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Replicator Equations and Space

Published online by Cambridge University Press:  28 May 2014

A. S. Bratus ,
V. P. Posvyanskii  and
A. S. Novozhilov
A. S. Bratus
Affiliation:
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
Affiliation:
Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow 127994, Russia
A. S. Novozhilov*
Affiliation:
Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA
*
⋆ Corresponding author. E-mail: artem.novozhilov@ndsu.edu
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Abstract

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.

Keywords

replicator equationreaction-diffusion systemsstabilitypersistence
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 3: Biological evolution , 2014 , pp. 47 - 67
Copyright
© EDP Sciences, 2014

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