Mean extinction time for birth-death processes failure of the Fokker-Planck approximation

摘要

We consider extinction times for a class of birth-death processes commonly found in applications where there is a control parameter that defines a threshold. Below the threshold the population quickly becomes extinct and above the threshold it persists for an exponentially long time. We derive an asymptotic expression for the mean time to extinction in the discrete population case for large values of the population scale, i.e., in the continuous population limit. The Fokker-Planck (Markov diffusion process) approximation does not generally predict the true behavior of the extinction time for large values of the population scale. This is surprising because the continuum limit is precisely where it could be anticipated that the Fokker-Planck approximation should be most accurate. Rather, it is valid only near the threshold. This constitutes an interesting example of the delicate relationship between discrete and continuum treatments of the same problem.