Stability of the replicator equation with continuous strategy space

We extend previous work that analyzes the stability of evolutionary dynamics on probability distributions over continuous strategy spaces. The stability concept considered is that of “neighborhood” convergence to a rest point (i.e. an equilibrium distribution over the strategy space) under the dynamics in the weak topology for all initial distributions whose support is close to that of the rest point. Stability criteria involving strategy domination and neighborhood superiority are developed for monomorphic rest points (i.e. the equilibrium distribution is supported on a single strategy) and for distributions that have finite support.