Abstract
A method is presented to analyse the long-term stochastic dynamics of a biological population that is at risk of extinction. From the full ecosystem the method extracts the minimal information to describe the long-term dynamics of that population by a stochastic logistic system. The method is applied to a one-predator-two-prey model. The choice of this example is motivated by a study on the near-extinction of a porcupine population by mountain lions whose presence is facilitated by mule deer taking advantage of a change in land use. The risk of extinction is quantified by the expected time of extinction of the population.
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References
DeAngelis, D. L. (1992). Dynamics of Nutrient Cycling and Food Webs, London: Chapman and Hall.
Foley, P. (1994). Predicting extinction times from environmental stochasticity and carrying capacity. Conserv. Biol. 8, 214–237.
Grasman, J. (1996). The expected extinction time of a population within a system of interacting biological populations. Bull. Math. Biol. 58, 555–568.
Grasman, J. (1998). Stochastic epidemics: the expected duration of the endemic period in higher dimensional models. Math. Biosci. 152, 13–27.
Grasman, J. and R. HilleRisLambers (1997). On local extinction in a metapopulation. Ecol. Modelling 103, 71–80.
Grasman, J. and O. A. van Herwaarden (1999). Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications, Berlin, Heidelberg: Springer-Verlag.
Grzimek, B. (1967–1974). Grzimeks Tierleben: Enzyklopaedie des Tierreiches, Zürich: Kindel.
Ludwig, D. (1996). The distribution of population survival times. Am. Nat. 147, 506–526.
May, R. M. (1973). Stability and Connectance in Model Ecosystems, Monographs in Population Biology 6, Princeton: Princeton University Press.
Nåsell, I. (1999). On the time to extinction in recurrent epidemics. J. R. Stat. Soc. B 61, 309–330.
Nisbet, R. M. and W. S. C. Gurney (1982). Modelling Fluctuating Populations, New York: Wiley.
Nowak, R. M. and J. L. Paradiso (1983). Walker’s Mammals of the World, Baltimore: The John Hopkins University Press.
Pimm, S. L. (1982). Food Webs, London: Chapman and Hall.
Roozen, H. (1989). An asymptotic solution to a two-dimensional exit problem arising in population dynamics. SIAM J. Appl. Math. 49, 1793–1810.
Roughgarden, J. (1979). Theory of Population Genetics and Evolutionary Ecology: An Introduction, New York: Macmillan.
Sweitzer, R. A., S. H. Jenkins and J. Berger (1997). Near-extinction of porcupines by mountain lions and consequences of ecosystem change in the Great Basin Desert. Conserv. Biol. 11, 1407–1417.
Yodzis, P. P. (1981). The stability of real ecosystems. Nature 289, 674–676.
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Grasman, J., van den Bosch, F. & van Herwaarden, O.A. Mathematical conservation ecology: A one-predator-two-prey system as case study. Bull. Math. Biol. 63, 259–269 (2001). https://doi.org/10.1006/bulm.2000.0218
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DOI: https://doi.org/10.1006/bulm.2000.0218