Solutions with a bounded support promote permanence of a distributed replicator equation

Abstract

The now classical replicator equation describes a wide variety of biological phenomena, including those in theoretical genetics, evolutionary game theory, or in the theories of the origin of life. Among other questions, the permanence of the replicator equation is well studied in the local, well-mixed case. Inasmuch as the spatial heterogeneities are key to understanding the species coexistence at least in some cases, it is important to supplement the classical theory of the nondistributed replicator equation with a spatially explicit framework. One possible approach, motivated by the porous medium equation, is introduced. It is shown that the solutions to the spatially heterogeneous replicator equation may evolve to equilibrium states that have a bounded support, and, moreover, that these solutions are of paramount importance for the overall system permanence, which is shown to be a more commonplace phenomenon for the spatially explicit equation if compared with the local model.

Keywords:

• replicator equation
• reaction–diffusion systems
• stability
• permanence
• uniform persistence

AMS Subject Classifications:

• Primary: 35K57
• 35B35
• 91A22
• Secondary: 92D25

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

ASB and VPP are supported in part by the Russian Foundation for Basic Research [grant number 13-01-00779].