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Robust permanence and impermanence for stochastic replicator dynamics. (English) Zbl 1140.92025

Summary: B. M. Garay and J. Hofbauer [SIAM J. Math. Anal. 34, No. 5, 1007–1039 (2003; Zbl 1026.37006)] proposed sufficient conditions for robust permanence and impermanence of the deterministic replicator dynamics. We reconsider these conditions in the context of 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.

MSC:

92D40 Ecology
37A50 Dynamical systems and their relations with probability theory and stochastic processes
92B05 General biology and biomathematics
37A99 Ergodic theory

Citations:

Zbl 1026.37006
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