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Replicator Dynamics With Alternate Growth Functions, Delay, And Quasiperiodic Forcing
Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics. The growth rate of each strategy is its excess fitness: the deviation of its fitness from the average. The gametheoretic aspect of the model lies in the choice of fitness function, which is determined by a payoff matrix. Two well-known replicator systems are the threestrategy Rock-Paper-Scissors game and the two-strategy Hawk-Dove game. In this work, we analyze the dynamics of replicator systems with three different types of modifications. The first generalization of the replicator model is given by considering alternate growth functions. We find that in the Rock-Paper-Scissors game with a logistic growth function, there are several fixed points that do not exist in the standard replicator model. The system exhibits both periodic motion and convergence to attractors. We also analyze replicator systems with delayed interactions between strategies. We consider a symmetric delay model, in which the fitness of each strategy is its expected payoff delayed by a time interval; and an asymmetric model, in which same-strategy terms appearing in the fitness of a given strategy are not delayed. In both cases, limit cycles arise that cannot occur in the usual replicator model. Finally, we examine Rock-Paper-Scissors systems with quasiperiodic forcing of the payoff coefficients. This model may represent systems in which the competition is affected by cyclical processes on different time-scales. We find that the stability of the equilibrium state depends sensitively on the two forcing frequencies; in fact, the region of stability has fractal boundary.
evolutionary dynamics; dynamical systems; perturbation
Guckenheimer,John Mark; Steen,Paul Herman; Strogatz,Steven H
Ph.D. of Applied Mathematics
Doctor of Philosophy
dissertation or thesis