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Stability classification of a Ricker model with two random parameters

Part of: Markov processes

Published online by Cambridge University Press:  19 February 2016

Henrik Fagerholm  and
Göran Högnäs
Henrik Fagerholm*
Affiliation:
Åbo Akademi University
Göran Högnäs*
Affiliation:
Åbo Akademi University
*
∗ Postal address: Department of Mathematics, Åbo Akademi, FIN-20500 Åbo, Finland.
∗ Postal address: Department of Mathematics, Åbo Akademi, FIN-20500 Åbo, Finland.
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Abstract

We consider a stochastic version of the Ricker model describing the density of an unstructured isolated population. In particular, we investigate the effects of independently varying the per capita growth rate and the parameter governing density dependent feedback. We derive conditions on the distributions sufficient to guarantee different forms of stochastic stability such as null recurrence or positive recurrence. We find, for example, that null recurrence appears in two widely different scenarios: when there is a mean-zero growth rate or via a growth-catastrophe behaviour.

Keywords

Markov chainspopulation dynamicsinvariant measuresnonlinear stochastic difference equationsrecurrence

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 112 - 127
Copyright
Copyright © Applied Probability Trust 2002 

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References

Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques. Jones and Bartlett, London.Google Scholar
Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, Redwood City, CA.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Gyllenberg, M., Hanski, I. and Lindström, T. (1996). A predator–prey model with optimal suppression of reproduction in the prey. Math. Biosci. 134, 119–152.Google Scholar
Gyllenberg, M., Högnäs, G. and Koski, T. (1994). Population models with environmental stochasticity. J. Math. Biol. 32, 93–108.Google Scholar
Gyllenberg, M., Högnäs, G. and Koski, T. (1994). Null recurrence in a stochastic Ricker model. In Analysis, Algebra, and Computers in Mathematical Research (Lecture Notes Pure Appl. Math. 156), eds Gyllenberg, M. and Persson, L.-E., Marcel Dekker, New York, pp. 147–164.Google Scholar
Kersting, G. (1986). On recurrence and transience of growth models. J. Appl. Prob. 23, 614–625.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
Ricker, W. E. (1954). Stock and recruitment. J. Fisheries Res. Board Canada 11, 559–623.CrossRefGoogle Scholar
Ripa, J. and Lundberg, P. (1996). Noise colour and the risk of population extinctions. Proc. R. Soc. London B 263, 1751–1753.Google Scholar
Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.Google Scholar
Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis. Chapman and Hall, London.Google Scholar
Vellekoop, M. H. and Högnäs, G. (1997). Stability of stochastic population models. Studia Sci. Math. Hung. 33, 459–476.Google Scholar
Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral—An Introduction to Real Analysis. Marcel Dekker, New York.Google Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.Google Scholar