On the asymptotic behavior of the solutions to the replicator equation

TL;DR

This paper introduces novel analytical methods for studying the asymptotic behavior of solutions to the replicator equation, reducing complex high-dimensional problems to simpler escort systems and analyzing their stability properties.

Contribution

It applies singular value decomposition and Newton diagram methods to analyze replicator equations with low-rank interaction matrices, providing new insights into their asymptotic states.

Findings

Full analysis of rank 1 interaction matrices and conditions for polymorphic equilibrium.

Demonstration of a stable equilibrium on the 1-skeleton for rank 2 systems.

Reduction of high-dimensional problems to lower-dimensional escort systems.

Abstract

Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm the interaction matrix of the replicator equation should be transformed; in particular the standard singular value decomposition allows us to rewrite the replicator equation in a convenient form. The original $n$-dimensional problem is reduced to the analysis of asymptotic behavior of the solutions to the so-called escort system, which in some important cases can be of significantly smaller dimension than the original system. The Newton diagram methods are applied to study the asymptotic behavior of the solutions to the escort system, when interaction matrix has rank 1 or 2. A general replicator equation with the interaction matrix of rank 1 is fully analyzed; the conditions are provided when the asymptotic state is a polymorphic equilibrium. As an example of the system with the interaction matrix of rank 2 we consider the problem from [Adams, M.R. and Sornborger, A.T., J Math Biol, 54:357-384, 2007], for which we show, for arbitrary dimension of the system and under some suitable conditions, that generically one globally stable equilibrium exits on the 1-skeleton of the simplex.