  Understanding the conditions ensuring the persistence of a population is an
issue of primary importance in population biology. The first theoretical
approach to the problem dates back to the 50's with the KiSS (after Kierstead,
Slobodkin and Skellam) model, namely a continuous reaction-diffusion equation
for a population growing on a patch of finite size $L$ surrounded by a deadly
environment with infinite mortality -- i.e. an oasis in a desert. The main
outcome of the model is that only patches above a critical size allow for
population persistence. Here, we introduce an individual-based analogue of the
KiSS model to investigate the effects of discreteness and demographic
stochasticity. In particular, we study the average time to extinction both
above and below the critical patch size of the continuous model and investigate
the quasi-stationary distribution of the number of individuals for patch sizes
above the critical threshold.
