The Annals of Applied Probability

Coexistence and extinction for stochastic Kolmogorov systems

Alexandru Hening and Dang H. Nguyen

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Abstract

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.

The analysis is done by a careful study of the properties of the invariant measures of the process that are supported on the boundary of the domain. To our knowledge this is one of the first general results describing the asymptotic behavior of stochastic Kolmogorov systems in non-compact domains.

We are able to fully describe the properties of many of the SDE that appear in the literature. In particular, we extend results on two dimensional Lotka-Volterra models, two dimensional predator–prey models, $n$ dimensional simple food chains, and two predator and one prey models. We also show how one can use our methods to classify the dynamics of any two-dimensional stochastic Kolmogorov system satisfying some mild assumptions.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 3 (2018), 1893-1942.

Dates
Received: October 2016
Revised: August 2017
First available in Project Euclid: 1 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1527840035

Digital Object Identifier
doi:10.1214/17-AAP1347

Mathematical Reviews number (MathSciNet)
MR3809480

Zentralblatt MATH identifier
06919741

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] 60H10: Stochastic ordinary differential equations [See also 34F05] 60J05: Discrete-time Markov processes on general state spaces 60J99: None of the above, but in this section

Keywords
Kolmogorov system ergodicity Lotka–Volterra Lyapunov exponent stochastic environment predator–prey population dynamics

Citation

Hening, Alexandru; Nguyen, Dang H. Coexistence and extinction for stochastic Kolmogorov systems. Ann. Appl. Probab. 28 (2018), no. 3, 1893--1942. doi:10.1214/17-AAP1347. https://projecteuclid.org/euclid.aoap/1527840035


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