The following abbreviations are dened in and used throughout the text:
CA Cellular Automata CML Coupled Map Lattice
ESS Evolutionarily Stable Strategy IPS Interacting Particle System IPD Iterated Prisoner's Dilemma ODE Ordinary Dierential Equation PA Pair Approximation
PDE Partial Dierential Equation
Andreasen, V. and Christiansen, F.B. (1995)
Slow Coevolution of a Viral Pathogen and its Diploid Host.
Phil. Trans. R. Soc. Lond. B348: 341-354
Anita, A., Koella, J.C. and Perrot, V. (1996)
Models of the Within-host Dynamics of Persistent Mycobacterial Infections.
Proc. R. Soc. Lond. B263 : 257-263
Ardito, A. and Ricciardi, P. (1995)
Lyapunov Functions for a Generalized Gause-type Model.
J. Math. Biol. 33 : 816-828
Altmann, M. (1995)
Susceptible-Infected-Removed Epidemic Models with Dynamic Partnerships.
J. Math. Biol. 33 : 661-675
Anderson, R.M. and May, R.M. (1992)
Infectious Diseases of Humans: Dynamics and Control.
Oxford Science Publications, O.U.P. Axelrod, R. and Dion, D. (1988)
The Further Evolution of Cooperation.
Science242: 1385-1390
Axelrod, R. and Hamilton, W.D. (1981)
The Evolution of Cooperation.
Science211: 1390-1396
Bascompte, J. and Sole, R.V. (1995)
Rethinking Complexity: Modelling Spatiotemporal Dynamics in Ecology.
Trends in Ecology and Evolution10: 361-366
Batali, J. and Kitcher, P. (1995)
Evolution of Altruism in Optional and Compulsory Games.
Boerlijst, M.C. and Hogeweg, P. (1995)
Attractors and Spatial Patterns in Hypercycles with Negative Interactions.
J. Theor. Biol. 176: 199-210
Bolker, B. and Grenfell, B.T. (1993)
Chaos and Biological Complexity in Measles Dynamics.
Proc. R. Soc. Lond. B251 : 75-81
Bolker, B. and Grenfell, B.T. (1995)
Space, Persistence and Dynamics of Measles Epidemics.
Phil. Trans. R. Soc. Lond. B348: 309-320
Bolker, B. and Pacala, S. (1996)
Understanding Stochastically Driven Spatial Pattern Formation in Ecological Systems using Moment Equations.
Princeton Preprint.
Boyd, R. and Lorberbaum, J.P. (1987)
No Pure Strategy is Evolutionarily Stable in the Repeated Prisoner's Dilemma Game.
Nature327: 58-59
Connor, R.C. (1995)
Altruism Among Non-relatives: Alternatives to the Prisoner's Dilemma.
Trends in Ecology and Evolution10: 84-86
Cressman, R. (1990)
Strong Stability and Density-dependent Evolutionarily Stable Strategies.
J. Theor. Biol. 145: 319-330
DeRoos, A.M., McCauley, E. and Wilson, W.G. (1991)
Mobility Versus Density-limited Predator-Prey Dynamics on Different Spatial Scales.
Proc. R. Soc. Lond. B246 : 117-122
Dickman, R. (1986)
Kinetic Phase Transitions in a Surface Reaction Model: Mean Field Theory.
Physics Review A34 4246-4250
Durrett, R. and Levin, S. (1994a)
The Importance of Being Discrete (and Spatial).
Durrett, R. and Levin, S. (1994b)
Stochastic Spatial Models: A User's Guide to Ecological Applications.
Phil. Trans. R. Soc. Lond. B343: 329-350
Dusho, J. (1996)
Incorporating Immunological Ideas in Epidemiological Models.
J. Theor. Biol. 180: 181-187
Edmunds, W.J., O'Callaghan, C.J. and Nokes, D.J. (1997)
Who Mixes With Whom? A Method to Determine the Contact Patterns of Adults that may lead to the Spread of Airbourne Infections.
