Evolutionary stability of dimorphic population states

We consider a dimorphic population state, P, which is a convex combination of two Dirac measures δx and δy, in evolutionary games with a continuous strategy space. We first establish necessary and sufficient conditions for this dimorphic population state, P, to be a rest point of the associated replicator dynamics. We provide sufficient conditions for the replicator dynamics trajectory to converge to P when it originates from the line L= { ηδx+ (1 − η) δy: 0 < η< 1 }. If the trajectory emanates from a point outside L, then we derive sufficient conditions for the trajectory to converge to L in the special case where each point in L is a rest point. We have, also, obtained condition for the trajectory to stay away from the line L in the limit. Furthermore, main results are illustrated using examples.