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, in "Artificial Life" (eds. C. G. Langton, C. Taylor, J. D. Farmer and S. Rasmussen), Addison-Wesley, 2 (1991), 255-276.

[2]

M. C. Boerlijst and P. Hogeweg, Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites, Physica D, 48 (1991), 17-28. 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), 1564-1576.

[4]

A. S. Bratus, A. S. Novozhilov and A. P. Platonov, "Dynamical Systems and Models in Biology," (Russian), Fizmatlit, 2010.

[5]

A. S. Bratus and V. P. Posvyanskii, Stationary solutions in a closed distributed Eigen-Schuster evolution system, Differential Equations, 42 (2006), 1762-1774.

[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), 1897-1917, arXiv:0901.3556.

[7]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003.

[8]

R. Cressman, "Evolutionary Dynamics and Extensive Form Games," MIT Press Series on Economic Learning and Social Evolution, 5, MIT Press, Cambridge, 2003.

[9]

R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory, Journal of Theoretical Biology, 184 (1997), 359-369. 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), 31-49. 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, IIASA, Laxenburg, Cambridge University Press, Cambridge, 2005.

[12]

M. Eigen and P. Schuster, The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle, Naturwissenschaften, 64 (1977), 541-565. doi: 10.1007/BF00450633.

[13]

R. Ferriere and R. E. Michod, Wave patterns in spatial games and the evolution of cooperation, in "The Geometry of Ecological Interactions: Simplifying Spatial Complexity," (eds. U. Dieckmann, R. Law and J. A. J. Metz), Cambridge University Press, (2000), 318-339. doi: 10.1017/CBO9780511525537.020.

[14]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 353-369. 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, Springer-Verlag, New York, 1990.

[16]

K. P. Hadeler, Diffusion in Fisher's population model, Rocky Mountain Journal of Mathematics, 11 (1981), 39-45. doi: 10.1216/RMJ-1981-11-1-39.

[17]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, 1998.

[18]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of American Mathematical Society, 40 (2003), 479-519. 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), 393-404. 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), 107-129, arXiv:0812.4280.

[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), in press, arXiv:0906.4986.

[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), 15-18. doi: 10.1038/246015a0.

[24]

S. G. Mikhlin, "Variational Methods in Mathematical Physics," A Pergamon Press Book, The Macmillan Co., New York, 1964.

[25]

P. Schuster and K. Sigmund, Replicator dynamics, Journal of Theoretical Biology, 100 (1983), 533-538. 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, Kluwer Academic Publishers, Dordrecht, 1990.

[27]

P. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.

[28]

G. T. Vickers, Spatial patterns and ESS's, Journal of Theoretical Biology, 140 (1989), 129-135. 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), 623-638.

show all references

References:
[1]

M. Boerlijst and P. Hogeweg, Self-structuring and selection: Spiral waves as a substrate for prebiotic evolution, in "Artificial Life" (eds. C. G. Langton, C. Taylor, J. D. Farmer and S. Rasmussen), Addison-Wesley, 2 (1991), 255-276.

[2]

M. C. Boerlijst and P. Hogeweg, Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites, Physica D, 48 (1991), 17-28. 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), 1564-1576.

[4]

A. S. Bratus, A. S. Novozhilov and A. P. Platonov, "Dynamical Systems and Models in Biology," (Russian), Fizmatlit, 2010.

[5]

A. S. Bratus and V. P. Posvyanskii, Stationary solutions in a closed distributed Eigen-Schuster evolution system, Differential Equations, 42 (2006), 1762-1774.

[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), 1897-1917, arXiv:0901.3556.

[7]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003.

[8]

R. Cressman, "Evolutionary Dynamics and Extensive Form Games," MIT Press Series on Economic Learning and Social Evolution, 5, MIT Press, Cambridge, 2003.

[9]

R. Cressman and G. T. Vickers, Spatial and density effects in evolutionary game theory, Journal of Theoretical Biology, 184 (1997), 359-369. 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), 31-49. 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, IIASA, Laxenburg, Cambridge University Press, Cambridge, 2005.

[12]

M. Eigen and P. Schuster, The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle, Naturwissenschaften, 64 (1977), 541-565. doi: 10.1007/BF00450633.

[13]

R. Ferriere and R. E. Michod, Wave patterns in spatial games and the evolution of cooperation, in "The Geometry of Ecological Interactions: Simplifying Spatial Complexity," (eds. U. Dieckmann, R. Law and J. A. J. Metz), Cambridge University Press, (2000), 318-339. doi: 10.1017/CBO9780511525537.020.

[14]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 353-369. 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, Springer-Verlag, New York, 1990.

[16]

K. P. Hadeler, Diffusion in Fisher's population model, Rocky Mountain Journal of Mathematics, 11 (1981), 39-45. doi: 10.1216/RMJ-1981-11-1-39.

[17]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, 1998.

[18]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of American Mathematical Society, 40 (2003), 479-519. 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), 393-404. 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), 107-129, arXiv:0812.4280.

[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), in press, arXiv:0906.4986.

[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), 15-18. doi: 10.1038/246015a0.

[24]

S. G. Mikhlin, "Variational Methods in Mathematical Physics," A Pergamon Press Book, The Macmillan Co., New York, 1964.

[25]

P. Schuster and K. Sigmund, Replicator dynamics, Journal of Theoretical Biology, 100 (1983), 533-538. 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, Kluwer Academic Publishers, Dordrecht, 1990.

[27]

P. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.

[28]

G. T. Vickers, Spatial patterns and ESS's, Journal of Theoretical Biology, 140 (1989), 129-135. 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), 623-638.

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