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Replicator Equations and Space

Published online by Cambridge University Press:  28 May 2014

A. S. Bratus
Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State University, Moscow 119992, Russia Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow 127994, Russia
V. P. Posvyanskii
Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow 127994, Russia
A. S. Novozhilov*
Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA
Corresponding author. E-mail:
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A reaction–diffusion replicator equation is studied. A novel method to apply the principle of global regulation is used to write down a model with explicit spatial structure. Properties of stationary solutions together with their stability are analyzed analytically, and relationships between stability of the rest points of the non-distributed replicator equation and the distributed system are shown. In particular, we present the conditions on the diffusion coefficients under which the non-distributed replicator equation can be used to describe the number and stability of the stationary solutions to the distributed system. A numerical example is given, which shows that the suggested modeling framework promotes the system’s persistence, i.e., a scenario is possible when in the spatially explicit system all the interacting species survive whereas some of them go extinct in the non-distributed one.

Research Article
© EDP Sciences, 2014

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G. I. Barenblatt, V. M. Entov, V. M. Ryzhik. Theory of fluid flows through natural rocks. Kluwer Academic Publishers, 1989.
A. S. Bratus, C.-K. Hu, M. V. Safro, A. S. Novozhilov. On diffusive stability of Eigen’s quasispecies model. Journal of Dynamical and Control Systems. arXiv:1212.1488 (2013).
A. S. Bratus, A. S. Novozhilov, A. P. Platonov. Dynamical systems and models in biology. Fizmatlit, 2010. in Russian.
Bratus, A. S., Posvyanskii, V. P.. Stationary solutions in a closed distributed Eigen–Schuster evolution system. Diff. Eq., 42 (2006), 17621774. CrossRefGoogle Scholar
Bratus, A. S., Posvyanskii, V. P., Novozhilov, A. S.. Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates. Nonl. Anal. Real World Appl., 11 (2010), 18971917. CrossRefGoogle Scholar
Bratus, A. S., Posvyanskii, V. P., Novozhilov, A. S.. A note on the replicator equation with explicit space and global regulation. Math. Bios. Eng., 8 (2011), 659676. CrossRefGoogle ScholarPubMed
R. S. Cantrell, C. Cosner. Spatial ecology via reaction-diffusion equations. Wiley, 2003.
Cressman, R., Dash, A. T.. Density dependence and evolutionary stable strategies. J. Theor. Biol., 126 (1987), 393406. CrossRefGoogle Scholar
Cressman, R., Vickers, G. T.. Spatial and Density Effects in Evolutionary Game Theory. J. Theor. Biol., 184 (1997), 359369. CrossRefGoogle Scholar
U. Dieckmann, R. Law, J. A. J. Metz. The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge University Press, 2000.
Eigen, M., McCascill, J., Schuster, P.. The Molecular Quasi-Species. Adv. Chem. Phys., 75 (1989), 149263. Google Scholar
L. C. Evans. Partial Differential Equations. American Mathematical Society, 2nd edition, 2010.
R. Ferriere, R. E. Michod. Wave patterns in spatial games and the evolution of cooperation. In U. Dieckmann, R. Law, and J. A. J. Metz, editors, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge University Press, (2000), 318–339.
Hadeler, K. P.. Diffusion in Fisher’s population model. R. Moun. J. Math., 11 (1981), 3945. CrossRefGoogle Scholar
J. Hofbauer, K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, 1998.
Hofbauer, J., Sigmund, K.. Evolutionary game dynamics. Bull. Am. Math. Soc., 40 (2003), 479519. CrossRefGoogle Scholar
Hutson, V. C. L., Vickers, G. T.. Travelling waves and dominance of ESS’s. J. Math. Biol., 30 (1992), 457471. CrossRefGoogle Scholar
Hutson, V. C. L., Vickers, G. T.. The Spatial Struggle of Tit-For-Tat and Defect. Phil. Trans. Royal Soc. Ser. B: Biol. Sc., 348 (1995), 393404. CrossRefGoogle ScholarPubMed
Karev, G. P., Novozhilov, A. S., Berezovskaya, F. S.. On the asymptotic behavior of the solutions to the replicator equation. Math. Med. Biol., 28 (2011), 89110. CrossRefGoogle ScholarPubMed
P. Knabner, L. Angerman. Numerical methods for elliptic and parabolic partial differential equations, vol. 44. Springer, 2003.
S. G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964.
Novozhilov, A. S., Posvyanskii, V. P., Bratus, A. S.. On the reaction–diffusion replicator systems: spatial patterns and asymptotic behaviour. Russ. J. Num. Anal. Math. Mod., 26 (2012), 555564. Google Scholar
M. A. Nowak. Evolutionary dynamics: exploring the equations of life. Harvard University Press, 2006.
K. Rektorys. Variational methods in mathematics, science and engineering. Springer, 1980.
Vaidya, N., Manapat, M. L., Chen, I. A., Xulvi-Brunet, R., Hayden, E. J., Lehman, N.. Spontaneous network formation among cooperative rna replicators. Nature, 491 (2012), 7277. CrossRefGoogle ScholarPubMed
Vickers, G. T.. Spatial patterns and ESS’s. J. Theor. Biol., 140 (1989), 12935. CrossRefGoogle Scholar
Vickers, G. T.. Spatial patterns and travelling waves in population genetics. J. Theor. Biol., 150 (1991), 329337. CrossRefGoogle Scholar
Vickers, G. T., Hutson, V. C. L., Budd, C. J.. Spatial patterns in population conflicts. J. Math. Biol., 31 (1993), 411430. CrossRefGoogle Scholar
Weinberger, E. D.. Spatial stability analysis of Eigen’s quasispecies model and the less than five membered hypercycle under global population regulation. Bull. Math. Biol., 53 (1991), 623638. CrossRefGoogle Scholar