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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, RussiaApplied 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

Abstract

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A reaction–diffusion replicator equation is studied. A novel method to apply theprinciple of global regulation is used to write down a model with explicit spatialstructure. Properties of stationary solutions together with their stability are analyzedanalytically, and relationships between stability of the rest points of thenon-distributed replicator equation and the distributed system are shown. In particular,we present the conditions on the diffusion coefficients under which the non-distributedreplicator equation can be used to describe the number and stability of the stationarysolutions to the distributed system. A numerical example is given, which shows that thesuggested modeling framework promotes the system’s persistence, i.e., a scenario ispossible when in the spatially explicit system all the interacting species survive whereassome of them go extinct in the non-distributed one.

Keywords

replicator equationreaction-diffusion systemsstabilitypersistence

Information

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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