We review the critical patch size problem, already classic in the mathematical biology literature. We consider a logistic population
living in a finite patch of length L and undergoing random dispersal. The patch presents good conditions for life, while the conditions are so harsh outside that they lead to certain extinction. The usual mean field approach leads to a critical patch size Lc, such that if the actual length of the patch is smaller than Lc the population becomes extinct with certainty, whereas a longer patch leads to certain survival. We study the fluctuations in the population due to its low density near extinction and analyze their effects on the probability of extinction. We find that there is no patch size that can be considered absolutely safe for the population and that, under certain circumstances, the population is under risk of extinction for any patch size.
We present an analysis of two features that generalize the original
model for the spread of the Hantavirus introduced by Abramson and Kenkre [Phys. Rev. E Vol. 66, 011912 (2002)]. One, the effect of seasonal alternations, may cause the virus to spread under conditions that do not lead to an epidemic under the action of either season alone. The other, the effect of internal fluctuations, modifies
the distribution of infected mice and may lead to extinction of the infected population even when the mean population is above epidemic conditions.