110' t;6 fur.lllzJ In, any event, it swms char that, if Iegalizatio~is to avoid the long-run problem s u e d above, it may be necessav tO increase educalclon, vefy substa~tiafly
It should be emplrwized that;, in addition to all sorts of implicit assumptians concerning the? factors dhcumd above, positions an legalization reiimt hndmental a-titieudes on the appropriate role of government in. remtating individual: choicm generally, a crucial question not addressed here. Obviousl;y, this thwretical exercise does not purport to resolve any policy issue. R,stther, iil; to focus the debate by helping to identify the empirical imuw that d w w e highest priority, by encouraging aplicitness in the statement; of assumptions, and by oEering very prelhinary, but testable, made18 of the dynamic iXlf;erwtionof care variabl?~. Finally, fi-om a purdy irntellectuaj standpaint, the discussion sugesds the relevance to social science of scrtemingly distant areas like matlkematicaj, epidemiiolom and eeasystem modeling. And it tries to illustrate how simple wnjlinear models are built and axzalyzed. E1l1I'rhf: id= that the individual will comume where the indifference curve and blidget con~traint; axe tangent is develop4 as follows. W inragine that an iirtdividud berivm utility from the consumption of quantity qd of drugs airtd quantity q3 of some alternative commodity, and that total utility is a function (with all nice behaviors) of thaw, %(gd, qS). On an indiEerenee, or isoutility, = 0,memkg that the slope of the indBerence came, dzl r 0,m that ( i ) (au/aqd)dqd3-(du/aq3)dq, cume in (Q, qS) spwe is given by (ii) (&d/&3) = -(@zl/bq3)[(bu/@qd). But i f the individual mmi m i m utility subject to a budget constraint B = Pdqd P,qS, then a t the constrained ( i i ) must equal: the price ratio, making the indiBerence curve tangent to the budget constraint. To see this, form the Lapangian for the comtrdnsd problem: L =; %(qr,qz)- X j B - &(ld - F)3q3). The first-order conditions far a maximum being (aL;/Bgd) == ( a f ; / d q 3 = ) 0, we immdiately obtain that f&/aqd) = X& and (au/i)gj)= XP,, making the right-hand side of (ii) equal the price ratio on division. Qbviousliy, for the tangency point to be unique, the indiEerenee c u r v a must be strictly c onva, which is among the tzlthaviora of a nice u-function. il1211ndeed,one might go farther and argue that the goal is to ensurethrough education--that indifference c u r v a for harmful commodities are positively sIoped. In other words, people wouId have to be compensated with higher q u a n t i t i ~of some other good t o induce them t o consume the harmful commodity I thank Steven McCmo11 for this idea.
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LECTURE 6 ntroductionto Non inear Dynamical Systems
In this lecture, I want to collect some bmic, and very powerful, results from the quditatiw thmry of nonlinear autonomous diEertmlia1 dynamical systems, primarily in the plane. In a field as vast as nanliaest-r dynamics, any essay of the prest?nt len@h must be selective. In this ease, the story begins with liaearized &ability andysis for hyperbolic equilibria and proceds t o develop s o w diagnostic tools for nonhyperbolic ernes (including the use of polar coordinattes, Lyapunov Eunctions, and Hmiltonian formulations). The distinctly nonlinear fienomenon of the limit cyde is then discussed and Hilbertk-&ill unsolved-16th Problem is stated. The Poincar&Bendkson and Hopf Bifurcation Theorems are presented, as well as an introduction to Paincar6 maps, which beautifully connect the world of continuous system (Bows) to that of discrete systems (maps), Tools for precluding periodic orbits-the Bend n and Bendhan-Dulw negative tests-are then pre8ent;ed and applied to a Kolmogorov system which, naturally, proedes a forum for Kolmogorov's Thearem on cycks-. Powerhi as they are, none of these methods give much insip;ht concerning k m the local equilibria and limit eyelw fit together glabdly-in the phase plane as a whole. Irrdrirx theory penetrates dmply into this quest;ion, to reveal topological "con~ervationlaws" of great interest. 1 present some of the hadamenLa1 rwults in this area, and give an index theoretic proof s f Brouwerk famous fixed poixtt dhmrern on the disk. Extending these ideas
108
Nonlinear mnamics, Mathematical Birolwy, and Social Science
ta closed surfaem (tw-manifolds) like the sphere ibnd torus, the lecture conclude8 with an informal pre~mkationof the m1ebrat;ed Pairreclr&Napf Index Theorem from diRerentiaf.t a p o l o ~ ~
NONLINEAR AUTONOMOUS PUNAR SYSTEMS We cornider the nonlinear autonomous systernf11131
where fi and fi are C' on 'R2. The system (6.1) defines a vector field F = ( f 1 ( 2 1 , 2 2 ) ~ f 2 ( 2 1 , ~ 2 ) ) fmm R2 to R2.Recall that the Jacobian of F at a point
In the linear homogeneous ease where F(%)=: Az,DF(z) iS just A. One major difXIerence bewwn linear and nonlinear systenns in the plaae is that the latter may i1'31~uch of the orbit theoq dmelopd below r w b on the wumption that the nonlinw firstorder initial value problemAg/dz = f ( z , g), f continuous on SZ = fa,bj x [c,dj, y(zo) = go-hw a unique mlution y, defined and continuous on a cl subinternal of [a,bj containing a;.a, A basic thwrem is that if f is Lipf~chitzon R, such is the case. A full dkcussion wouXd rquire development of metric s p s e raults quite foreign to the rest of this may, notably (a) that the space cfa,61 of functions continuous on an intern1 fa,b]--the space that would contain m y mlutlon-is complete with metric p(s, v> = m m
LEIa,Y
- y(gZf;
and (b) Banachk thwrem that a eontrmtion mapping on a complete m&rie space a unique f i x 4 point;. T h m theorem in hand, however, aistenceuniquenm is direst, Fkst, one obmmm that y is the unique solution to the initial value probliclsm above if m d only if it is the unique fixed point of the i n t e p d operator, P : e[a,bf efa,b] defined by
The proof then consista in eablishing simply that if / is Lipchitz; on Q, then P (for Picturd) is a cantrm%ionon e[a,bj. As Kreyazig puts it, '"the idea of the approach is quite simple: [the iaitiai wjue probbmf will be eonve&d to an integral equation which definm a mpping T , irnd the eonditione of the thmrem will imply that 2" is a contraction such that its fixed point becomm the solution af our problem." Our P is just Kreyssig's %. For detailed proofs of dl the many s u b s i d i q claim involved, me, for aaxnpb, Krqszig (1978), Naylor and Sell (1982), Waltman (1986),Groetseh (19801, Gamelin and Grmne (1983), or Marden (39741,
have multiple equilibria whereas the linear homogenmus system & == A=c hats the origin as its unique equilibrium. Suppom no-rv that; = (21, z2)is an ewilibrium of the nonlinear systtem (6.1). Assuming fr and fi have partial derivativm of atf arders on an open set containing 2, W can. expand ewh in a Taylar serim about 3. Retaining only the linear termg, we obtain the so-called linear variational equations:
Since (Zl,Zz) is an equilibrium of (6.1), we have fi(51,Z2) = f2(z1,z2) = 0. Defining the deviations yl = (sl- f 1) and yz =t (z2- z ~ ) (6.3) , becomes
the familiar linear hornogenmus problern. If, far apositowy ease, we assume D F ( z ) to have disCinet real namero eigenvalucts, Xi and Xz, with (perhrce linearly independent) eigenvectors vl and e,the general solution of Ctj.4) is
Given the vector initial condition y(0) = 90, we can determine the c-va1ur;s; indwd, with the matrisr exponential, we abfain the succinct form:
Uou will doubtless recall from previous tnrork that if both eigenvalues are negative reds, or have negative real pmt in the cornflex (conjugate) c=, then the origin is a globally ~ y a n p t o t i c d ystable equilibrium sf (6.4).ffi4If both eigenvalues are, in fact, negative (positive) reals, then the origin is a stable (unstable) node. fn the complex ewe, if real parts are negative fpositivct),then the origin is a stable (unstable) spiraL It is a center if the eigenvalum are purely imaginary, and a saddle if they ase real with XI < 0 < Xz. All of this is summarized in figure 6.1, whose axes are y = Tr[DF(%)jand P = Det[DF(%)]. You daubtlws dsct reed1 that there are the fu&her, repeat& eigenvalue, cases wbere issues of multiglicity mise,~ll~~ As an exercise, you might e@oy &monstrating f11410nstability and wymptotk &ability, see Him& and Smale (1974; Swtion 9.2). [f"bm for example, Hirwh ;and S m b (X9'14), Waltman (1986), Braun (19831, or Borelfi and Golemibn (198'1).
f"M3
Nonlinear Dynamics, MathematicalBiology, and Social aienw
fornndly a bmdy fact that emerges from this figure, nmely that the origin is stable if the B w e is negative and the Determinant is po~litive. FIGURE @ Global ,I Stability of the Origin In the Linear Casa (Distinct Nonirero
Eigenvafues)
Source: Bassd on Edetstein-ashat (1988, p. 190). Of course, all of this hasa to do with (5,4), when we are re;tiIy interated in (6.1). If, somehow, we knew that the behevior of the linearhation (6.4) faithfully repremnled the behavbr of (6.2) in the vicinity of Z,then we could conclude from the gEaBal stabil* of (6.4) at the ori@n,the bomb stability of (6. l) E&I.Xndwd, at; e x h equilibrium Z of (6.1), we could siimply inspect; the eigendum of the Jaeobian D F ( z ) and classify exactly as in the linear case. Amazingly, for s certain class of equilibria we can do just th&! With th& thowgh in mind, we m a b the follotving definition. Definition. An equilibrium 2 of the, (linear or nonlinear) system 2 =. F(%)is hyperbatie if and only if nll eigenfticlua of the Jwobian evaluated at, 2, DF(je), have nonzero real parts. Now id is a very useful fw&th& for hmerbofic equilibria, finea;riza,t;ionis refhble. Zf f is a hyperbolic equilibrium of (tj.l),then its type (mde, focus, saddle) ltnd its stability cormspond exactly to the eype and stability of the zero equilibrium of (6.41, its local linearkation. This is a consequence of the Hartman-Grobman Theorem.
