A general classification of the replication dynamics with a unique fixed point in the interior of simplex $S_N$

TL;DR

This paper provides a comprehensive classification of replication dynamics with a unique fixed point inside the simplex for any number of strategies, extending the understanding beyond the well-studied cases of two and three strategies.

Contribution

It establishes necessary and sufficient conditions for the existence of a unique fixed point in the interior of the simplex for n-strategy replication dynamics.

Findings

Derived conditions for unique fixed points in interior of simplex for n strategies

Classified types of replication dynamics with a unique fixed point

Extended classification from known cases to general n strategies

Abstract

The replication dynamics (differential equation system) is the foundation of evolutionary game theory. When n=2, there are four possible types of replication dynamics. When n=3, there are 49 possible types of replication dynamics. However, when n>3, the classification of replication dynamics has not been solved. In this article, the sufficient and necessary conditions of the replication dynamics equation with a unique fixed point in the interior of simplex $S_n$(Int$S_n$) for $n\geq 2$ are presented. Furthermore, the different types of replication dynamics equations with a unique fixed point in IntSn is discussed.