# American Institute of Mathematical Sciences

May  2012, 11(3): 981-1002. doi: 10.3934/cpaa.2012.11.981

## A hyperbolic model of spatial evolutionary game theory

 1 Dipartimento di Scienze Applicate - Università di Napoli "Parthenope", Via A. De Gasperi, 5 - 80133 Napoli, Italy 2 Dipartimento di Scienze di Base e Applicate per l’Ingegneria (SBAI), Università degli Studi “Sapienza” di Roma 3 Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, c/o Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica, 1; I-00133 Roma

Received  October 2010 Revised  February 2011 Published  December 2011

We present a one space dimensional model with finite speed of propagation for population dynamics, based both on the hyperbolic Cattaneo dynamics and the evolutionary game theory. We prove analytical properties of the model and global estimates for solutions, by using a hyperbolic nonlinear Trotter product formula.
Citation: Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
##### References:
 [1] A. L. Amadori, A. Boccabella and R. Natalini, A one dimensional hyperbolic model for evolutionary game theory: numerical approximations and simulations, Commun. Appl. Ind. Math., 1 (2010), 1-21. doi: DOI: 10.1685/2010CAIM494. [2] D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumour growth, Math. Models Methods Appl. Sci., 12 (2002), 737-754. [3] A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, 20 (2000). [4] M. C. Cattaneo, Sur une forme de l'equation de la chaleur liminant le paradoxe d'une propagation instantane, Comptes Rendus L'Acad. Sci. Ser. I-Math., 247 (1958), 431-433. [5] K. N. Chueh and C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. [6] R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory, J. Theor. Biol., 184 (1997), 359-369. [7] R. Ferriere and R. E. Michod, The evolution of cooperation in spatially heterogeneous populations, The American Naturalist, 147 (1996), 692-717 [8] F. Fu, M. A. Nowak and Ch. Hauert, Invasion and expansion of cooperators in lattice populations: Prisoner's dilemma vs. Snowdrift games, J. Theor. Biol., 266 (2010), 358-366 doi: DOI: 10.1016/j.jtbi.2010.06.042. [9] T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034. [10] T. Hillen, On the $L^2$-moment closure of transport equations: the Cattaneo approximation, Discrete Cont. Dyn. Syst., B4 (2004), 961-982. [11] J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, 1998. doi: DOI: 10.1090/S0273-0979-03-00988-1. [12] J. Hofbauer and K. Sigmund, Evolutionary game dynamics, B. Am. Math. Soc., 40 (2003), 479-519. [13] V. C. L. Hutson and G. T. Vickers, Travelling waves and dominance of ESS's, J. Math. Biol., 30 (1992), 457-471. [14] J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, 1982. [15] J. Maynard Smith and G. R. Price, The logic of animal conflicts, Nature, 246 (1973), 15-18. [16] J. Fort and V. Mendez, Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment, Rep. Prog. Phys., 65 (2002), 895-954. [17] V. Mendez, S. Fedotov and W. Horsthemke, "Reaction-transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities Series: Springer Series in Synergetics,", 2010., (). [18] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. xxvi+811 pp. [19] J. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295. [20] M. A. Nowak, "Evolutionary Dynamics," Harvard University Press, 2001. [21] M. E. Taylor, "Partial Differential Equations III. Nonlinear Equations," Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. xxii+715 pp. [22] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (2001), 145-156. [23] G. T. Vickers, Spatial patterns and ESS's, J. Theor. Biol., 140 (1989), 129-135.

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##### References:
 [1] A. L. Amadori, A. Boccabella and R. Natalini, A one dimensional hyperbolic model for evolutionary game theory: numerical approximations and simulations, Commun. Appl. Ind. Math., 1 (2010), 1-21. doi: DOI: 10.1685/2010CAIM494. [2] D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumour growth, Math. Models Methods Appl. Sci., 12 (2002), 737-754. [3] A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, 20 (2000). [4] M. C. Cattaneo, Sur une forme de l'equation de la chaleur liminant le paradoxe d'une propagation instantane, Comptes Rendus L'Acad. Sci. Ser. I-Math., 247 (1958), 431-433. [5] K. N. Chueh and C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. [6] R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory, J. Theor. Biol., 184 (1997), 359-369. [7] R. Ferriere and R. E. Michod, The evolution of cooperation in spatially heterogeneous populations, The American Naturalist, 147 (1996), 692-717 [8] F. Fu, M. A. Nowak and Ch. Hauert, Invasion and expansion of cooperators in lattice populations: Prisoner's dilemma vs. Snowdrift games, J. Theor. Biol., 266 (2010), 358-366 doi: DOI: 10.1016/j.jtbi.2010.06.042. [9] T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034. [10] T. Hillen, On the $L^2$-moment closure of transport equations: the Cattaneo approximation, Discrete Cont. Dyn. Syst., B4 (2004), 961-982. [11] J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, 1998. doi: DOI: 10.1090/S0273-0979-03-00988-1. [12] J. Hofbauer and K. Sigmund, Evolutionary game dynamics, B. Am. Math. Soc., 40 (2003), 479-519. [13] V. C. L. Hutson and G. T. Vickers, Travelling waves and dominance of ESS's, J. Math. Biol., 30 (1992), 457-471. [14] J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, 1982. [15] J. Maynard Smith and G. R. Price, The logic of animal conflicts, Nature, 246 (1973), 15-18. [16] J. Fort and V. Mendez, Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment, Rep. Prog. Phys., 65 (2002), 895-954. [17] V. Mendez, S. Fedotov and W. Horsthemke, "Reaction-transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities Series: Springer Series in Synergetics,", 2010., (). [18] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. xxvi+811 pp. [19] J. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295. [20] M. A. Nowak, "Evolutionary Dynamics," Harvard University Press, 2001. [21] M. E. Taylor, "Partial Differential Equations III. Nonlinear Equations," Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. xxii+715 pp. [22] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (2001), 145-156. [23] G. T. Vickers, Spatial patterns and ESS's, J. Theor. Biol., 140 (1989), 129-135.
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