Issue |
Math. Model. Nat. Phenom.
Volume 9, Number 3, 2014
Biological evolution
|
|
---|---|---|
Page(s) | 47 - 67 | |
DOI | https://doi.org/10.1051/mmnp/20149304 | |
Published online | 28 May 2014 |
Replicator Equations and Space
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
⋆
Corresponding author. E-mail: artem.novozhilov@ndsu.edu
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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