# American Institute of Mathematical Sciences

2011, 8(3): 659-676. doi: 10.3934/mbe.2011.8.659

## A note on the replicator equation with explicit space and global regulation

 1 Applied Mathematics–1, Moscow State University of Railway Engineering, Obraztsova 9, Moscow, 127994, Russian Federation, Russian Federation, Russian Federation

Received  July 2010 Revised  December 2010 Published  June 2011

A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.
Citation: Alexander S. Bratus, Vladimir P. Posvyanskii, Artem S. Novozhilov. A note on the replicator equation with explicit space and global regulation. Mathematical Biosciences & Engineering, 2011, 8 (3) : 659-676. doi: 10.3934/mbe.2011.8.659
##### References:
 [1] M. Boerlijst and P. Hogeweg, Self-structuring and selection: Spiral waves as a substrate for prebiotic evolution,, Artificial Life, 2 (1991), 255. [2] M. C. Boerlijst and P. Hogeweg, Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites,, Physica D, 48 (1991), 17. doi: 10.1016/0167-2789(91)90049-F. [3] A. S. Bratus and E. N. Lukasheva, Stability and the limit behavior of the open distributed hypercycle system,, Differential Equations, 45 (2009), 1564. [4] A. S. Bratus, A. S. Novozhilov and A. P. Platonov, "Dynamical Systems and Models in Biology,", "Dynamical Systems and Models in Biology,", (2010). [5] A. S. Bratus and V. P. Posvyanskii, Stationary solutions in a closed distributed Eigen-Schuster evolution system,, Differential Equations, 42 (2006), 1762. [6] A. S. Bratus, V. P. Posvyanskii and A. S. Novozhilov, Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates,, Nonlinear Analysis: Real World Applications, 11 (2010), 1897. [7] R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley Series in Mathematical and Computational Biology, (2003). [8] R. Cressman, "Evolutionary Dynamics and Extensive Form Games,", MIT Press Series on Economic Learning and Social Evolution, 5 (2003). [9] R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory,, Journal of Theoretical Biology, 184 (1997), 359. doi: 10.1006/jtbi.1996.0251. [10] M. B. Cronhjort and C. Blomberg, Hypercycles versus parasites in a two dimensional partial differential equation model,, Journal of Theoretical Biology, 169 (1994), 31. doi: 10.1006/jtbi.1994.1128. [11] U. Dieckmann, R. Law and J. A. J. Metz, editors, "The Geometry of Ecological Interactions: Simplifying Spatial Complexity,", Cambridge Studies in Adaptive Dynamics, 1 (2005). [12] M. Eigen and P. Schuster, The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle,, Naturwissenschaften, 64 (1977), 541. doi: 10.1007/BF00450633. [13] R. Ferriere and R. E. Michod, Wave patterns in spatial games and the evolution of cooperation,, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, (2000), 318. doi: 10.1017/CBO9780511525537.020. [14] R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics, 7 (1937), 353. doi: 10.1111/j.1469-1809.1937.tb02153.x. [15] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, 42 (1990). [16] K. P. Hadeler, Diffusion in Fisher's population model,, Rocky Mountain Journal of Mathematics, 11 (1981), 39. doi: 10.1216/RMJ-1981-11-1-39. [17] J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998). [18] J. Hofbauer and K. Sigmund, Evolutionary game dynamics,, Bulletin of American Mathematical Society, 40 (2003), 479. doi: 10.1090/S0273-0979-03-00988-1. [19] V. C. L. Hutson and G. T. Vickers, The spatial struggle of tit-for-tat and defect,, Philosophical Transactions of the Royal Society. Series B: Biological Sciences, 348 (1995), 393. doi: 10.1098/rstb.1995.0077. [20] G. P. Karev, On mathematical theory of selection: Continuous time population dynamics,, Journal of Mathematical Biology, 60 (2010), 107. [21] G. P. Karev, A. S. Novozhilov and F. S. Berezovskaya, On the asymptotic behavior of the solutions to the replicator equation,, Mathematical Medicine and Biology, (2010). [22] J. Maynard Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). [23] J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15. doi: 10.1038/246015a0. [24] S. G. Mikhlin, "Variational Methods in Mathematical Physics,", A Pergamon Press Book, (1964). [25] P. Schuster and K. Sigmund, Replicator dynamics,, Journal of Theoretical Biology, 100 (1983), 533. doi: 10.1016/0022-5193(83)90445-9. [26] Y. M. Svirezhev and V. P. Passekov, "Fundamentals of Mathematical Evolutionary Genetics,", Mathematics and its Applications (Soviet Series), 22 (1990). [27] P. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics,, Mathematical Biosciences, 40 (1978), 145. doi: 10.1016/0025-5564(78)90077-9. [28] G. T. Vickers, Spatial patterns and ESS's,, Journal of Theoretical Biology, 140 (1989), 129. doi: 10.1016/S0022-5193(89)80033-5. [29] E. D. Weinberger, Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation,, Bulletin of Mathematical Biology, 53 (1991), 623.

