Persistence for stochastic difference equations: a mini-review

• Published in: Journal of difference equations and applications Vol. 18; no. 8; pp. 1381 - 1403
• Main Author: Schreiber, Sebastian J.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis Group 01.08.2012; Taylor & Francis Ltd
• Subjects: applications of Markov chains and discrete-time Markov processes on general state spaces; Asymptotic methods; Boundary conditions; Differential equations; Mathematical analysis; population dynamics (general); Random variables; stochastic difference equations; Stochastic models
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236198.2011.628662

Understanding under what conditions populations, whether they be plants, animals or viral particles, persist is an issue of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic forces is the construction and analysis of stochastic difference equations , where represents the state of the populations and is a sequence of random variables representing environmental stochasticity. In the analysis of these stochastic models, many theoretical population biologists are interested in whether the models are bounded and persistent. Here, boundedness asserts that asymptotically tends to remain in compact sets. In contrast, persistence requires that tends to be 'repelled' by some 'extinction set' . Here, results on both of these proprieties are reviewed for single species, multiple species and structured population models. The results are illustrated with applications to stochastic versions of the Hassell and Ricker single species models, Ricker, Beverton-Holt, lottery models of competition, and lottery models of rock-paper-scissor games. A variety of conjectures and suggestions for future research are presented.