On the quasi-stationary distribution of a stochastic Ricker model

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.