Theorem 1 (Hartman-Grobman). If 3 is a hyperbolic equilibrium of j. = F(%), ithen there is a neighborhaod o f 2 ia zuhieh F is tol~ologicalkyequivalent to the linear vector field f = DF(%)z, Here, two dynamical sy&ems $ = f (X) and i -. $(S), degned on open sets U and V of R~ are topologically equivalent if there exists a homeomorphism (a oneto-one conlinuous mapping with continuous inverse) h : U -4 V mapping the orbits of f onto those of g md preserving direction in time. This gives us a "recipe," if you will, for local stability analysis of hyperbolic equilibria of nonlinesr systems: For eaeh such equilibrium 2, compute the eigenvalues of the Jacobim, DFCZ), and classi& as you would classify the origin in the linear system i TL. Dlj-((g)z.Neighborhod stability analysis, as the technique is a h know, is among the most ubiquitous tools in all of applied mathematics, It appears frequently in, physics and enginering and is a cornerstone of mathematical e c o l o ~ epidemiology, , and population pnetics. Linearized stability analysis also pemades important subfields af miJ1Chematical economics, ev~lutianarygame the-. ory, and other meas of social science, including the thwry of arms racw, wars, and revolutions, which topics I discuss in other lectures. Precisely because examples are so abundant in the literature, X thought a "nonstock" application would be nice. A particulasly ingenious one arises in connection with traveling wave solutions to reaction-diEusion equations, Since these figure in two other lectures of mine (on revolutions and drugs), X thought ""kill two birds with one stone.""
TRAVELING WAVES, HETEROCLINIC ORBITS, AND LINEARIZED STABILIW ANALYSIS, Among the m&famous reaction-diffusion equations is Fisher's (1937) equation governing the spread of an advantageous gene in a populdion. "With D a difision constant, a a parameter memuring the intensity of mlcctiazr, and p(%,t) the g;ene9s frequency at paint IC at time t , Fisher's equation is
W wish t o e&&lish whether f 6.7) admits traveling wave solutions consistent with bioXogically realistic assumptions. We posit travelirrg wave solutions of the forxm[1161
P(x) = p($, 1) with z
==
z -d .
f l l ( E I ~ h diSCuw60~ is paralXels Melstein-Kfsshet (1988).See &so Mumay (1989).
'l32
Plfanlinaat Dynamics, Mathematical Biolof,ly, and Social Science
By the chain rule,
BP -a ~ - aap ~so (6.7) becomes t t ~a~ a~
a nonlinear m a d - o r d e r o r d i n ~ ydigefential equation. Defining -S r a c e (6.8) sls the sysLern
==
U/&, we
Notice that s uaw plszys the role normaJiy play& by tt so that, in phatse slpwe, z is changing dong-is parmetrizing-traj~toriw m d eau w ~ u m e(ZXl real valuw. By coatrmt, we place some definite conditions on P . First, since P is the relative kequency of our advantageous gene, we require that it be bounded: (i) O < P ( z ) < 1f o r a f l z , ( - o o < z < + o o ) . Second, when we say the gene is advmtagwus, we mem that it will ultimately dominate the pool. In &her wads, (ii) P ( z ) -, 1 as z
-+ -oo(since
t
-+
m).
Symmetrically, and hdEy, we m u m e that
= How, the dynamical system (6,9) has two equilibria in Lhtt PS plane: z2 = (h,%) = 0,o). T~US, P = : o %e 2 , and P = I a-t z2. (p1,s1) = (o,a) Hence, if we rzre to satis@ conditions (ii) and (iii), we umd an orbit connecting these equillbria. Such orbits are termed hetemlinie. Technically, a heterocbnic orbit is one whose a and u limit sets (defined below) are distinct equilibrium points, Two centers clearly cannot be heteroefinie. NeiLkrer can a pair of s i n k (aterae tors) or a pair of sources (repetlors). h t h e r , we nwd one unstable and a ~ stable equilibrium, so %fiat%het r a j e c b ~ yemanating born one may be attrwted to the ogher. We will h a w that Fkherk equation (6.7) d m i t s a %ravelingwave solution p r e cisely if we can show that; (6.9) admiLs a cert;ain hetemclinic orbit. Enter linearizd stablity a~alysis!
The Jwabian of (6.9) is
and at
32,
Now, the eigenvalues in each case are hncdions of the parameters a, c, a d D. One can. e ~ i l yshow, by computing the eigenvalues, that if (6.m)
the origin, gl,is a &&h node (XI, Xz < 0) and 22 is a saddle goink ((X1 < O < Xz), which is just the sort af configuration we need. Thme equilibria are indeed connected by the desired hetemclinic orbit, shoarn in 6 p e 6.2.1113 FIGURE 6.2 Weteroclinic Orbit
one u ~ t & l eequif1171~obe precise, t h e f& that the system (6,9)hw exatly one stabb librium does not alone atabIish that the two are connect& by a heteraclinic orbit, The unstable equilibrium could be sumoundd by a, mmi-stable limit cycle (see below). Xn fmt, for the system (16.9),ttrk c m Be ruld out by BendiarsonJsnegative test (sec? below).
"l4
Nontinear Dynamics, Marthamaticticaf Bi~togy,and Social Sciiencs
Findly, interpreing (6.10), we have dsa learned that; the travelillg wave" spwd a very nice byproduct of the analysh, This value is boundcld below by 2 is a ""bifurcation" paint (we discussion belw) alw, in that the phase portraits are not topologically orbitally equivalent; for orbit. There are a a m b e r of things to relish in this example, quik wide from the mathematical propagating wave iLself. The reiilsoning, originally due to F ~ h e rKol, mogorov, and others, is a mawel of indireclion. We sta& with a nonlinear partial diEerentiaI equation which we never solve; we posit a traveling wave solution whose substitution into the original equation converts it into a nonlinear system of ordinary differential quations, which we also never solve. Rather, we make a small number of remned wsuntptiom about the bounds and wppkotic behavior of P, deduce $)h& there mwt be a heteroclinic orbit, and use fhearized stability analysis to establish the requisite parametrsr rdationship, which happens to provide the minimum wave s p d as a byproduct! ~ the more general qu&ion For a fulfctr account aE pomibili t i under
SW Fife
(1979), Britton (1986), and Smaller (1983). Finally, befare l e a ~ a this g exmple, notice how the nonlinectrity sf f is emential to the $raveling wave sdution. (For in~tance,if yau aga-in b@n by positing such a wave solution aad try to carry through the same derivation as above, but BOW with f linear, you will obtain a Inear analowe of the system (6.9), with a, single equilibrium and, hence, no prosptxt of a heteroclinie orbit.) Returning to the main plot, however, the central point is that linewization is an extremely powerhl toof whom applications are really quite far Bung, as this example suggats. PowerEul ens it is, linearized &ability andysis is not omnipotent. Xf it were, nonlinear dynamics would be a pretty small field.
WHEN LtMEARIZATIQN FA1LS: SOME NQNHVPERISQLIC EQUlLIBRIA As m exmple of a failure of lineasizatian, cornider the syskrnil1B~
Lecture 6
What is the aart;ure of the zero equilibrium? Applying the recipe, we compute the Jwabim matrix and evduate at; 2 = 0 to yield:
The linearized systtem at 2 is thus tht: classk harmonic oscillator fmiliar from elementary p h ~ i c s : =: 22;k2 =. - X I . The charwteristie polynomid is
whose solutions w e the purely ima@nary eigenvaluw Xr,z == f i. Since Lbe eigeavdues have zero real parts, they are ozo&yperbolic. Now,in tbe linear case, imaginary eigenvdues would indicate a center, neutral stability (like the harmonic oscillator). Is that true h e ? Xf we convert to polar coordinates, the system (6.12) becomes
with equifibriurn p = 0. Clearly f is gloM1y ~ p p t o t i c a l l i ystable if a < 0, neutrally stable (a center) if a = 0, and unstable if a > 0,as indicated in figure 6.3.
Clearly, then, as a generd propositiort, linearkation is not rdiable for noahmerbolic fixed points. However, for critical points with purely innaginay (as agaimt, foir exampb, zero) eigeavalues, we do have the follawirtg result of PoincetrB's.
l16
Nonlinear Dynamics, Malhstnatiiticat Biofagy, and Social Science
Theorem 2 (PoincmB). A enter equilibriwm of the Iinw&xed system f6.4) .is eiMer a center or a foeas of the of;,ginal nonknew system (6.2).[1191 This is understandable ixr 1ig.b of our prweding results. Since the eigenvalua are b a g i n a y , nod= md saddles are precludeb. We have ju& displayed a fscus, So a corntruetitre proof will be done owe we find a center quilibrium of a nanlinear system that is also a cenwr of its linearhation (try the Loth-Volterra predator-prey model, equations (4.6) above). AB a nrethodelogicd point, %hisexample illustrates the usehlnms of polar COordinates in some cases (the presence of the term st s; is always suggestive). Andher way to analyzc! the nonhyperbolic equilibrium 3 = O of (6.12) is to reason as follows, We ;itre worrying about whether the representative paint on the solution (zl ($1,z2(t))moves tovanl the equilibrium, the origin, over time. So let us look at the Euclidean dbtanee
+
/4I= or, quivalently, the square of the difjtilllce,sin= if the s q w e approac_kt;s zero, so mmt the distance itself, Accordingly, define
How do= V change with t i m e m y the chain rule:
But, with iland k2 fmm (6.12), we have
wbieh a g r m with our prwious re~uft:the equilibrium (F = 0)is R global m p p t o t i c attrator far a < 0, a center for a = 0, and a repellor for a > 0, The fitaction V , above, is an example of a Lyapunov function. And, in fact, m have just used Lyapunav's Direct Method, so-called b e c a m m avert3 able to determine the stabiliw of equilibrium directly, th& is, without having to salve (6.12). The mom general result is given in the fallawing:
lal@l~untley and Johnson (1983, p, 117).
