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Coexistence and extinction for stochastic Kolmogorov systems. (English) Zbl 1410.60094
Summary: In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $$n$$ populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). Our models are described by $$n$$-dimensional Kolmogorov systems with white noise (stochastic differential equations-SDE). We give sharp conditions under which the populations converge exponentially fast to their unique stationary distribution as well as conditions under which some populations go extinct exponentially fast.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J05 Discrete-time Markov processes on general state spaces
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