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

A hyperbolic model of spatial evolutionary game theory

Anna Lisa Amadori 1, , Astridh Boccabella 2, and  Roberto Natalini 3,

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.
Keywords: Prisoner's dilemma., hyperbolic equations, Evolutionary game theory, Cattaneo model.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure & 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.  doi: DOI: 10.1685/2010CAIM494.  Google Scholar

[2]

D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumour growth,, Math. Models Methods Appl. Sci., 12 (2002), 737.   Google Scholar

[3]

A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem,, Oxford Lecture Series in Mathematics and its Applications, 20 (2000).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[6]

R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory,, J. Theor. Biol., 184 (1997), 359.   Google Scholar

[7]

R. Ferriere and R. E. Michod, The evolution of cooperation in spatially heterogeneous populations,, The American Naturalist, 147 (1996), 692.   Google Scholar

[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.  doi: DOI: 10.1016/j.jtbi.2010.06.042.  Google Scholar

[9]

T. Hillen, Hyperbolic models for chemosensitive movement,, Math. Models Methods Appl. Sci., 12 (2002), 1007.   Google Scholar

[10]

T. Hillen, On the $L^2$-moment closure of transport equations: the Cattaneo approximation,, Discrete Cont. Dyn. Syst., B4 (2004), 961.   Google Scholar

[11]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998).  doi: DOI: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[12]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics,, B. Am. Math. Soc., 40 (2003), 479.   Google Scholar

[13]

V. C. L. Hutson and G. T. Vickers, Travelling waves and dominance of ESS's,, J. Math. Biol., 30 (1992), 457.   Google Scholar

[14]

J. Maynard Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982).   Google Scholar

[15]

J. Maynard Smith and G. R. Price, The logic of animal conflicts,, Nature, 246 (1973), 15.   Google Scholar

[16]

J. Fort and V. Mendez, Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment,, Rep. Prog. Phys., 65 (2002), 895.   Google Scholar

[17]

V. Mendez, S. Fedotov and W. Horsthemke, "Reaction-transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities Series: Springer Series in Synergetics,", 2010., ().   Google Scholar

[18]

J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications,", Third edition. Interdisciplinary Applied Mathematics, (2003).   Google Scholar

[19]

J. Nash, Non-cooperative games,, Ann. of Math., 54 (1951), 286.   Google Scholar

[20]

M. A. Nowak, "Evolutionary Dynamics,", Harvard University Press, (2001).   Google Scholar

[21]

M. E. Taylor, "Partial Differential Equations III. Nonlinear Equations,", Second edition. Applied Mathematical Sciences, (2011).   Google Scholar

[22]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics,, Math. Biosci., 40 (2001), 145.   Google Scholar

[23]

G. T. Vickers, Spatial patterns and ESS's,, J. Theor. Biol., 140 (1989), 129.   Google Scholar

show all references

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.  doi: DOI: 10.1685/2010CAIM494.  Google Scholar

[2]

D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumour growth,, Math. Models Methods Appl. Sci., 12 (2002), 737.   Google Scholar

[3]

A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem,, Oxford Lecture Series in Mathematics and its Applications, 20 (2000).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[6]

R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory,, J. Theor. Biol., 184 (1997), 359.   Google Scholar

[7]

R. Ferriere and R. E. Michod, The evolution of cooperation in spatially heterogeneous populations,, The American Naturalist, 147 (1996), 692.   Google Scholar

[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.  doi: DOI: 10.1016/j.jtbi.2010.06.042.  Google Scholar

[9]

T. Hillen, Hyperbolic models for chemosensitive movement,, Math. Models Methods Appl. Sci., 12 (2002), 1007.   Google Scholar

[10]

T. Hillen, On the $L^2$-moment closure of transport equations: the Cattaneo approximation,, Discrete Cont. Dyn. Syst., B4 (2004), 961.   Google Scholar

[11]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998).  doi: DOI: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[12]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics,, B. Am. Math. Soc., 40 (2003), 479.   Google Scholar

[13]

V. C. L. Hutson and G. T. Vickers, Travelling waves and dominance of ESS's,, J. Math. Biol., 30 (1992), 457.   Google Scholar

[14]

J. Maynard Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982).   Google Scholar

[15]

J. Maynard Smith and G. R. Price, The logic of animal conflicts,, Nature, 246 (1973), 15.   Google Scholar

[16]

J. Fort and V. Mendez, Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment,, Rep. Prog. Phys., 65 (2002), 895.   Google Scholar

[17]

V. Mendez, S. Fedotov and W. Horsthemke, "Reaction-transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities Series: Springer Series in Synergetics,", 2010., ().   Google Scholar

[18]

J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications,", Third edition. Interdisciplinary Applied Mathematics, (2003).   Google Scholar

[19]

J. Nash, Non-cooperative games,, Ann. of Math., 54 (1951), 286.   Google Scholar

[20]

M. A. Nowak, "Evolutionary Dynamics,", Harvard University Press, (2001).   Google Scholar

[21]

M. E. Taylor, "Partial Differential Equations III. Nonlinear Equations,", Second edition. Applied Mathematical Sciences, (2011).   Google Scholar

[22]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics,, Math. Biosci., 40 (2001), 145.   Google Scholar

[23]

G. T. Vickers, Spatial patterns and ESS's,, J. Theor. Biol., 140 (1989), 129.   Google Scholar

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