Theorem 3 (Lyapunov). Let Z be a frzed point of k = f ( X ) , z E RZ and let V : W C R2-+ R be difl'erentiable on some neighbarhood W o f 2 and satisfy: ( i ) V @ )= 0;
) 0 if a; # Z; (ii) V ( % > (iii) p(,) < O on W - ( g ) . Then 5 is stable. If
< O in (iii), then 2 is asymptotiw~llystable.[1201
+
In the previous example, V = zf +S:. Quadratic forms (V = azf bslsz+W;) are often good canditdatm, Wiile the method is very pawerEuZ, there is no hrrnula for constructing Lyapulnov functions; this is a bid of an 81%. The typical Lyapunov function is a bowl-lib surface w h w level sets lie in a subset of the plane, Gmme-trically, the theorem says simply that m p p t o t i e sdabifity requirm trajectorim to cross these level sets in the inward direction, EM shown in fiwre 6.4. FIGURE 6.4 kyrnptatic Stability
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Nontinear @lynamics,Matt-rernaticalBiolagy, and Social Seisncs
FIGURE 6.5 Close Up
This swms eminexltly reasonable. Can we get from there to a more Eorrnal argument"?I2lfLet us "zoom in" on a paint where this inwad crossing takm place, zls shown in figure 6.5. Let Q, be the mgle betwwn V V ( n o m d to the level set of V) and the trajwb e ' s tangent, ( k l ,J2f. Rwa11 that for two vectors a and b, a . b = llalf - llbtl cos@, where 6 is t;he angle bet wee^ a and b. As long as the flow points inward, we must have r/2 < QI < 3 ~ 1 2which , implies cos sP < 0. But since I 'TV11 m d Il(kl,kz>lt are positive, V11 * ( k l ,k2) < 0,but V I I (kl, k2)= and we are through. As a second exmpfe of Lyaprtnovk direet method, let us establish a nice general property of gradient systems, which, are impaftant in many ~plications.By way of defi~tian,let U(%)be a real-valued digeremttiable Eunetion on Rn, Then
v,
t -dU/8zi; the veloci-ty of G equals the. is a gmdielat system. For eveq i, d ~ i / d = negative partial af U with respect to Z i . Far physicists, U iS most; naturally interpreted as a potential of some sort. But., one can think of (6.17) as stipulating a rule of acljustment for each. xi. For example, in the backpropaga-tian neural nteb work, connection weights are adjust& in proportion La ;an error grizdienteix22JIn evolutianary game theory, phenotmie frequencies are asfijusted (by selection) in the direction of fitnms grdiemts. Gradient systems have the following:
Propem A gradient minimum them.
s.jlstem is asymptotieaklg stable at
if U has an isolated local
Lecture 6
Proof. We demonstrate that V = U ( I ) - U ( Z ) is a Lyapunov function. We need only establish the properties given in Theorem 3. First, V ( $ ) = O by construction. Now, to call 3 an isolated minimum is to assert the existmm of some neigbborhood W of in which V(%)> O for z f 2, which is the second Lyapunov property. Third, we show that p(z) < O on W - {z). Computing,
(by the Chain Rub) (since gradient)
Since they we d a t e d to Lyapunov functiam aad arise in some of thrr other lectures, f will brieAy &cuss Harnaknian flows. A planar d p m i e a t systenn 2 I=. f (z) is said to sdmit an autonomous Hamiltonian formulation if there exists a C' h n c t b n H : R2-+ R such thaL
In that case, H is said to be a Hamzlloaian of the systern.fa24fAs a pneral proposition, we can consider H ( z ( t ) )along trajectories of (6.18). We differentiate H with mpeet t o time exwtly EIS m did the Lyzrpunov funetion &m:
Hence H is constant, or consermd, d o n g trajectories of (6.18). Total mechanical energy is the archetypal Hamiltonian from classical physics. Notice that, with F = (ft, fi), V H * F = 0, wheress in the gradient case V U F < 0. For this reason, wt.1 speak, af gradienk fields as dksipatiare in mntrast to consernative Hamiltonia-n. fields. 112.a]~or the definition of n dimmsional Hamilton be even, we J d w n (1989, val. 1, p. 20).
3 20
Nonfinear Dynamics, Mathematical Biology, and Social Science
For another perspective, recall that the divergence of a vector field can be interpreted as the rate of expansion per unit volume of a fluid whose Row is modeled by F. In the event F is Hamiltonian, we find: div E'=Vel"'
since H is Cl and so the mked partials are equal. Wmi1toni;zn flows are volume gre~rving,a result sometimes known as LiouGlte's Theorem,[12@ Now, it is evident Erom "Ce above considerations that f-ramiltonim Aows cannot have sinks or sources as equililaria since constancy along trajectories m u l d then clearly be violated. Centers, by corttrmt, are admissible, and, less obviously, so are (certain) saddla.[126j &turning, then, to the issue of local stability analpis, suppow we encounter a system we know to be Hamiltonian +hether or not we can display H)and we have an equilibrium where the eigenvalues are purely irna@nary. The equilibrium is nonhyperbolic, so lineariz&ion fails, but we can irtstantly conclude that it is a center. m y 3 From Poincarh's Thmrem above, we know it is a center or a focus. And, since the system is HarniXtonian, are h o w it is a center or a saddle. Hence, it is a center. Nonlinear systems e h i b i t behavisrcs quite unlih those we haw discussed to this point.
LIMIT CYCLES To introduce the central, and (for autonomous planar systems) uniquely nonlinear, phenome~oaof the limit cycle md some of the waciated thary, csnsider the fallowing system-a vmiatisn on f 6.12).
In polar coordinates, this %&W the form:
gX251~uckenheimer and IEEolmm (1983, p. 47). i 1 2 S I ~~~ W ~ Q(1989, X L vol. 1, p. 237).
For X 5 0,i. < O and. solutiom spiral to the origin as t are t h r e cm- to camider:
+ m.
But, if X
> 0,there
This tells us that trajectorim beginning outside the circle, rZ = X, wind inward while traj&ofia (the origin aside) be@nning inside that circle wind ou-rd, and that as t -,W, all these trajectories spiral toward the circle r2 = X, itself a periodic orbit of (6.21). Bwause it is, in this =me, an attrwtor as t ---,oo and a cycle, the orbit r = 6is called a stable limit cycle. For the time reversed system, the same object is aa unstable limit cycle, for obvious rewons. While the more modem dmhnical definition (in %er- of w-lidt sets) involvw hrther appar&us, Mianorskrlyk s a k w immediately clear the di&inction betwwn limit cycles and the orbits surrounding neutrally stable centers. (I italicize the mlevaat phr~tse.) ""A limit eycb is a closed trajectary (hence the trajectory of a perladic solution) such that no trajec;tory suficientty near it i s also cdosd. In other words, a limig cycb is an isolatd closed trzljectory*Every trajectov beginning sufficiently near a lim& cycle approaches it either far E -+ oo or h r t --+ --m, that, is, it either winds itself upon, the limit cycle, or unwinds from it. If all near& trajcxtories approwh a l h i t cycle C as t -+cm,we say that C is stable; if they approach C as L + --so we say that C is u~stable, If the trajectories on one side of C approwh it while t h a s on the other depart from it, we sometimes say that C is semi-stable although fmm a practical pain$ of view C must be considered unstalble" "inorsky, 1962, pp, 71-72).
The ftabie Xidt cycle iis the bwic model h r all self-sustained oscilators-those which return, or recover, $0 some Eu~dannentalperiadic orbit when perturbed from it. As Hirsch md Smde put it, "hr a periodic soluti~nto be viable in. applied mathematics, this or wme relatd stability property. must be satisfied." "271 The stable oscillations, ""b&i@ af %hehuman heart (which returns to some normal rrate &er we raise it by spriding), eyclm of predator-prey systems, and various electrical circuits are thrert among myriad examples. Business eyclw and certain periodic ciutbreah of social unrest (SW lecture 4) are, quite possibly, athers.
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Nonlinear Dynamiss, Mathematttimt Biotagy, and Social Science
HILIEIERTS "1H PROBLEM Quite aside from their prwtical importance, limit cycles also occupy a venerable , the Second position in the history of We~tieth-centuqmathematics. Xn XNQat International Congress of Mathmaticiw in Paris, David Hilbert presented his famous 23 prsblew. The second part of Hilbert" 16th problem may be &ated as follows: Deternine the mainrum number and position of Eirnit cyclm for the syst;em
where f and g are nth-degre polyxlomids
Defining the Hilbert numbers H(n) =
{number of limit cycles of (6.22))
,
tke problem is to deterrnhe H(%)for arbitrary n. f t is not hard to estabXish that; H(O) = H(1) = 0.But, for n 2 2, we know precious little. Il'yashenko (1991) has shown that H(%)is finite. We also know that H ( 2 ) > 4 and that H ( 3 ) >_ 6. And, a&er nearly a century; that's about iLr1I2q Small wander that in 1947, Minorsky could write, "Perhizps it, is not too great an exaggeration to say that, t b principal line d endeavor of nonlinear mechanics at present is a search for limit In today's world of chaotic dynamics and strange attrwtors, this would be m exaggceratian, But limie w l e s remdn eminentity tvczrthy quarry and-by way of Poin~aremags-they lead to the! study of discr&e dynamical system^, where low-dimensional &m c m be found. (In mtoaomaus digerentiabte systems, chaos only arises i s dimensions three or peater.) The main $hwret,icd tool in the search for limit cycles is the Paincar&Bendhrtn. ~ statement requires us to define w (omega) limit Thmrem, w h 6 contemporary points. The bwic idea of w and rw limit points is simple, Axly point to which a trajec-y converges in forwrd time is an w limit point and any point to which the timereversed trajectory converges is an a Xirnit point. The twhnical definition is just a bit m m discriminaling. Let r(t)= (zlt),y ( t ) ) be a trajedory of
Definition. A point P E RZ is an w limit point of 'l if there exists a sequence (t,) such that t ,
4
oo as .n
m and
The set of aH such P s is called an w limit set; cr limit paints and limit s&s are andogously deGned, with t, appromlking --m rtzLher than +ca. A limit cycle, tlren, is rigarausly defined as a peodie olrbit that is the w or a limit set of other orbits.(l301 Denoting the w limit set of a trajectory 'I by w ( r ) , we state the celebrded Poincm&Bend n Thmrem, perhaps the centerpiece of nonlinear dynamics in the pla~1e.tl3~j Theorem 4 (Poincar6Benbkon). Let I"@) Z= ( z ( t ) ,y(t)) be a trajectory of (6.29) such that, far t 2 to, f ( t ) remains in a closed and bounded =@ionof the plane containing. no eqztilibrpltzcm pints. Then either I' or ~ ( f " )is a p e ~ o d i corbit.
