  We study a generic reaction-diffusion model for single-species population
dynamics that includes reproduction, death, and competition. The population is
assumed to be confined in a refuge beyond which conditions are so harsh that
they lead to certain extinction. Standard continuum mean field models in one
dimension yield a critical refuge length L_c such that a population in a refuge
larger than this is assured survival. Herein we extend the model to take into
account the discreteness and finiteness of the population, which leads us to a
stochastic description. We present a particular critical criterion for likely
extinction, namely, that the standard deviation of the population be equal to
the mean. According to this criterion, we find that while survival can no
longer be guaranteed for any refuge size, for sufficiently weak competition one
can make the refuge large enough (certainly larger than L_c) to cause
extinction to be unlikely. However, beyond a certain value of the competition
rate parameter it is no longer possible to escape a likelihood of extinction
even in an infinite refuge. These unavoidable fluctuations therefore have a
severe impact on refuge design issues.
