Geodesic
Replicator Equations and Space
A. S. Bratus1, 2 ; V. P. Posvyanskii2 ; A. S. Novozhilov3
1 Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State University, Moscow 119992, Russia
2 Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow 127994, Russia
3 Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 3, pp. 47-67.

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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.
DOI : 10.1051/mmnp/20149304

A. S. Bratus 1, 2 ; V. P. Posvyanskii 2 ; A. S. Novozhilov 3

1 Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State University, Moscow 119992, Russia
2 Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow 127994, Russia
3 Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA 
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A. S. Bratus; V. P. Posvyanskii; A. S. Novozhilov. Replicator Equations and Space. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 3, pp. 47-67. doi : 10.1051/mmnp/20149304. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149304/

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