The proof rests on the fundamental Jordm Cuwe Tkorem: a simple closd cume divides the plane into two wnnected rqions and is their common boundary; one region (catled "the inside" ') is bounded and the other (citlled "the outside") is unbounded. On the face of it, nothing could appear more obvious. Yet, the proof is In other words, it is hard to really put the basic concepts of "inside'" and "outside" on a firm footing. How many other ""obvious" things must we not understand? The main difieul-l;y in. applying the Poinear&Bendima Theorem lies in. - t a b lhhing an equilibrium-free closed a ~ bda u n d d "trapping" region. Sometimes it is emy, as in the system (6.20) above. fn this c m , we first imagine a circle eenterd at the origin with radius ro < 6.Since i > 0, all trajectories cross this circle outward. For a circle of radius rl > 6,i < 0 and all trajectories cross inward. Hence, every trajecbry that, a t t = to, is inside the closed and bounded region betwen the= circles, the annulus ro 5 r r l , remains there for dI t > to, The annulus ~ontaiasno cquilihria of (6.20). Hence, by the PoinearBBendixson Theorem, it must contain a periodic orbit. In ewes of this sort, the thmrem makes good intuitive sense. A trajectory starting inside the circle r = is spiraling out. It cannot intersect itself (by uniqueness) and it call% '"Lscap$" so it winds out to a periodic orbit. As 1 said, establishing a compactf13331trapping region can be hard. The other limitation is that the Poincar6-Bendixson Theorem is false for autonomous systems
0 . dp
That is to say, at the real part of the eigenvalue is zero, but its rate of change, the slope, is pmitive; SO, the value must be negativt3 a lialt? to the left, and positive a, little to the. right, of p*. On either side, then, the equilibrium is hyperbolic and sa, by linearbed stability analysis, we have the stable and unstable foci predicted in (h) and (c). Overall, then, we would expect a change of stability as ReX1(p) "crosses the imaginary axis." It is, however, the birth of a stable limit cycle in particular that is surprising, and harder to prove. Indeed a big-league proof requires material (center manifold and normal form theory) beyond the scope of this lecture. But, there is a "physical" way to think about it. Compare the earlier dynamical system (6.12) with the Hopf case. In the first, the origin is not asymptotically stable and the unstable focus (to the right of the critical point in this case) simply spirals out; no limit cycle takes shape. But, in the swand, Wopf, c-, where a limit cycle does form, the origin is mymploeiedly stable. &s 'Yorce of attraction,'Yif you will, while too weak to bbck the passage of &(X) across the imaginary axis (it has f"']~his statement parallels hrrorvsmith and Place (W90, p. 205). See also Marden and MCCrxken (19761, and Guckenheirnm and Holm= (1983, pp. 15142).
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Nonlinear Dynamics, Mathematical Biofogy, and Smiaf Science
'"m~mentum"ReAf(p*) r 0 reedf), is sufficient to prevent an unbounded escape of the orbit and so (by uniqueness %&in, as in PoinearbBend emerges. The thwrem is e a y to use. HOPF EMMPLE #l,Csrrsider the dynamical system
The Jacobian at p is
Evduated at the equiEibrium Z .= 0,the p-dependent J~ztebianis
The cbaraeteristie equation is
Let us uow cheek if the Hopf bifurcation conditions are met, First, the eigenvalues of the Jacobian at zero are purely imwinary if p -- Q, So, p* = 0. Second*s i n e ReX(p) = p, its derivtktiw with mpact to p is 1, so
Finally, we need to d e c k thtzt, for p .= p* = 0, the origin, 2 == Q, is nsymptoticdly stable, Since the eige~valuesat p r= p* are purely imagfnary, the equilibrium is nonh~erboXicand Iinearized stability analysis fails. tyapunw k digme m & b d
Then, with r2 = sz + zg, we obtain
And, ad p = p* = O we have
Hence, by the Hopf Bifurcation. Theorem, p = p" i s bifurcation point and for wme p > p", the origin is an unstable focus surrounded by a limit cycle who= size grows wikh p, m shown in figure 6.6, FIGURE 6.6 Bifurcation to a Limit Cycle.
Source: Bwed on Wiggins (1990, p. 2741,
HOPF EXAMPLE ## 2 (The Van der Pol asclllat;~).Another, very famous, example is the Van der PoI osciXla.tor
or, equiv&lently, k l = 22,
At 2 = 0, we have
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Nantincaar Dynamiw, Mathematical Biotczgy, and Swial Science
with characteristic equation
P ( X ) = X ~ - ~ ~ X + ~ = ~ .
=p f i . Regarding the H ~ p bifurca;tioa f require The eigenvalues are meats, the eigenvalues of the Jacobian at 3 = 0 are again purely imaginary if p = 0. So define p' = 0.ReX(p) = p once more, so we again have
Finally, at p = p*, the bifurcation point, the origin is nonhyperbolic, so linesrization fails. But, once more, it's Lyapunav to the r ~ s c w !T&&ng
And, a&p = p* = Q t
P = -z:z$ < O on
- {Q).
Hence, 3 is asymptotically stable. Just so you won't think all limit cycles are circular, the Van der Pol oscillator is shown in figure 6.7. FIGURE 6.7 Van der Pal Qscittator
Lecture 6
129
Por the Van der Pal equation and its vmim$s (in the family of so-called Lienard equations), the st~tualcon,structio~ of a eomp& invariarrt (trapping) R$ eoxltaining no equilibria, as called for by PaincarkBendhon, is fdrly arduous. The Hopf bifurcation thwrem demom%ratmthe wktmet; of a stable limit; cycle quite patinlegsly (though, unlike aa explicit construction, it says little about its shape). Now, the examples above are all cases of stable limit cycles in R2; orbits wind bward them-they are a t l r ~ d o r sThere . is a very ingenious way to rc3prc;sent these orbits he-d points of discrete maps in a lower dimensional space. The method, like so much else, is due to Paincard, and bears his name.
POINCARE MAPS The bwic idea is easily caaveyed in. the plane. Far a soplristica$ed trc?;atment see Wiggins (1990). Fbr illustr&ive purposes, imagine a circular stable limit cycle centered at the origjn surrounding an unstable focus, as shown in figure 6.8. FIGURE 6.8 Crossings
Given some initial point zo on, say?the positive z-axis, we can follow the trajecbry s say, at zt. We call zx the point af fir& around until it crass- the z - ~ again, return, of zo. Then, zz is the- first return of zl , and so forth. For a given cross seekion 7.2 ( t h w arc: edled hinear4 sec~ioas),here the gositive z-
f 30
Nonlinear Dynainics, MaUlsrnaticai Biology, and Ssciat Science
map, which. we denate TT, is called the Poinear6 map.f1381Tbe idea is to reduce the study of continuous time flows in two dimensions to the study of wsociated discrete time syskems (maps) in one dirnmsian. Very elegand simpfifieations retiult. Far example, the trajectory through a point z* is a periodic orbit if and only if s* returns to itself under the Poincard map;that is to say, nfz*) =: s*. Demonstrating the existenm of a limit cyele or ather priodic orbit (a continuous euwe) thus rducm ta exhibiting a bed pint of the discrete Pairzcar8 map, Xn turn, a limit cycle if stable if the fixed point of the Paincar4 map is stable. &re formally, a periodic o r b i f l with z* E l? is iaspptoticdly stable if X* is an aymptoticaily 2 I , l? is unstable.[sQJ stable k e d point of n-that is, if I7"(2*)< 1. If flE(s*) I. To see this, imagine a &able Emit Conversely? if 'I is stabb, then cycle whose Poincard map has the axed point z*. majectories beginning outside the limit eyele w(l") are winding dawn onto it. By the uniqueness of orbits, the crossings z, are apgromhing z* manotonicdly kern above. Thus,
n'(z*)
0).
Now,W seek some real Eunction B(%,y) such that. Bendkon-.Dulac applies. Caming up, with such Eunetioxls is a bit of an art, like ixrventing tyapunov functions. Following Het;h~ste,I'~~1 we try a function of the farm:
Specificalfy, let; j
=.
-1. Then, &&era little algebra, we find that
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Nonlinear Dynamics, Mathematical Biology, and Social Science
Since X and y are assumed positive, so are 11% and lly. But, Bf l a x and 8glBy are negative by hypothesis. Hence
and we are through. U Clearly, the same reasoning applies to Kolmogorov systems in which each species is everywhere self-amplifying, or autocatalytic:
Then, the system (6.31) has no cycles in the population quadrant. In summary, as a corollary to Bendixson-Dulac: for Kolmogorov systems, if either (6.32) o r (6.33) obtain, there can be no cycles. Notice that the generalized Lotka-Volterra equations of lecture 1 are of Ko1mogorov type. X1 = X l ( ~ l +a11x1 012x2) , ~2 = x2(~2 a2lxl+ a2zx2). If a11 and a22 are negative, each species is self-regulating; if all and a22 are positive, each species is self-amplifying. In either event, there are no cycles. (In the predatorprey variant where cycles do occur, a l l = a n = 0.) There is a powerful theorem of Kolmogorov governing the stability of those systems that bear his name. These, recall, are
+
+
= x f (X,Y)? Y = Y ~ ( xY), Following May (1974), we state the result as follows: Kolmogorov systems have either a stable equilibrium point or a stable limit cycle provided f and g are C' functions of X and y on X, y 2 0 and the following relations h0ld[145]: (i) -Sf 0, (viii) g(C, 0) O with C > 0 , (ix) B > G . =l.
