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On the quasi-stationary distribution of a stochastic Ricker model

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  • Högnäs, Göran

Abstract

We model the evolution of a single-species population by a size-dependent branching process Zt in discrete time. Given that Zt = n the expected value of Zt+1 may be written nexp(r - [gamma]n) where r > 0 is a growth parameter and [gamma] > 0 is an (inhibitive) environmental parameter. For small values of [gamma] the short-term evolution of the normed process [gamma]Zt follows the deterministic Ricker model closely. As long as the parameter r remains in a region where the number of periodic points is finite and the only bifurcations are the period-doubling ones (r in the beginning of the bifurcation sequence), the quasi-stationary distribution of [gamma]Zt is shown to converge weakly to the uniform distribution on the unique attracting or weakly attracting periodic orbit. The long-term behavior of [gamma]Zt differs from that of the Ricker model, however: [gamma]Zt has a finite lifetime a.s. The methods used rely on the central limit theorem and Markov's inequality as well as dynamical systems theory.

Suggested Citation

  • Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
    • Handle: RePEc:eee:spapps:v:70:y:1997:i:2:p:243-263
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    References listed on IDEAS

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    1. Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
    2. Klebaner, Fima C., 1993. "Population-dependent branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 115-127, May.
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    Cited by:

    1. Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    2. Aziz Khanchi, 2012. "Asymptotics of Markov Additive Chains on a Half-Plane: A Ratio Limit Theorem," Journal of Theoretical Probability, Springer, vol. 25(1), pages 62-76, March.

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