Proc. R. Soc. Lond. B264 : 949-957
Farrell, J. and Ware, R. (1989)
Evolutionary Stability in the Repeated Prisoner's Dilemma.
Theor. Pop. Biol. 36 : 161-166
Ferriere, R. and Michod, R.E. (1995)
Invading Wave of Cooperation in a Spatial Iterated Prisoner's Dilemma.
Proc. R. Soc. Lond. B259 : 77-83
Ferriere, R. and Michod, R.E. (1996)
The Evolution of Cooperation in Spatially Heterogeneous Populations.
The American Naturalist147: 692-717
Frean, M.R. (1994)
The Prisoner's Dilemma Without Synchrony.
Proc. R. Soc. Lond. B257 : 75-59
Glendinning, P. (1994)
Island Chain Models and Gradient Systems.
J. Math. Biol. 32 : 171-178
Granovsky, B.L. and Rozov, L. (1994)
On Transient Behaviour of a Nearest Neighbour Birth-Death Process on a Lattice.
J. App. Prob. 31: 549-553
Grenfell, B.T., Bolker, B.M. and Kleczkowski, A. (1995)
Seasonality and extinction in chaotic metapopulations.
Proc. R. Soc. Lond. B259 : 97-103
Grim, P. (1995)
The Greater Generosity of the Spatialized Prisoner's Dilemma.
Guckenheimer J. and Holmes P. (1983)
Nonlinear Oscillations, Dynamical Systems and Birufcations of Vector Fields.
Springer Verlag
Hamilton, W.D., Axelrod, R. and Tanese, R. (1990)
Sexual Reproduction as an Adaptation to Resist Parasites (review).
Proc. Nat. Acad. Sci. USA87: 3566-3573
Harada, Y., Ezoe, H., Iwasa, Y., Matsuda, H. and Sato, K. (1995)
Population Persistence and Spatially Limited Social Interaction.
Theor. Pop. Biol. 48 : 65-91
Harada, Y. and Iwasa, Y. (1994)
Lattice Population Dynamics for Plants with Dispersing Seeds and Vegetative Propagation.
Res. Popul. Ecol. 36(2): 237-249
Herz, A.V.M. (1994)
Collective Phenomena in Spatially Extended Evolutionary Games.
J. Theor. Biol. 169: 65-87
Hethcote, H.W. and van den Driessche, P. (1996)
An SIS Epidemic Model with Variable Population Size and a Delay.
Math. Biol. 34: 177-194
Hofbauer, J. (1996)
Evolutionary Dynamics for Bimatrix Games: A Hamiltonian System?
J. Math. Biol. 34 : 675-688
Hofbauer, J. and Sigmund, K. (1987)
Dynamical Systems and the Theory of Evolution.
Cambridge University Press. Hofstadter, D.R. (1983)
Mathematical Themas: Computer Tournaments of the Prisoner's Dilemma Sug- gest how Cooperation Evolves.
Scientic American248: 14-20
Hsu, L., Hsu, T., Mortimer, J., Panju, M. and Schroeder, S. (1995)
Dynamically Stable Multiple Strategy States of the Iterated Prisoner's Dilemma.
Physica D85: 296-303
Hutson, V.C.L. and Vickers, G.T. (1995)
The Spatial Struggle of Tit-For-Tat and Defect.
Jonansen, A. (1996)
A Simple Model of Recurrent Epidemics.
J. Theor. Biol. 178: 45-51
Keeling, M.J. (1995)
The Ecology and Evolution of Spatial Host-Parasite Systems.
Ph.D.Thesis, Warwick University, U.K. Keeling, M.J. and Grenfell, B.T. (1997)
Disease Extinction and Community Size: Modelling the Persistence of Measles.
Science275: 65-67
Keeling, M.J. and Rand, D.A. (1996)
Two Methods to Model the Persistence of Measles in a Population.
Warwick Preprint.
Keeling, M.J., Rand, D.A. and Morris, A.J. (1997)
Correlation Models for Childhood Epidemics.