Plwing the theorem in context, May writes,
""Inmare biological terms, Kolmogarov's conditions are rough& that (i) for any given populatim size (as rnewured by numbers, biomass, etc.), the per capita rate of increse of the prey species is a decreming function of the number of predators, and similarly (iii) the rate of inerease of predat;ors decremes with their population size. For any given ratio between the two species, (ii) the rate of i n c r e ~ eof the prey is a decreasing funetion of population size, while converse& (iv) %h&Of the predat;ors is an iwreasing Eunction. It is also reguired that (v) when b&h popufations are small the prey have a positive rate of increase, and that (vi) there e m be a preddor population size suEciently 1 x 8 to stop further prey increase, even when the prey are rare, Condition (vii) requires a critical prey popul&ion size B, beyond which they emnot; increwe even in the absence of predators (a resource or other self limitation), and (viii) requires a critical prey size C that stops fusther increase in predators, even if they be rase; unlas (h) B r C, the system will coXfacpse9"CMay, 1974, pp. 88-89). For rt @ven ECo1xxrog;orov system, then, the programme is clear. First, establish whekfier the system sat&&= the thwrem's Irypothwm, an c3ve~tudiil;yMay accounts as quite likely. He writes, "What h a b e n lacking in the literature is not the derivation. of the above theorem, but rather the realization that it applia to essentiauy all the conventional models people urn." If the system meets all the conditions, then it posmses either a stable equilibrium point or a stable limit; cycle, Our sdandiard linearized stabililty analysis at the equilibrium will then reveal whether that point is stable, "whereupon we have the complete global &abiii$y character of this system faid hare."^^^^^
INDEX THEORY We have seen that nonlinear vector fields xnw have multiple k e d poi~w.And we
have developed a theofy of stability allowing us to elmsie hmerbotic and (to some
f 36
Nontinear Dynamics, Mathematical Biolcrgy, and Social Science
extent) nonhyperbolic fixed points as stable or unstable nodes, foci, saddles, and centers, and to rule out and (with some cre&ivity) to d e h d limit cycles. But, the theory is completely local; we have dewloped no theary of how these entities "can combine, or "fit together." Far instance, could one evter encounter a limit cycle with a sixlgle focus and tovo saddlm inside? There is a beautiful and penetrating theory that allows us to answer many such questions. It is called index theory and originates, once more, with Poincar4.[1471 Among other things, index Lhwry reveals what amount to toyalogkd cansew* tion laws-people even speak of the ""consemation of topological charge'? !It also reveals that, from a particular topological perspective, centerg, nodes, and foci are equivalent! Ta begin, we recall the planar autoxlomous system
The w&or field (f,g) is tangent to the solution Bow in phase spwe at each point (s,g). Now, let C be a simple closed curve (not necessarily a trajectory) that does not intersect any fixed points of (6.34). This last proviso ensures th&, on C, J and g are never simultaneously zero, so that f gZ # 0,a condition whose importance will soon be evident. Since the field (f,g) is tangent to the flow for all (z, y), it is tarmgent to the flow for all ( X , g) on C. So imagine basing a small arrow at some point of C, letting the arrow point in the direction (f,g). Now9 slide the arrow's base exactly once around C coun%e?relochise,allowing the arrow to w u m e the (continuously changing) direction ( f , g ) at each point;, Since the terminal point is the initial point, the terminal dirmtion of the iarrow eoineidiw with the, initial direction and, thus, having returned to its initial position, the arrow must have rotated through 27r radians m irrtegral number of times: that imteger is the indez of C. Qhiously it depends on, the field. For example, if the field is evervhere flowing right to left, the arrow will not rotate at all and, for m y cloed curve C, the index wilE be zero. By contrast, if the curve C is itself a simple closed trajectory of (6.34), then, at any point an C, the tang@&to the flow i s the tangent to C, so that the index is 1, as illwtrated in Ggure 6.9, which shows iz variety sf singularities and their indices (in pwentheses).
+
[147j~incar~ index k is defind in b i n c a r d (1881, 1882, 1885, and X886),. For an x c a u n t , see Lefsclnetz (1977, pp. 195-96). For developments of index theory, see d%oJzteksan (1989, vol. 1, pp, 243-501, Jordan and Smith (1981, pp. 65-14), Cuckenheimer and Holm= (1983, pp. 5&53), and Amof" (1991, pp, 30S18).
FIGURE 8.9 Fbld Dirstions and Indices.
"--%
/
--..
*
--m
'-%
/
%--"
/ \
Circulation ( + l )
Siah (+l t
Source (4-1)
Saddle (-1)
Spiral (+l)
(~2)
Source: Based on Guiltemin and hilack (1974, g. 133). To put this on a more formal fasting, we will m u m e that f and g and their first p;artids are continuous on the relevant sts-bmicalliy ss there is no problem invoking Grwn's Theorem, which looms large here. At any point ( X , g), the slope! of the field =tor is &/dz =. g / f . Hence, relative to the X-=is, the angle of inclination, @, of the field vector must satkEy
Defining zc = g f f : we find
+
whence our concern that f 2 92 # 0. Now, it is clear that the total change in if2 over one circuit around a simple ccIosd cuwe G is given by the line integral.
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Nonlinear Dynamics, Mathsmiatical Biology, and Social Science
But, as argued above, thh Is an integer multiple of 2n,the integer being the index of C. Hence, denoting the index I(G), we arrive at the relation
Rom here, it is a short step to the next basic result:
Theorem 8. het C be a simple closed eume that neither contains nor interswb ang eqailib+.tsnz point8 of (6.34). men I(C) = 0. Proof. Let R be the (simply connected) interior of C. Then (assuming f and g to be G') Green's Theorem yields
But, biEerentiatioa will quicHy show the integrmd on the right to be mro. Cl So, the index of a closed curve containing (and inter~cting)no equilibria is zero. Now, wh& happens to the index if we deform C? If the index Es invariaxld under (certain) d&rma;tions of C, then W really sbuld no&think of the index W characteristic of G at all; but, then, what is the ixldex really about? We begin to answw the qumtion .with a corollav to Theorem 8.
Corollary 1, Let G be a simple dosed eecsue and let C& be a sim*
ctased came sumozmding T f them i s 720 qailibrizlm of (6.34) on either carme or i~7~ the region between them, then I(Cl) = I(Cz). Deformation has no eflect.
a.
Proof. The proof is reminiscent of the standard complex variables proof of Cauchy'~ Tlsmrem for multiply connected domains; we "cut" a multiply connected dorndn, leil-ving a simply connected one3 to which the more bltsk theorem applies. So, as at point A with a s h w n in figure 6.10,let us cut the annulus between Cl and segment; K, Sta&ing A, we intepale counlerclochise around G ,then down K - , around C2 (now clockwise, reversing sign), and out K+ to the start. The clo~ed['~8] curve just trwed neither intersects nor eontdne any equilibria, so its index is zero by Theorem 8. That is,
[2481~hroughout, f will ask your indulgence for a slight mathematical, indiscretion. Strictly s p e d ing, if vcre cut the annulus with a lino mgment, then the path traced in the proof is not a simple cXos& curve since K+ intemmts K - m e w h e r e . To mske things right, ane couM take ewntialty the above approach, but s t a by ~ snipping ouf a swath of width e 3 0, with sidm K- ~ n K"'", d and then m&@ a riprouts limiting cwe Z ~ SE 4 0.
Lecture 6
The integrals along K- and K+ cancel, and we have our result. O FtGURE 6.10 Deforrnatian
We now h a w that the index of a ebsed curve containing rro equilibris is invariant under deformations, sc, long as equilibria are not intersected. Since the index, %herefore,h a little to do with the cume, what; is it truly refiecdirzgUt is really the fixed point structure inside the eume, not the eume itself, that; is reAwted. We make the following definition;
Definition, The index, I b ) , of a singulx point, p, is the index af m y simple closed curve surrounding p that neither intersects nor enclsm equilibria other than p. Using an, argument similar to that emplayed above, we shall prove:
Theorem 9, The index of a closed curue equals the sum of the indices of the equzE i b ~ u mpoints it encloses, Proof. We prove the result for $WO equilibria, p1 and B. With C the outer closed curve, surround p1 and p2 with eurms Cl 8nd C2,m d make cuts K and L, as in figure 6.1 l.
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Nonlinear Dynamics, Mathematical Biology, and Social Science
FIGURE 6.11 Sum of indices
If we integrate Rom A to B, down L-, around C2,and out L+; then from B to A, down K - , around Cl, and out K" to the start, the cfasd curve described neither ialersctcls nor enciosw any equilibria? so its index is zero. Writing this out in fulX, but noting that .LCcaneels L- and that K+ caneels K-, we are bft with
which quite evidently generafizes to the rault we seek;
where the fi's are singularities e n e l o d by. C. B With one more theorem uve will be able to extrwt s o m very unexpmted rexufts.
