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 $(c_k)$ be a sequence of real numbers in $(0,1)$. Let $X_n$ be a size of the population at time $n\geq 0$. Then, $X_{n+1}=X_n - Y_{n+1}+1$, where the conditional distribution of $Y_{n+1}$ given $X_n=k$ is a binomial random variable with parameters $(k ,c(k))$. We assume that $\lim_{k\to\infty}kc(k)=\rho$ exists. If $\rho<1$ the process is transient with speed $1-\rho$ (so yes a single migrant per generation may rescue a dying population!) and if $\rho>1$ the process is positive recurrent. In the critical case $\rho=1$ the process is recurrent or transient according to how $k c(k)$ converges to $1$. When $\rho=0$ and under some regularity conditions, the support of the increments is eventually finite.