Math. Model. Nat. Phenom.
Volume 9, Number 3, 2014Biological evolution
|Page(s)||47 - 67|
|Published online||28 May 2014|
Replicator Equations and Space
1 Faculty of Computational Mathematics
and Cybernetics Lomonosov Moscow State University, Moscow
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
Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 35K57 / 35B35 / 91A22 / 92D25
Key words: replicator equation / reaction-diffusion systems / stability / persistence
© EDP Sciences, 2014
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