Long-Run Analysis of the Stochastic Replicator Dynamics in the Presence of Random Jumps

Mathematical Biology Seminar
Wednesday, January 30, 2013 - 11:05am for 1 hour (actually 50 minutes)
Skiles Bld Room 005
Andrew Vlasic – Indiana University
Leonid Bunimovich
For many evolutionary dynamics, within a population there are finitely many types that compete with each other. If we think of a type as a strategy, we may consider this dynamic from a game theoretic perspective. This evolution is frequency dependent, where the fitness of each type is given by the expected payoff for an individual in that subpopulation. Considering the frequencies of the population, the logarithmic growth is given by the difference of the respective fitness and the average fitness of the population as a whole. This dynamic is Darwinian in nature, where Nash Equilibria are fixed points, and Evolutionary Stable Strategies are asymptotically stable. Fudenberg and Harris modified this deterministic dynamic by assuming the fitness of each type are subject to population level shocks, which they model by Brownian motion. The authors characterize the two strategy case, while various other authors considered the arbitrary finite strategy case, as well as different variations of this model. Considering how ecological and social anomalies affect fitness, I expand upon the Fudenberg and Harris model by adding a compensated Poisson term. This type of stochastic differential equation is no longer continuous, which complicates the analysis of the model. We will discuss the approximation of the 2 strategy case, stability of Evolutionary Stable Strategies and extinction of dominated strategies for the arbitrary finite strategy case. Examples of applications are given. Prior knowledge of game theory is not needed for this talk.