American Institute of Mathematical Sciences

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

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

Alexander S. Bratus 1, , Vladimir P. Posvyanskii 1, and  Artem S. Novozhilov 1,

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.
Keywords: evolutionary stable state, reaction-diffusion systems., Nash equilibrium, Replicator equation.
Mathematics Subject Classification: Primary: 35K57, 35B35, 91A22; Secondary: 92D2.
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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

G. P. Karev, On mathematical theory of selection: Continuous time population dynamics, Journal of Mathematical Biology, 60 (2010), 107-129, arXiv:0812.4280.  Google Scholar

[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. Google Scholar

[22]

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

[23]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0.  Google Scholar

[24]

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

[25]

P. Schuster and K. Sigmund, Replicator dynamics, Journal of Theoretical Biology, 100 (1983), 533-538. doi: 10.1016/0022-5193(83)90445-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

G. P. Karev, On mathematical theory of selection: Continuous time population dynamics, Journal of Mathematical Biology, 60 (2010), 107-129, arXiv:0812.4280.  Google Scholar

[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. Google Scholar

[22]

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

[23]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0.  Google Scholar

[24]

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

[25]

P. Schuster and K. Sigmund, Replicator dynamics, Journal of Theoretical Biology, 100 (1983), 533-538. doi: 10.1016/0022-5193(83)90445-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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