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 Z_t in discrete time. Given that Z_t = n the expected value of Z_t+1 may be written nexp(r ? γn) where r> 0 is a growth parameter and γ > 0 is an (inhibitive) environmental parameter. For small values of γ the short-term evolution of the normed process γZ_t 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 γZ_t is shown to converge weakly to the uniform distribution on the unique attracting or weakly attracting periodic orbit. The long-term behavior of γZ_t differs from that of the Ricker model, however: γZ_t 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.