Lecture fi
"Cl
Theorem IQ. The indea of a ctosed periodic orbit is 1, Proof. We nzerely formalize the plausibility a r g u e a t made earlier. Were, C is a closed prsriadic orbit (mshown in figure 6.9 above). At every point of 6, therefore, the field veetor is precisely $=gent to C. In one circuit around C,the tdal change in the tangexl(;vector%-and hence the field vector"-angle of elevation, @, to the z-axis is 271"radiam, or
But this line inlegraf is, by definition, 2x1(6). CJ Thmrerns 8 and 10 imply:
Corollary 2. A closed p ~ o d i ctmjeetory mwt szlwovnd an quilib~um, Proof, Posit, to the contrary, a closed periodic orbit C enclosing no equilibria. Since closed, its index must be zero, by Theorem 8. But, since C is a periodic trajectory, its index i s is, by Thwrem 10, yielding a contradiction. O More general is the following '%conserv&i.~nlaw," Gorollary 3. Suppose a closed qoerz'odie Cmjeetory slsmuncds C centem, N nodes, F foci, and S saddles. Then,
Proof. 1E3y Thwrern 9, the index of the surrounding orbi-t;equals the sum of the indices of interior h e d points; m d as obsewed above, every center, node, and focus explaining the l@&-hand contributlcss 3-1 to the sum; every s d d l e contributa ---l, side. But, by Thmrem f 0, tfre surrouding orbit" index i s 1. O So, for example, one will not encounter a limit cycle with exactly one saddle and one focus inside, or a center and a node, as thwe arrrangements viafate (g.37). Interpmed slightly dierent15 (6.37) o@ema sewnd negative eriterion far limit cycles. If a region is papulated by mnters, nodes, foci, and s d d k and (6.31) is violated, then there cannot p ~ g ~ i b lbe y a surrounding periodic trajectory; Or, assuming C to be a periodic orbit, is the ghme poI-trait shown in, figure 6.12 impossible?
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Nonlinear Dynamics, Matfiernatia1Biatogy, and Social Science
FIGURE 6.12 lmpossibls Phase Portrait?
Another very unexpected result, with applications in mathematical economics,g'mj is:
Corollary 4. A closed periodic trajectory must enclose an odd total number of centew, nodes, foeri, and saddles. f roof. Suppose to the contrary, an even number, 272, n 2 1. Then, in the notation of Corollary 3, C+N+r;"+S=2rz.
But, by (6.37),ure have Cit-N+F-S=1. Adding, we obtain 2 ( 6 $- N
+ F) = 272 + l , an obvious comxtradic'eion.
CT
As a final result of this sart, aassume, as usud, that l" is a, closed periodic orbi* interseetiag no critical points. Then, from the conmrvation law (6.311, the falbwing corollary rdatiag index thmry to bifurcations bums trivially. Corollary 5. If, @at a bibreation point, a saddle is created (destroyed) inside I?, then a node, fclcus, or eenter must be created (destroyed),
To conserve ""dopologicaE charge" (i.e., the sum of the hdicw) new sinplarities mu& arise or disappear in pairs having opposite indices. (The parallel to particles and anf;iparticlesin high energy physics has been suggated). In faet, the coroHary
goes through if r( is a simple clmed curve, not nwessztrily an orbit* Far this, much stronger, result, see Birhoff and ftslta (1989). While t k s coroflitry flows eEordbssly from the basic conservation thmrem, stop and consider how utterly unapproachable it would be without index theory,
BROUWERS FIXED POINT THEOREM As demaastrated in Theorem 9, the index of a curve I(C) is a topological invariant in that continuous deformations of C do not aEect the index so long as no equilibria are intersected. Here, we imagine G as being deformed over same spwific underlying vector field V, An equivdent, but digerently powerful, perspwtive is to fix C and continuously deform the vector field V , ensuring as usual that no singularities are brought i&o contact with C. Qf course, we have not defined "con_tinuousdeformation of a vector field." This is a hndamental notion in topolom and has a special. name.
Definition, Let f ,g : X -,Y be, Then f is ham6 topic to g if t h e ~ eexists a continuous map H ( s , t ) , O 5 1 5 1, such that H ( z , 0 ) = f ( 2 ) and H(%, 1) = y(z). With this $efinition,[l5" it can be shown--and
it s e m s entirely plausib
Fact: The index of a curve is invariaat under continuous deformations-horn* E the vector field so tong as no equilibria are brought in$o contact with the curve in. the course af the deforrxz~tion. Accepting this faet, index theory provitclw an efegant way to prove Brauwer's celebratd fixed paint theorem on the closd disk in the plane. Although he does not use quite this terminoltog, the proof faflows Arnoldd.[1521
Theorem 11. Every smooth mapping f : t)4
of the ctosed disk
into itself has a
&ed point, Proof. We imagine the disk-whwe closed boundary we denote BD-to at the ori@n. Define now the vector field V(.) = f ( a )- 2 .
be centered (6.38)
Clearly, the k e d points of f are, identically; the equikibria of V. The Tltmrenr will therefore be proved if we c m establish thaL the index of aD in Ir is I. Why? [lso]~erctX and Y are topatogiocal spwm. See hydctn (1988). ilGL1SO=@ tratment~rquim sm06thnm. See Guilledn and P a l l d (1974). ffml~rnol"(1973, p. 257).
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Nonlinear Bynamics, Mathematical Biologysand Social Science
Because then, by Tbwrem (10), L1 =e int(8D) must contain an equilibrium of V (else I(i3D) would be zero). But th& equilibrium, as just noted, is a fixed paint; of f . SO,we slate; Claim, The index of d D in V is 1.
Now, by the Fact above, the claim will be proved if we can provide a vector field homotopic to V in which I(aD) = l. lCiQ that endl define
H defines ;;l, hamotopy b e w e n the fields H(%,Q) = -s and W ( z ,X) = V(LE),and for no t E [Q,11 does H(%,t ) have singularities on OD.But I(@D)in the field -s is obviously 1 since -s is just the field with all vectors pointing to the center of D. D To summarize the proof, f : b -+ b ha4 a fixed point if and only if V has an equilibrium (by construction of V). In turn, V has an equilibrium in D if I(aD) in V is 1 (by Thwrem 10). And I ( a D ) in V is 1 if I ( 3 D ) is 1 in any field homotopic to V. With H(%,t)defined as above, H ( z 2Q) = ---X is homotopic to V and I(aD) in -z is obviously 1. Penetrating rzs the kfieory is when applied to vector fields in the plane, even more startling results emerge when. we consider vector fields on more general objects, notably closed surfaces like the sphere m d torus.
A GLIMPSE BEYOND THE PLANE In pa&icular, X will conclude with a very infarmd presrsnt;ztian of the beautiful hincar6Hopf index theorem, which nicely connects our work on index theory to digereatial t o p o l o ~ . to begin is with Eulerb sell-hewn formula Perhaps the most e l ~ s i c a place l (which was apparently known to Descartes)il54 that for closed convex polyhedral ~urfaees"likC?'' the pyramM and cube,f1Sd1if V, E, and F are the numbers of vertices, edges, and faces, then V-E+E"=2. Now, think of a sphere surrounded by a closed ""approSmatixre;"' polyhdral surface, all of whose hces are triangles that fit together '"nicely"-so that trimgles intersect only h a, common vatex or an entire common edge. By (6.39), if you count vertices, subtract edges, and adcl fma, you will again find that the sum is 2. This number, of 11531~eeFrkbet and Fan (1961,g, 2 l). ilriclI~n toplolawt we do not urn the word "like" fifigh%ly.Far a punetiXious chwwt;eriz~ttionof ex&Iy thme polybdm ta which the rmult applia, see Armstrong (2983).
course, does not change if we diow ""rubber trianglesD-which we imagine pressing . is the down to exwtly cover the spher and count curved fwes and d g e ~This idea of a t~angulationof a surfze. You could &so i m q i m etching curved triangles all over the spherical surface in the ""nice" w w 1 menitioned above, as shown in figure 6-23. FIGURE 6.33 The Triangulations of the Sphere.
T'o be precise, hawever, a triangulation af ia (compact) surfme S is a covering of S by a finik family of dosed sets (G), e x h of whi& is the homwmorphic image of a triawle in the plane,i3=l And, again, we require that ttvo distinct "triangles" (irnaga) be disjoint or have in common either a single "vertex" aor one entire "edge?" where thme are aXsa understood to be image objects. Lewing out the quotes, on this understanding, we can count V - E+F just as before. It is a deep fact that for a given surfwe, S, the number thus obtained is independent of the triangul~tioxl;im it is a ehnracteristic of the surfwe itselg-the so-called Euier cXzmwteri&ie % ( S ) . The n-dimensional definition is
, and a2 where a, is the nurnber of "fwes af dimension i."[157j For n = 2, a ~al, are, rapeetivelly; the numbers of vefiiees (dimension 0 ), edges (dimension l), and ~ z s 5h/lmwY ] (1967).
['"l~t is1in fact, a thmrem (WO, 1925) that a triangulation iu pmible,
i-simplexm.