Proc. R. Soc. Lond. B264 : 1-8
Krakauer, D.C. and Pagel, M. (1995)
Spatial Structure and the Evolution of Cost-free Signalling.
Proc. R. Soc. Lond. B260 : 365-372
Kubo, T., Iwasa, Y. and Furumoto, N. (1996)
Forest Spatial Dynamics with Gap Expansion: Total Gap Area and Gap Size Distribution.
J. Theor. Biol. 180: 229-246
Lema^itre, A., Chate, H. and Manneville, P. (1995)
Cluster Expansion for Collective Behaviour in Discrete-Space Dynamical Sys- tems.
Preprint
Levin, S.A. and Durrett, R. (1996)
From Individuals to Epidemics.
Princeton/Cornell Preprint. Lindgren, K. and Nordahl, M.G. (1994)
Evolutionary Dynamics of Spatial Games.
Physica D75: 292-309
Lloyd, A.L. and May, R.M. (1996)
Spatial Heterogeneity in Epidemic Models.
Lorberbaum, J. (1994)
No Strategy is Evolutionarily Stable in the Repeated Prisoner's Dilemma.
J. Theor. Biol. 168: 117-130
Lotka, A.J. (1925)
The Elements of Physical Biology.
Williams and Williams Co., Baltimore. Malthus, T. (1798)
Essay on the Principle of Population.
In: The Faber Book of Science, ed. John Carey. Faber and Faber, London. Matsuda, H., Ogita, N., Sasaki, A. and Sato, K. (1992)
Statistical Mechanics of Population - The lattice Lotka-Volterra model.
Prog. Theor. Phys. 88: 1035-1049
May, R.M. (1974)
Biological Populations with Non-overlapping Generations: Stable Points, Sta- ble Cycles and Chaos.
Science186: 645-647
May, R.M. (1976)
Simple Mathematical Models with Very Complicated Dynamics. (review)
Nature261: 459-467
May, R.M. (1987)
More Evolution of Cooperation.
Nature327: 15-17
May, R.M. (1994)
Spatial Chaos and its role in Ecology and Evolution.
Frontiers of Theoretical Biology (Lecture Notes in Biomathematics vol. 100). Springer, New York.
Maynard Smith, J. (1974)
The Theory of Games and the Evolution of Animal Conflicts.
J. Theor. Biol. 47 : 209-221
Maynard Smith, J. (1982)
Evolution and the Theory of Games.
Cambridge University Press
Maynard Smith, J. and Price, G.R. (1973)
The Logic of Animal Conflict.
Mesterton-Gibbons, M. (1992)
On the Iterated Prisoner's Dilemma in a Finite Population.
Bull. of Math. Biol. 54 : 423-443
Molander, P. (1985)
The Optimal Level of Generosity in a Selfish, Uncertain Environment.
J. Conict Resolution29 : 611-618
Murray, J.D. (1990)
Mathematical Biology
Springer-Verlag Nowak, M. (1990)
Stochastic Strategies in the Prisoner's Dilemma.
Theor. Pop. Biol. 38 : 93-112
Nowak, M. and May, R.M. (1992)
Evolutionary Games and Spatial Chaos.
Nature359: 826-829
Nowak, M., May, R.M. and Sigmund, K. (1995)
The Arithmetics of Mutual Help.
Scientic AmericanJune1995: 50-55
Nowak, M. and Sigmund, K. (1992)
Tit-For-Tat in Heterogeneous Populations.
Nature355: 250-252
Nowak, M. and Sigmund, K. (1993)
A Strategy of Win-Stay, Lose-Shift that Outperforms Tit-For-Tat in the Pris- oner's Dilemma.
Nature364: 56-58
Nowak, M. and Sigmund, K. (1994)
The Alternating Prisoner's Dilemma.
J. Theor. Biol. 168: 219-226
Nowak, M., Sigmund, K. and El-Sedy, E. (1995)
Automata, Repeated Games and Noise.
