Benaïm, Michel; Hofbauer, Josef; Sandholm, William H. Robust permanence and impermanence for stochastic replicator dynamics. (English) Zbl 1140.92025 J. Biol. Dyn. 2, No. 2, 180-195 (2008). 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. Cited in 26 Documents MSC: 92D40 Ecology 37A50 Dynamical systems and their relations with probability theory and stochastic processes 92B05 General biology and biomathematics 37A99 Ergodic theory Keywords:permanence; replicator dynamics; stochastic differential equation; recurrence; transience Citations:Zbl 1026.37006 PDFBibTeX XMLCite \textit{M. Benaïm} et al., J. Biol. Dyn. 2, No. 2, 180--195 (2008; Zbl 1140.92025) Full Text: DOI References: [1] DOI: 10.1007/BF01450011 · Zbl 0022.02304 · doi:10.1007/BF01450011 [2] Conley, C. Conference Board of the Mathematical Sciences. Isolated Invariant Sets and the Morse Index, Vol. 38, American Mathematical Society. · Zbl 0397.34056 [3] Duflo M., Random Iterative Models (1997) · Zbl 0868.62069 · doi:10.1007/978-3-662-12880-0 [4] Durrett R., Stochastic Calculus: A Practical Introduction (1996) [5] Eigen M., The Hypercycle: A Principle of Natural Self-Organization (1979) [6] DOI: 10.1016/0040-5809(90)90011-J · Zbl 0703.92015 · doi:10.1016/0040-5809(90)90011-J [7] DOI: 10.1016/0022-0531(92)90044-I · Zbl 0766.92012 · doi:10.1016/0022-0531(92)90044-I [8] DOI: 10.1137/S0036141001392815 · Zbl 1026.37006 · doi:10.1137/S0036141001392815 [9] DOI: 10.1007/BF01301790 · Zbl 0449.34039 · doi:10.1007/BF01301790 [10] DOI: 10.1007/BF01049740 · Zbl 0729.34032 · doi:10.1007/BF01049740 [11] DOI: 10.1016/0022-5193(79)90058-4 · doi:10.1016/0022-5193(79)90058-4 [12] Hofbauer J., Evolutionary Games and Population Dynamics (1998) · Zbl 0914.90287 [13] DOI: 10.1007/BF01540776 · Zbl 0542.34043 · doi:10.1007/BF01540776 [14] DOI: 10.1016/0025-5564(92)90078-B · Zbl 0783.92002 · doi:10.1016/0025-5564(92)90078-B [15] Ikeda N., Stochastic Differential Equations and Diffusion Processes (1981) · Zbl 0495.60005 [16] DOI: 10.1214/105051604000000837 · Zbl 1081.60045 · doi:10.1214/105051604000000837 [17] DOI: 10.1142/S0219493706001712 · Zbl 1100.60030 · doi:10.1142/S0219493706001712 [18] Meyn S. P., Markov Chains and Stochastic Stability, 2. ed. (2005) · Zbl 0925.60001 [19] Montgomery, R. 2001. ”A Tour of Subriemannian Geometries, Their Geodesics and Applications”. Providence: American Mathematical Society. · Zbl 1044.53022 [20] DOI: 10.1006/jdeq.1999.3719 · Zbl 0956.34038 · doi:10.1006/jdeq.1999.3719 [21] DOI: 10.1016/j.jtbi.2006.04.024 · doi:10.1016/j.jtbi.2006.04.024 [22] DOI: 10.3934/dcdsb.2007.7.457 · Zbl 1133.37345 · doi:10.3934/dcdsb.2007.7.457 [23] DOI: 10.1016/0022-0396(79)90039-1 · Zbl 0384.34029 · doi:10.1016/0022-0396(79)90039-1 [24] Stroock, D. W. and Varadhan, S. R.S. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. 1970–1971. On the support of diffusion processes with applications to the strong maximum principle, Vol. 3, pp.333–359. Berkeley: Berkeley. University of California Press [25] DOI: 10.1016/0025-5564(78)90077-9 · Zbl 0395.90118 · doi:10.1016/0025-5564(78)90077-9 [26] Zeeman E. C., Global Theory of Dynamical Systems (Evanston, 1979) pp 472– (1980) · Zbl 0409.58008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.