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Nonlinmr Dynamics, Mathematical Biology, and Social Science
faces (dimension 2) in the trian(;ul&ion. So in two dimensions, we reeover Euler's alternating sum: X(S)==V-E-~-E'. And, the generstfized Euler theorem is that; for all surfaces hornearnorphic to the spheresuefit as the cube and pyramid ab he Euler characteristic is 2. Naturally, we will be interested in the Euler characteristies of other smooth closed surfaces like the torus. And we expect the Euler characteristic, being topolo@cal, to depend on the conneetedness of the surfwe, a prope&y captured in the secalled genus. Mie met the notion of eorzncsctedness in proving that the index of a closed euwe equals the sum of the indices of enclosed singularities-we used a line segment to "dice'" multiply connected domain, ereating a simply connecbd one to which a more basic theorem applied. But, the number of slices nwded w a clearly a topologicai invariant having to do with the coanwtedness of the domain. Similarlx for closd two-manifolds (which we will define shortly) like the sphere, torus, and so on, one e m ask: What is the m~xirnumnumber of, now, simple clost?d curves o ~ l ecan draw an the surface vrithozlt dissecting id. into disconnected parts? This number is the genus of the manifold. Far the sphere, it is clearly zero, since even one simple closed eume will dissect it. Two simple c l a d curves are neded to dissect the brus, so its genus is l, and sis on, ;ss shown in figure 6-14. FIGURE 6.14 Classification of Oriented Closd Two-Manifolds
Gem= O (hommmorpbicLo a sphere with no haadles)
Gmus l (homeomofpKcLo a sphere witH one hmdk)
&nut?,2 fhommmofp&cto ai sphere with two harxdfes)
a n u s 3 ( h a m o m a ~ to k a sphere with tkm haadlee)
Source: Based on Guillemin and Pollack (1974, p. 124). i r s a l ~ h Eufer e ~Gfxerizcrteristicillso depends an the an'entability of t h e surfwe, a topic into which we will not enter here. All surfwes are ws,sun?& to be orient&, even though X will o&en say so explicitly just to drive borne t h e point that i t matters. The cXwic nonofient8ble surfme is, of
course, the Moebius strip,
The figure rc?flec%~ a hndamental ctassification theorem in topalogy: every cornpact o ~ e n t e dboundqkss two-manifobd is honteonaorphic to one of Ihese surfscms-a sphe.au?*.t;h n 3 O hanrlkes,lf5@1We know from above that the Euler charwteristic of the sphere is 2. Its genus is zero. The general relationship is:
Theomm t 2 . Oeentd closed surJitceslX64 of gelazds g have Eulier cltaracterislie 2 - "2,fI"ZS We S-m to have wandered -Ear from index theory, which concerned singularities of vectar fields, To procmd h r t her and connect all of this to index thmry, we nwd to be able to define vector fields on sudwes of the sort we have discussd. And we are going to want to do calculus on them, which puts us in the world of diEerential tspalogy. Obviously, we all learned calculus on the Euclidean plane. And we will be able to do calculus on objrtets that are "looelly Euclidean" which is certainly how practitioners think of manifolds: surfwes that we, locdly, smooth dtlformatiom of the plane. In fact, all the closed surfwes we have been discussing are twenranifolds. ~ ~ each I poirrL of Technically, a two-manifold is a connected Hausdorff s p ~ e , i M, which has a neighborhood homeomorphic to an open set in R2.This is what one means by "looclly Eucfidean." And, in turn, a vector field on M is simply a smooth (C') assignment of a tangent vector to each point z. E M, just as in vector calculus. But, it will pay to be a bit Dore painstaking. Following Smde (19691, we iassociate the so-called tangent to each point a: E M a twedimensiond wetor space, %(M), spwe of M at z, For a twemanifold, like the sphere, T z ( M )is the plane t a n e n t to M at s. A vector field V ( z )on M is a C' assignment to each z E M of a tangent vector-that is, a vector lying in % ( M ) . At singularities I where V ( f )= 0, the field may exhibit sinks, saurces, centers, saddlm, and more cmplex behaviors-ph~ portraits-on M , Technically, we do not know how to compute indices on M. But, M is m object th& is 1ocaIly horn* morphic to the Euclidean space, R', where we do know how to compute indices. So, we imagine doing the same thing on a small surficGe element of the tangent, space T z ( M )E& As Guilfemin and Polfaek put it, '"~ooking at; the manifaid localXy, ~259~Xnded, for two-mnifalds, diffmnnorphic. i1""11'21hough, again, we will nat delve into it, the general relationship between genus g, EuIer chig~~ttcteri~tic X , and oritsntabilib is
fzsxfl;w ie~aurantand b b b l n s (1963)for a very nice intuitiw development. [ l c j z ftapolag~cal ~ space is Hausdorff if it siltisfim the f o l b w ~ n gcondition: Given two distinct points z and g , there exist apen sets Q1 and O2such that zr E Q1 and y E Q z and Ox n Qz = Q1 See &yden (1988, p. 178). ~ ' " ~ ~ b i again, s, 1s standard procedure in surface integration, far example.
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Noniinear Dynamics, Mathematical Biology, and Sr>cialScience
we m emefiidly a piece of Euclidean spMe, so we simply r e d off the index as if
the vector field were Euclidean." With all this in pfme, we can state the Paincw6HopE index thmrem. AS you d g h t imagine, there are mmy formulations; we fallow Ateksmdrav".
lt?eorem13. (PaincarBHapf). 1Tf, on a given o ~ e n t e dtwo-manifold, a eontin~ous vector field is defined hauzng only a f i ~ i t enumber of siwular points, then the sum of their indices equab &fie Euler charneterzstic of ithe manifold. This is remslrkabte because the Euler charaederist;ic ia a bpologicaf prapefiy of the manibfd only, and would appear ts have absolutely nothing ta do with flows, or vector gelds, on the manifold. How does the manifold, M, already "how" 9-0 much about the singularity structure ctf vwtar fields definable on M? C l e ~ f ydwp ~ things are afoot, One immediate consequctnice of the theorem is %hatvector fiel& having no singularities are passible only an manifolds with Euler characteristic: zer 1; namely, the toms (and, in fact, the Klein batdfe, a nonorientable "'one-sidedn torus). Notably, it is hpossible to construct a nonvanishing fidd on the sphere. TMs result is sometimes aflFeetionately called "the hairy ball theorem," the interpretation being that any attempt to wmb smooth a "hdry balf" must leave at lertst one babfd spot, Bdbnms can be avoided on, the brus, as shown in figure 6.15. FIGURE 6,15 Hairy Batt Theorem,
Twa bald spots
Qna bald @pot
Hetry torus
Source: Bamd on Armstrong f 1gm, p. 198). [lci416uililerninand PoIlwk (2974, p. 133). i1551~ee Abhandrov, et al. (1963, p. 216)
Another interprf:tat;ion on the sphere is that "somewhere on the surfwe of the earth the wind isn't btoaring." Ebr m interwting application of the thmrern to chemical and ecological aetworks, see Glass (1975). Mlk certalniy cannot prove the hincar&Hopf index theorem here. But, if you will grant that there is mmethiw generic &out a certain map, the su-called Lefschets, map, then at least; a vefy suggestive emmple will be in hand. Following Guilternin and Pollack (f974), consider the genus 4 twumnifold in figure 6-16, FIGURE 6.1fi A Two-Manifold of Genus 4.
Source: Based on Guillemin and Polltack ($9'74, p, 125).
Their dmcription, of the map indicated by arrows in the figure is clear, Bnd appetizing! ' W e keonstructl a Lehchetz map on the surfwe of genus k: m follows, Stmd the surface vertically on one end, and coat it evenly with hot fudge topping. Let ft(z) denote the oozing; trajechry of the point r?: of hdge as time t pwsm. At time 0, fo is just the identity. At time t 3 Q, f t is a Lefs~kedz
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Manlinear Dynamics, Mathematical Biology, and Social Science
map with one source at the top, one sink at the bottom, and satddlepoints ~t the top rand bottom of each hold"(Guillernin and P o l l ~ k1974, , p. 125).
As 1said, this oriented maaifoEd is obviously of genus 4. Notice that in this case, the genus (the number of handles) is also the number of holes. Now, let us sum the indices. By our previous w r k (and the crucial fwt that a manifloid is locally Euclidean), we can directly count plus two for the sink and the source, and minus axle for each sddle; that is, minus tTwo for each ""hof$\of which there are, in general, h. So, the global sum of the indices is 2 - 2h. But, the number af holes i~ the genus of the manifold. So, by Theorem 12, 2 - 212 is also the Euler chaaeteristic, as predicted by Poincar&Wopf!
Many of the topics treated in t h m lecture are far more dvanced than the mathematics applied .in other ledurm, While certain of the G O ~ ~ C ma;v B seem highly abstract;, the history of science shows that applications of pure mathematics are hard ta anticipate. Who imagined that complex numbers, non-Euclidean geometries, a ~ d infinitedimemiond spaces wauld find powerful applications in physies?E1"F.l Against that background, it; muid be naive to preclutctc: the applicability of even the most abstrwt; mathematics t;o social scieace. And meanwhile, of course, the mathematics is its awn reward.
VVigner9s%say, "The Unremnable ERwtivenw of Mathern~%it;ies in the Natural Scit-mcw." "igner, 1963). I I s s j ~ nthis, see Eugene
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Jordan, D. W., and P. Smith. 1987. Nonlinear Odinary Difle~pttialEqsations, 2nd ede Mew York: Oxford Universi.t;y Press. Kaufmann, William W. 1983. "The Arithmetic of Force Blmning," h Alliance Sec.u&ty: NATO and the No-First-Ulse Question, edited by 3. D, Steinbruner and L. V. SigaX. Wwhia@on, DC: Brookings Institution, Kermack, W. Q., ancl A. G. MeKendrick. 1927. ""Contributions to the M2tt;hernaticd Thmry of Epidemics." Pmc. Rog. fiat. Sac., Ser. A 115: 70+721. Kritsner, Stephen. 1983. International Regimes. Ithxa, NU: Cofnefl University Press. Kreyszig, Erwin. 1978. Int~oductoryFunctional Analysis &th Applimtiow New York: John. Wiley &, Sons. Reprinted in 1989. Kupch~n,Charles A., and CliRord A. Kupchan, 1995. "The Promise of Collective Security;" Internat!. $eearity 20(1) (Summer): 52-70. Kupehan, Charles A., and CliEord A. Kupehan. 1991. ""Conce&-t;s,Collective Security, and the h t u r e of Europe." hinternag. Securitg Lfi(3) (Summer): 114-162. Lanehmter, F. W. 1916, AircraJJtin Wrjare: The Dawn af the Fourth A m , London: Cowtable, Lanchester, F. W. 1956. "Mathematics ixl Warfare." In 7"he World oj kthernat;zlcs, ediGed by James R. Nwman, VOX. 4, 2136-2137, New York: Sirnon & Sehustw. Lefschetz, Solomon. 1977, Diflerentid Equations: Geometric Theory, New York: Dover Publication8, Inc. Loreaz, Hans Wltlter. 1989. Nonlinear Dynamical Economies and Chaotic MOtz'on. Berlin: Springer-Verlag. Mansfidd, Edwin. 1961, "Technical Changes and the Rate of Imitation." Emnom e t ~ e a29(4) (October): 741-766. Marsden, Jerrald E. 1974. Zlementary Classical Analysis, New York: &',H, Reeman and Gonnpmy* Marsden, Jerrold E., and M. MeCrwken. 5976. The Hopf Bifircation and Its Appliwtions. New York: Springer-Verlag, Marsden, Jerrold E., and Anthow J. n0mb8. 119713. Vector Calculus. Sm Raneisco: W. H. Reeman. MW-Colell,Andreu. f 985. The Theory of Geneml Economic Equilzbeum. New York: Cambridge University P r w . Massey, W, S. f 96"7 ,Algebraic Toplogy: An Introduction. New York: SpringerVerlag. MW, Robe& M. 1974. Stability m d Conzple&tg in &del Bcosystem, 2nd ed. Princeton, PJJ: Princeton Universi.t;y Pregs, May, Robert M, 1981. "Models for Two Intermting Populati~m.'~ In Theomtical Ecology, edited by 1Rober.t M. May. London: Blackelf Scientific Publications, May, R, M. 1983, "Parasitic Xnfectiom as Regulators of Animal Populatiom." Amer. Sci. 71: 36-45. Mayer-Krms, Goktfiied. 2992. ""Naalineiizr Dynamics and Chaos in Arms Raee Models." h Inodeling Complez Phenomena, edited by Lui L m and Vladimir Naraditsky- New York: Springer-Verlag.
