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

TL;DR

This paper introduces a spatially explicit replicator equation inspired by the porous medium equation, demonstrating that solutions with bounded support enhance species coexistence and permanence in biological systems.

Contribution

It extends classical replicator dynamics by incorporating spatial heterogeneity and bounded support solutions, highlighting their role in promoting permanence.

Findings

Solutions can evolve to bounded support equilibrium states.

Spatial heterogeneity increases the likelihood of species permanence.

Bounded support solutions are crucial for species coexistence.

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 non-distributed 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.