Geodesic
Evolutionary Games in Space
N. Kronik1 ; Y. Cohen2
1 Department of Applied Mathematics, Holon Institute of Technology, Holon 58102, Israel
2 Department of Fisheries, Wildlife, and Conservation Biology, University of Minnesota, St. Paul, MN 55118
Mathematical modelling of natural phenomena, Tome 4 (2009) no. 6, pp. 54-90

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The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies.
DOI : 10.1051/mmnp/20094602

N. Kronik 1 ; Y. Cohen 2

1 Department of Applied Mathematics, Holon Institute of Technology, Holon 58102, Israel
2 Department of Fisheries, Wildlife, and Conservation Biology, University of Minnesota, St. Paul, MN 55118 
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N. Kronik; Y. Cohen. Evolutionary Games in Space. Mathematical modelling of natural phenomena, Tome 4 (2009) no. 6, pp. 54-90. doi: 10.1051/mmnp/20094602

[1] P.A. Abrams Evol. Ecol. Res. 2001 369 373

[2] P.A. Abrams, H. Matsuda Evol. Ecol. 1993 465 487

[3] D. Alonso, F. Bartumeus, J. Catalan Ecology 2002 28 34

[4] H. Anton.and C. Rorres. Elementary linear algebra: applications version. 8th Edition. John Wiley Sons, New York, 2000.

[5] J. Apaloo. Revisting strategic models of evolution: The concept of neighborhood invader strategies. Theor. Pop. Biol.,52 (1997), 71–77.

[6] N.F. Britton. Reaction-diffusion equations and their applications to biology. Academic Press, New York, 1986.

[7] J.S. Brown, N.B. Pavlovic. Evolution in heterogeneous environments - effects of migtation on habitat specialization. Evol. Ecol. 6 (1992),360–382.

[8] J.S. Brown, T.L. Vincent Theor. Pop. Biol. 1987 140 166

[9] J.S. Brown, T.L. Vincent Evolution 1992 1269 1283

[10] R.G. Casten, C.J. Holland. Stability properties of solutions of systems of reaction-diffusion equations. SIAM J. Appl. Math. 33 (1977), 353–364.

[11] Y. Cohen, J. Pastor, T.L. Vincent Evol. Ecol. Res. 2000 719 743

[12] Y. Cohen, T.L. Vincent, J.S. Brown Evol. Ecol. Res. 1999 923 942

[13] R. Cressman, G.T. Vickers Math. Biol. 1997 359 369

[14] U. Dieckmann, R. Law J. Math. Biol. 1996 579 612

[15] R. Durrett, S. Levin Theor. Pop. Biol. 1994 363 394

[16] R. Durrett, S. Levin J. Theor. Biol. 1997 165 171

[17] I. Eshel. Evolutionary and continuous stability. J. Theor. Biol. 108 (1983), 99–111.

[18] I. Eshel. On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution. J. Math. Biol. 34 (1996), 485–510.

[19] I. Eshel, U. Motro. Kin selection and strong evolutionary stability of mutual help. Theor. Pop. Biol. 19 (1981), 420–433.

[20] G. Gause. The struggle for existence. Williams and Wilkins, Baltimore, 1934.

[21] S.A.H. Geritz, M. Gyllenberg, F.J.A. Jacobs, K. Parvinen J. Math. Biol. 2002 548 560

[22] S.A.H. Geritz, S.A.H. Kisdi, G. Meszéna, J.A.J. Metz Evol. Ecol. 1998 35 57

[23] S.A.H. Geritz, J.A.J. Metz. É. Kisdi, G. Meszéna. Dynamics of adaptation and evolutionary branching. Physical Review Letters 78, 2024–2027.

[24] A. Gorban Math. Model. Nat. Phenom. 2007 1 45

[25] P. Grindrod. The theory and applications of reaction-diffusion equations: patterns and waves. 2nd Edition. Clarendon press, Oxford, 1996.

[26] M. Gyllenberg, J.A. Metz J. Math. Biol. 2001 545 560

[27] K.P. Hadeler Rocky Mountain J. Math. 1981 39 45

[28] J. Haldane. The causes of evolution. Princeton University Press, 1932.

[29] W.G.S. Hines Theor. Pop. Biol. 1987 195 272

[30] V. C.L. Hutson, G.T. Vickers J. Math. Biol. 1992 457 471

[31] N. Kalev-Kronik. Evolutionary games in space. Ph.D. Thesis, University of Minneosta, 2006.

[32] W. Kaplan. Advanced calculus. Addison-Wesley, Reading, 1952.

[33] C.L. Lehman, D. Tilman. Spatial Ecology : The Role of Space in Population Dynamics and Interspecific Interactions,chapter: Competition in Spatial Habitats.. Princeton University Press, Princeton, 1997.

[34] J.L. Lions. Equations differentielles operationelles. Springer-Verlag, New-York, 1961.

[35] S. Lipschutz. Linear algebra. McGraw-Hill, New York, 1991.

[36] J. Maynard-Smith. Evolution and the theory of games. Cambridge University Press, Cambridge, 1982.

[37] J. Maynard-Smith, G. Price Nature 1973 15 18

[38] J.A.J. Metz, M. Gyllenberg Proc. Royal Soc. London B 2001 499 508

[39] J. Murray. Mathematical biology, 2nd Edition, Springer-Verlag, Berlin, 1993.

[40] C. Neuhauser J. Theor. Biol. 1998 445 463

[41] C. Neuhauser, S.W. Pacala Ann. Appl. Probab. 1999 1226 1259

[42] H.G. Othmer, L.E. Scriven Ind. Eng. Chem. Fund. 1969 302 313

[43] K. Parvinen Bul. Math. Biol. 1999 531 550

[44] H. Qian, J. Murray Appl. Math. Let. 2001 405 411

[45] A. Sasaki, I. Kawaguchi, A. Yoshimori Theor. Pop. Biol. 2002 49 71

[46] L.E. Segel, J.L. Jackson J. Theor. Biol. 1972 545 559

[47] J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 1983.

[48] T. Takada, J. Kigami J. Math. Biol. 1991 513 529

[49] P.D. Taylor Theor. Pop. Biol. 1989 125 143

[50] D. Tilman, P. Kareiva eds. Spatial ecology : the role of space in population dynamics and interspecific interactions. Princeton University Press, Princeton, 1997.

[51] A.M. Turing Phil. Trans. B. 1952 37 37

[52] G.T. Vickers, Spatial patterns and ESS's. J. Theo. Biol., 140 (1989), 129–135.

[53] G.T. Vickers J. Math. Biol. 1993 411 430

[54] T. Vincent, Evolutionary games. J. Optim. Theor. Appl., 46 (1985), 605–612.

[55] T. Vincent, J. Brown Math. Model. 1987 766 771

[56] T.L. Vincent, J. Brown. Evolutionary game theory, natural selection, and Darwinian dynamics. Cambridge University Press, Cambridge, 2005.

[57] Vincent, T. Evolutionary stable strategies in differential and difference equation models. Evol. Ecol., 2, (1988), 321–337.

[58] T.L. Vincent, Y. Cohen, J.S. Brown Theor. Pop. Biol. 1993 149 176

[59] T.L. Vincent, M.V. Van, G.S. Goh Evol. Ecol. 1996 567 591

[60] V. Zakharov, V.S. L'Vov, S.S. Starobinets Soviet Physics Uspekhii 1975 896 919

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