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A
Adaptive Glynmic Model, 2, 19, 28-30 adaptation in, 34 adtzptive withdrawl in, 32 attrition in, 33 c s e s of, 35-36 overview of, 30-31 prowcution in, 32-33 simul;ationsof, 37-40 Azmrafl Z T ~ Warfare, 20 altruism, 3, 42-43 animal behavior, 2' arms racm, 2, T-8, IQ, 69, 1x1 a;nalio&;yto drug epidemic, 94 m d c o f l ~ t i v e~wuri@,41 bisfo@caf, 52 nanfhear models of, fib66 see dso Richardson model itttrilian, 20, 29 stalemate, 56 Axtell, b b e r t , 4 Axelrod, Robert, 41, $4
B Bendixson-Dulac negative t s t s , 107, 131134
bifurcation, 84, 142 to a limit cycle, 324-128 bre&point-s, 25-26 Brouwer's fixed point thmram, 4, 143-144
C casualty-exchange ratio, 27, 56 catastrophe thmry, 2 e h a s , 57, 80, 87, 122 Clark, Calin, 25 clmification theorem, 147 coevolulion, X2 rnatftematical theory of, 14 moth-bat vs. air war tmtics, 12-14 collective seurily, 2, 41-44, 47-48, 50-51, 64
eonnwtionism of, 53-55 effect on competition, 66 rigorous degaitirsn of, 41
community xnalrix, 95 competition, 14, 69 Complex System Stmmmer School, l
D damping, 46, 97 deformations, 138, 143 demity; 22-23, 27 drug &diction, 3 drug epidemic am race component, 94 model, 89-99 drug wm, 97-98, f 11, 103 and ducation, 103-2 0 4 model W epidemic, 89, 91. Y3, 95 SW also interdiction see also Eegdizatian dynmical malogiw, 69-71 epidemics, 711 revolutions, 71 dynamic& system, 1, 3, 41, 107 and collwdifr~lwuriky, 42 E wonomics, 70, 108-105, 111, 242 epidemics, 3, 9, 15, 69 mdogy for revolutions, 15-11, 71, 74-75 analogy to explssive socid ~ h m g e71 , drug model, 89-93 d y n m i c d aandogies, 6911. herd immunity, 116-17, 82-83 infeelivers in, 1516, 12, 90 Kermxk-McKendrick; threshold model, 15, 72-73, 91-92 model of, 72-73, 90-91 Ptaqztes and Peoples, 69 social change, 71 technology, 80 traveling wavw, 75 vitd dynamics, 76 muation of logistic gror;vlh, f 5, 131 Euler, 144-147 external thre;zts, 46
G Gause, G. F., 14 Gawek flour beetle experiment, 32 Gell-Mann, Mufray, 71 Glwnost, 17 glob& stability, 109110 gfoboeop, 47, 65 vs, Rieharctson's sodeE, 48, 52, 65 grdient systems, 118-119 Growing Artifictal Soezeties, 4
1EI. Hamiltonian, 78, l f 9 Aows, 119-120 Hartman-Grobman Theorem, f 10-411 herd immunity, 82-83 Eilbertb 116th Problem, 4, 101, 222 Erfopf Bihrcsation Theorem, 3, 107, 125-129 1 Index Theory, 3-4, 107, 135-143, 147-150 caroXlmy on bifurcation, 142-143 Poinear6Hopf Index Thmrenr, 3, 108, 144-146, 148-1.19 indirection, l l 4 interdiction, IQ0^1Q3
J Jordan Curve Theorem, 123-124 K
Kermmk, 1S, 72-73, 92-92 Kolmogorov's Theorem, 107, 134-135
L Lanchmter, R d e r i c k William, 20 1,anchwter Square Law, 21 La~~ehwter Thtliary, 2
tanchester quation, 2, 8, 20 ambush varicsnt, 23-24 and Gause, 14, 2425 attrition in, 20, 513 demity in, 22-23 linear digerential, 22 reinforcement, 24 square model, 20-21 Langlois, Jean-Pierre, 84 tefxhetz map, 149-150 Xegdttzation (drug), 402, 104 limit cycle, 3, 107, 12&121, 1123, 129-131, 241 Hilbctrtk 16th problem, 122 Minorsky's definition, 121-122 see also negative tmts linear homogenctous problem, 109 linearization, 118114 failure of, I f 4- 115 Iineilrizd stability andysis, 3, 95, 101 LiouvilleTsTheorem, 120 Lipschitz, f 08 Loth-Volterra ecosystem equations, 2, 9, 17 predator-prey model, 76, 93, 96 vs, Richardson's mod&, 2, 47 Lyapunov function, 3, 1Lei-119
M nzallizemilticd biology, 1, 7-9, 15, 76 Loth-Volterra model, 47, 76, 93, 96 see Loth-Volterra c?qu&ions mathematical theories of arms r w a see Richardsoxlk model mathematical tools, 1 May, Robert M., 11, 1%-135 McKendrick, 15, 72-73, 91-92 model Adaptive Uynmic, 2, 19, 28, 36-35 agent-bed, 4 as illuminating abstraction, 9 class struale, 77 collective security, 48, 58 drug epidemic, 90, 97 globocop, 48, 58 Kermmk-McKendrick threshold, 15, 72-13, 92-92
model (contud) Lmchester nonginear attrition, 23 linear with variants, 48 Loth-Valterra, 47, 76, 93, 96 movement in war, 28-29 noniinem with variants, 58 of complex graces=, 1 reaction-diffusion , 75, 19, 97-99, 111-114 Richardsonian comaetit ion, 48 SIRS. 80 SIR epidemic, 73, 92, 95 two-counlry, 43 war, 19 see Rchardson model mutualism, 10, 69 mutualistic gopulstions, I l
N negative tests, 131-434, 141 nonltinear arms r x e models see arms races nonlinear autonamous system, 108 nonlinear dynamical systems, I , 3, 107 EIopf Bifurcation Theorem, 3, 107 125-129 Index Theory, 3-4, 107, 235-143, 147-150 limit cycles, 3, 120 Lyapunov hnctions, 3, 116-119 Paincar8 maps, 3, 307, 129-131 TFZaineizfBBendixson Theorern, 3, 107 122-123 PoincarBHopf Index Theorem, 3, 108, 144-146, 148-149 stability malysis of, 3, 109-120 theory, 1 nonlinear reitction-diRusion model, 75, 79, 97-99, 111-l f 4 P Poinearci? map, 3, 107, 129-131. PoincarBBendixson Theorem, 3, 107, f 22124 PoincarBPXopf Index Theorem, 3, 2 08,144 146, 148-149
Poincardk Theorem, 116 political gievanca, 45 principal of competitive exefusion, 14, 25
R b g o z i n t s negative test, 133 reation-diRusion equdions, 75, "1, 92-99, 111-114 see Fis;herk equation reciprocal, mtivation see mutualism reciprocal d t r u b m , 41 removal capacity, 79-80 repression, 80, 84-86 revdution, 2, 7, 15, 69, I11 as epidemic, f 5 1 7 , 71-72, 74 deeentrdzed totalitarianism, 82-83 models of, 1516, 12-86 removal capacity; 79-80 reprmion in, 80, 84-86 SEW model, 80 traveling wavw, 75 Richardsonk model, 2, 8, 20,4548, 58, $4 bmic, 45 wonomie fatigue in, 11, 46 externd threat t e r m , 46 gtobacap and, 48, 52, 58, 65 gievancw in, 45 simple analytics of, 46-41 twecountry, 43 vs. collective swurity, 48, 58 vs. Lot h-Voiterra wossysdem m d e l , 10, 47 Robimon , Miehaet , 2 2
S Samuelson, P a d A., 70-71, 78 SIR epidemic model, 73, 92, 95 socid change explosive, 3, 86 social dynmics, 1, 86 mciai, systems, 1 and nonlinear dynmics, 71 stability andysis, 3, 95, 107, 109-114, 117-119, 124-125, 134-135
stability criterion, 1X. St~km'Thearern, ?Q str~~ctural stability, 124
T
totdi tarimism, 82-83 traveling wave solution, 2 11-112, l14
V Van der Pol oscillator, 127- 128 Verdun, 36, 40 vital d,mmics, 76
W WilXtmsn, Paul, 72-73 war, 2, 7-8, 19, 28, 69, 111 attrition and withdrawl in, 20 attrition rates, 20, 30 cwudty-exchange ratio, 27 drug, 91-99, 103 fixing operations in, 36 grremina, 36 mathematicd modeh of, 19 pain thresholds in, 30 stmdoff in, 36 Weis, Herbert, 22-23 withdrawl rakes, 32 -35
