Robust permanence and impermanence for stochastic replicator dynamics

Garay and Hofbauer (SIAM J. Math. Anal. 34 (2003)) proposed sufficient conditions for robust permanence and impermanence of the deterministic replicator dynamics. We reconsider these conditions in the context of the stochastic replicator dynamics, which is obtained from its deterministic analogue by introducing Brownian perturbations of payoffs. When the deterministic replicator dynamics is permanent and the noise level small, the stochastic dynamics admits a unique ergodic distribution whose mass is concentrated near the maximal interior attractor of the unperturbed system; thus, permanence is robust against small unbounded stochastic perturbations. When the deterministic dynamics is impermanent and the noise level small or large, the stochastic dynamics converges to the boundary of the state space at an exponential rate.