  In [12] we formulated an evolutionary game theory model as a dynamical system
on 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-time
behavior of solutions to the model. In particular, we show that M(Q), the
finite signed Borel measures are asymptotically closed. This means that if the
initial condition is a finite signed Borel measure and if the asymptotic limit
of the model solution exists, then it will be a measure (note that function
spaces such as L1(Q) and C(Q) do not have this property). We also establish
permanence results for the full replicator mutator model. Then, we study the
asymptotic analysis in the case where there is more than one strategy of a
given fitness (a continuum of strategies of a given fitness), a case that often
arises in applications. To study this case our mathematical structure must
include the ability to demonstrate the convergence of the model solution to a
measure supported on a continuum of strategies. For this purpose, we
demonstrate how to perform completions of the space of measures and how to use
these completions to formulate weak (generalized) asymptotic limits. In
particular, we show that for the pure replicator dynamics the (weak) solution
of the dynamical system converges to a Dirac measure centered at the fittest
strategy class; thus this Dirac measure is a globally attractive equilibrium
point which is termed a continuously stable strategy (CSS). It is also shown
that in the discrete case of the pure replicator dynamics and even for small
perturbation of the pure replicator dynamics (i.e., selection with small
mutation) there exists a globally asymptotically stable equilibrium.

Dynamical Systems