期刊
ANNALS OF APPLIED PROBABILITY
卷 32, 期 2, 页码 893-931出版社
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/21-AAP1699
关键词
Kolmogorov system; ergodicity; Lotka-Volterra; Lyapunov exponent; random environmental fluctuations
资金
- NSF [DMS-1853463, DMS-1853467, DMS-1716803]
The paper investigates the classification problem of stochastic models of interacting species and proves a variant of Palis' conjecture. The long-term statistical behavior is determined by a finite number of stationary distributions, and there are three general types of behavior that could occur. The classification problem can be simplified by computing Lyapunov exponents. This research provides a rigorous foundation for ecology's modern coexistence theory.
The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis' conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis' conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka-Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology's modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.
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