On the quasi-stationary distribution of a stochastic Ricker model

Göran Högnäs

Stochastic Processes and their Applications, 1997, vol. 70, issue 2, 243-263

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.

Keywords: Size-dependent; branching; process; Quasi-stationary; distribution; Invariant; measure; Weak; convergence; Ricker; model; Stable; period; Markov's; inequality; Entropy; function (search for similar items in EconPapers)
Date: 1997
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Citations: View citations in EconPapers (2)

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