Extinction Times for Birth-Death Processes: Exact Results, Continuum Asymptotics, and the Failure of the Fokker--Planck Approximation

Abstract

We consider extinction times for a class of birth-death processes commonly found in applications, where there is a control parameter which determines whether the population quickly becomes extinct, or rather persists for a long time. We give an exact expression for the discrete case and its asymptotic expansion for large values of the population. We have results below the threshold, at the threshold, and above the threshold (where there is a quasi-stationary state and the extinction time is very long.) We show that the Fokker-Planck approximation is valid only quite near the threshold. We compare our analytical results to numerical simulations for the SIS epidemic model, which is in the class that we treat. This is an interesting example of the delicate relationship between discrete and continuum treatments of the same problem.

A master equation for a spatial population model with pair interactions

“…2 However, the extinction time becomes exponentially long as the system size increases and so in practice one observes a quasi-stationary state in which ARTICLE IN PRESS a * = 1, n(λt) = 562 a * = 4, n(λt) = 593 a * = 16, n(λt) = 789 a * = 64, n(λt) = 984 the population fluctuates about some mean level (e.g., Doering et al, 2005). The typical population fluctuations in these ma0 quasi-equilibria are qualitatively similar to those in Fig.…”

Continuum Approximation of Invasion Probabilities

“…The following argument for finding the exact solution is similar to one presented for birth-death processes by Norris [23] and has a similar form to the exact solutions for mean extinction times presented by Doering et al [11,12]. Proposition 2.2 (exact solution for extinction probability).…”

“…Exponential Approximation. Doering et al [11,12] demonstrated that making a WKB-type ansatz of the form q n \approx \sigma n e - Vn for some functions \sigma and V can be an accurate method for constructing a continuum approximation for solving Kolmogorov equations. In the motivation that follows we provide a different analytical justification for this observation than has been presented elsewhere.…”

“…However, multiple authors have shown that there are important differences between Diffusion Approximations and their corresponding continuous-time Markov chains (CTMCs). For example, Doering and coauthors [11,12] have shown that the mean extinction time for SDE models can differ substantially from that of the corresponding Markov chain model. Newby and Bressloff also documented the issue for neuronal models [16,22,8].…”

On a Versatile Stochastic Growth Model

“…We note that for particular instances of the stochastic model closed forms for the various transition probabilities can be obtained by using bivariate probability generating functions and by solving a Fokker-Planck partial differential equation. However, it is common knowledge that, in general, the resulting partial differential equation does not admit of a closed form solution and the only recourse is to employ various approximation schemes [3,13]. This is, no doubt, one of the main reasons that stochastic growth models have been less widely used, especially in the biological and biomedical communities [29,30].…”