Can a single migrant per generation rescue a dying population?

Abstract

We introduce a population model to test the hypothesis that even a single migrant per generation may rescue a dying population. Let (ck:k∈N) be a sequence of real numbers in (0,1). Let Xn be a size of the population at time n≥0. Then, Xn+1=Xn−Yn+1+1, where the conditional distribution of Yn+1 given Xn=k is a binomial random variable with parameters (k,c(k)). We assume that limk→∞⁡kc(k)=ρ exists. If ρ<1 the process is transient with speed 1−ρ. So for our model a single migrant per generation may rescue a dying population! If ρ>1 the process is positive recurrent. In the critical case ρ=1 the process is recurrent or transient according to how kc(k) converges to 1. When ρ=0 and under some regularity conditions, the support of the increments is eventually finite.