Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand, Neuchâtel, Switzerland.
J Biol Dyn. 2008 Apr;2(2):180-95. doi: 10.1080/17513750801915269.
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.
Garay 和 Hofbauer（SIAM J. Math. Anal. 34（2003））提出了确定性复制者动态稳健持久性和非永久性的充分条件。我们在随机复制者动态的背景下重新考虑这些条件，该动态是通过对收益进行布朗运动干扰从其确定性类似物中获得的。当确定性复制者动态是永久性的，且噪声水平较小时，随机动态存在一个唯一的遍历分布，其质量集中在未受干扰系统的最大内部吸引子附近；因此，持久性对小的无界随机干扰具有稳健性。当确定性动态是短暂的，且噪声水平较小或较大时，随机动态以指数速率收敛到状态空间的边界。
