A general classification of the replication dynamics with a unique fixed point in the interior of simplex $S_N$
This solves an open classification problem in evolutionary game theory for higher-dimensional systems.
The authors provide sufficient and necessary conditions for replication dynamics with a unique interior fixed point for any n≥2, and classify the possible types for n>3, extending previous classifications for n=2 and n=3.
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.