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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 |
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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