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Persistence in fluctuating environments for interacting structured populations

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Abstract

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below \(\delta >0\) approaches zero as \(\delta \) approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.

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Acknowledgments

GR was supported by the Swiss National Science Foundation Grant 137273 and a start-up grant to SJS from the College of Biological Sciences, University of California, Davis. SJS was supported in part by U.S. National Science Foundation Grants EF-0928987 and DMS-1022639. The authors thank two anonymous referees for their very useful comments on an earlier draft of this paper.

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  1. Department of Evolution and Ecology, One Shields Avenue, University of California, Davis, CA, 95616, USA

    Gregory Roth & Sebastian J. Schreiber

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Roth, G., Schreiber, S.J. Persistence in fluctuating environments for interacting structured populations. J. Math. Biol. 69, 1267–1317 (2014). https://doi.org/10.1007/s00285-013-0739-6

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