Continuum Approximation of Invasion Probabilities

In the last decade there has been growing criticism of the use of stochastic differential equations 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. In this work we investigate a related, but distinct, population dynamics property for which Diffusion Approximation is unreliable: invasion probabilities. We consider the situation in which a few individuals 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 a...