2012/02/16 by John D. Cleveland, Cleveland, John, Azmy S. Ackleh +2
Biochemistry, Genetics and Molecular Biology · Decision Sciences · Social Sciences · #Dynamical Systems (math.DS) #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Mathematics #Game Theory and Applications
paper · pdf · doi:10.48550/arxiv.1202.3689
openalex publication_date 2012/02/16 · openalex created_date 2022/09/16 · openalex updated_date 2026/07/28
In [12] we formulated an evolutionary game theory model as a dynamical system\non the state space of finite signed Borel measures under the weak* topology.\nThe focus of this paper is to extend the analysis to include the long-time\nbehavior of solutions to the model. In particular, we show that M(Q), the\nfinite signed Borel measures are asymptotically closed. This means that if the\ninitial condition is a finite signed Borel measure and if the asymptotic limit\nof the model solution exists, then it will be a measure (note that function\nspaces such as L1(Q) and C(Q) do not have this property). We also establish\npermanence results for the full replicator mutator model. Then, we study the\nasymptotic analysis in the case where there is more than one strategy of a\ngiven fitness (a continuum of strategies of a given fitness), a case that often\narises in applications. To study this case our mathematical structure must\ninclude the ability to demonstrate the convergence of the model solution to a\nmeasure supported on a continuum of strategies. For this purpose, we\ndemonstrate how to perform completions of the space of measures and how to use\nthese completions to formulate weak (generalized) asymptotic limits. In\nparticular, we show that for the pure replicator dynamics the (weak) solution\nof the dynamical system converges to a Dirac measure centered at the fittest\nstrategy class; thus this Dirac measure is a globally attractive equilibrium\npoint which is termed a continuously stable strategy (CSS). It is also shown\nthat in the discrete case of the pure replicator dynamics and even for small\nperturbation of the pure replicator dynamics (i.e., selection with small\nmutation) there exists a globally asymptotically stable equilibrium.\n