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In [12] we formulated an evolutionary game theory model as a dynamical systemon the state space of finite signed Borel measures under the weak* topology.The focus of this paper is to extend the analysis to include the long-timebehavior of solutions to the model. In particular, we show that M(Q), thefinite signed Borel measures are asymptotically closed. This means that if theinitial condition is a finite signed Borel measure and if the asymptotic limitof the model solution exists, then it will be a measure (note that functionspaces such as L1(Q) and C(Q) do not have this property). We also establishpermanence results for the full replicator mutator model. Then, we study theasymptotic analysis in the case where there is more than one strategy of agiven fitness (a continuum of strategies of a given fitness), a case that oftenarises in applications. To study this case our mathematical structure mustinclude the ability to demonstrate the convergence of the model solution to ameasure supported on a continuum of strategies. For this purpose, wedemonstrate how to perform completions of the space of measures and how to usethese completions to formulate weak (generalized) asymptotic limits. Inparticular, we show that for the pure replicator dynamics the (weak) solutionof the dynamical system converges to a Dirac measure centered at the fitteststrategy class; thus this Dirac measure is a globally attractive equilibriumpoint which is termed a continuously stable strategy (CSS). It is also shownthat in the discrete case of the pure replicator dynamics and even for smallperturbation of the pure replicator dynamics (i.e., selection with smallmutation) there exists a globally asymptotically stable equilibrium.

Evolutionary Game Theory on Measure Spaces: Asymptotic Behavior of Solutions