J. Math. Biol. 33 : 703-722
Olsen, L.F. and Schaer, W.M. (1990)
Chaos Versus Noisy Periodicity: Alternative Hypotheses for Childhood Epi- demics.
Poundstone, W. (1992)
Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb.
Oxford University Press. Rand, D.A. (1994)
Measuring and Characterising Spatial Patterns, Dynamics and Chaos in Spa- tially Extended Dynamical Systems and Ecologies.
Phil. Trans. R. Soc. Lond. B348: 497-514
Rand, D.A., Keeling, M., and Wilson, H.B. (1995)
Invasion, Stability and Evolution to Criticality in Spatially Extended, Artifi- cial Host-Pathogen Ecologies.
Proc. R. Soc. Lond. B259 : 55-63
Rand, D.A. and Wilson, H.B. (1991)
Chaotic Stochasticity - A Ubiquitous Source of Unpredictability in Epidemics.
Proc. R. Soc. Lond. B246 : 179-184
Rand, D.A. and Wilson, H.B. (1995)
Using Spatio-Temporal Chaos and Intermediate-Scale Determinism to Quantify Spatially Extended Ecosystems.
Proc. R. Soc. Lond. B259 : 111-117
Rhodes, C.J. and Anderson, R.M. (1996)
Persistence and Dynamics in Lattice Models of Epidemic Spread.
J. Theor. Biol. 180: 125-133
Roughgarden, J. (1979)
Theory of Population Genetics and Evolutionary Ecology: An Introduction.
Macmillan Publishing Co. Inc., New York. Rozanov, Y.A., translated by Silverman, R.A. (1977)
Probability Theory: A Concise Course.
Dover Publications Inc., New York. Ruxton, G.D. (1994)
Low Levels of Immigration between Chaotic Populations can Reduce System Extinctions by Inducing Asynchronous Regular Cycles.
Proc. R. Soc. Lond. B256 : 189-193
Sato, K. and Konno, N. (1995)
Successional Dynamic Models on the 2-Dimensional Lattice Space.
Sato, K., Matsuda, H. and Sasaki, A. (1994)
Pathogen Invasion and Host Extinction in Lattice Structured Populations.
J. Math. Biol. 32 : 251-268
Scheuring, I. and Janosi, I.M. (1996)
When Two and Two Make Four: A Structured Population Without Chaos.
J. Theor. Biol. 178: 89-97
Taylor, P.D. and Jonker, L.B. (1978)
Evolutionary Stable Strategies and Game Dynamics.
Mathematical Biosciences40 : 145-156
Tilman, D. (1994)
Competition and Biodiversity in Spatially Structured Habitats.
Ecology75: 2-16
Tome, T. and Drugowich de Felicio, J.R. (1996)
Probabilistic Cellular Automaton describing a Biological Immune System.
Physical Review E53 : 3976-3981
van Baalen, M. and Rand, D.A. (1997)
The Unit of Selection in Viscous Populations and the Evolution of Altruism.
Phil. Trans. R Soc. Lond. B (Submitted) van Herwaarden, O.A. and Grasman, J. (1995)
Stochastic Epidemics: Major Outbreaks and the Duration of the Endemic Pe- riod.
J. Math. Biol. 33 : 581-601
Volterra, V. (1926)
Fluctuations in the Abundance of a Species Considered Mathematically.
Nature118: 558-560
Weisser, W.W. and Hassell, M.P. (1996)
Animals 'on the Move' Stabilize Host-Parasitoid Systems.
Proc. R. Soc. Lond. B263 : 749-754
Wilson, D.S., Pollock, G.B. and Dugatkin, L.A. (1992)
Can Altruism Evolve in Purely Viscous Populations?
Evolutionary Ecology6: 331-341
Zeeman, E.C. (1981)
Dynamics of the Evolution of Animal Conflicts.
Zhou, J. and Hethcote, H.W. (1994)
Population Size Dependent Incidence in Models for Diseases Without Immunity.