Continuum Approximation of Invasion Probabilities

Abstract

In the last decade there has been growing criticism of the use of Stochastic Differential Equations (SDEs) to approximate discrete state-space, continuous-time Markov chain population models. In particular, several authors have demonstrated the failure of Diffusion Approximation, as it is often called, to approximate expected extinction times for populations that start in a quasi-stationary state.\n\nIn this work we investigate a related, but distinct, population dynamics property for which Diffusion Approximation fails: invasion probabilities. We consider the situation in which a few individual are introduced into a population and ask whether their collective lineage can successfully invade. Because the population count is so small during the critical period of success or failure, the process is intrinsically stochastic and discrete. In addition to demonstrating how and why the Diffusion Approximation fails in the large population limit, we contrast this analysis with that of a sometimes more successful alternative WKB-like approach. Through numerical investigations, we also study how these approximations perform in an important intermediate regime. In a surprise, we find that there are times when the Diffusion Approximation performs well: particularly when parameters are near-critical and the population size is small to intermediate.