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

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.