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##### References:
 [1] M. Boerlijst and P. Hogeweg, Self-structuring and selection: Spiral waves as a substrate for prebiotic evolution,, Artificial Life, 2 (1991), 255. [2] M. C. Boerlijst and P. Hogeweg, Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites,, Physica D, 48 (1991), 17. doi: 10.1016/0167-2789(91)90049-F. [3] A. S. Bratus and E. N. Lukasheva, Stability and the limit behavior of the open distributed hypercycle system,, Differential Equations, 45 (2009), 1564. [4] A. S. Bratus, A. S. Novozhilov and A. P. Platonov, "Dynamical Systems and Models in Biology,", "Dynamical Systems and Models in Biology,", (2010). [5] A. S. Bratus and V. P. Posvyanskii, Stationary solutions in a closed distributed Eigen-Schuster evolution system,, Differential Equations, 42 (2006), 1762. [6] A. S. Bratus, V. P. Posvyanskii and A. S. Novozhilov, Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates,, Nonlinear Analysis: Real World Applications, 11 (2010), 1897. [7] R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley Series in Mathematical and Computational Biology, (2003). [8] R. Cressman, "Evolutionary Dynamics and Extensive Form Games,", MIT Press Series on Economic Learning and Social Evolution, 5 (2003). [9] R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory,, Journal of Theoretical Biology, 184 (1997), 359. doi: 10.1006/jtbi.1996.0251. [10] M. B. Cronhjort and C. Blomberg, Hypercycles versus parasites in a two dimensional partial differential equation model,, Journal of Theoretical Biology, 169 (1994), 31. doi: 10.1006/jtbi.1994.1128. [11] U. Dieckmann, R. Law and J. A. J. Metz, editors, "The Geometry of Ecological Interactions: Simplifying Spatial Complexity,", Cambridge Studies in Adaptive Dynamics, 1 (2005). [12] M. Eigen and P. Schuster, The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle,, Naturwissenschaften, 64 (1977), 541. doi: 10.1007/BF00450633. [13] R. Ferriere and R. E. Michod, Wave patterns in spatial games and the evolution of cooperation,, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, (2000), 318. doi: 10.1017/CBO9780511525537.020. [14] R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics, 7 (1937), 353. doi: 10.1111/j.1469-1809.1937.tb02153.x. [15] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, 42 (1990). [16] K. P. Hadeler, Diffusion in Fisher's population model,, Rocky Mountain Journal of Mathematics, 11 (1981), 39. doi: 10.1216/RMJ-1981-11-1-39. [17] J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998). [18] J. Hofbauer and K. Sigmund, Evolutionary game dynamics,, Bulletin of American Mathematical Society, 40 (2003), 479. doi: 10.1090/S0273-0979-03-00988-1. [19] V. C. L. Hutson and G. T. Vickers, The spatial struggle of tit-for-tat and defect,, Philosophical Transactions of the Royal Society. Series B: Biological Sciences, 348 (1995), 393. doi: 10.1098/rstb.1995.0077. [20] G. P. Karev, On mathematical theory of selection: Continuous time population dynamics,, Journal of Mathematical Biology, 60 (2010), 107. [21] G. P. Karev, A. S. Novozhilov and F. S. Berezovskaya, On the asymptotic behavior of the solutions to the replicator equation,, Mathematical Medicine and Biology, (2010). [22] J. Maynard Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). [23] J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15. doi: 10.1038/246015a0. [24] S. G. Mikhlin, "Variational Methods in Mathematical Physics,", A Pergamon Press Book, (1964). [25] P. Schuster and K. Sigmund, Replicator dynamics,, Journal of Theoretical Biology, 100 (1983), 533. doi: 10.1016/0022-5193(83)90445-9. [26] Y. M. Svirezhev and V. P. Passekov, "Fundamentals of Mathematical Evolutionary Genetics,", Mathematics and its Applications (Soviet Series), 22 (1990). [27] P. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics,, Mathematical Biosciences, 40 (1978), 145. doi: 10.1016/0025-5564(78)90077-9. [28] G. T. Vickers, Spatial patterns and ESS's,, Journal of Theoretical Biology, 140 (1989), 129. doi: 10.1016/S0022-5193(89)80033-5. [29] E. D. Weinberger, Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation,, Bulletin of Mathematical Biology, 53 (1991), 623.
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