Abstract
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 alternative WKB-like approach. Through numerical investigations, we also study how these approximations perform in an important intermediate regime. Surprisingly, 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.
Highlights
Invasion events are fundamental in population biology
The same multiscale interest in invasions appears in the study of population genetics: at the multiorganism scale one studies the probability that a novel allele can fix in a population, possibly affecting the population's overall fitness [17, 21, 25]; at the cellular level, the invasion of a mutation in a stem cell population has been studied as an important first step in certain cancers [27]
We analyzed the Diffusion Approximation and a simple Exponential Approximation rigorously in the large population limit and numerically for finite size populations. We showed that both population size and the state of sub- vs. supercriticality play an important role in determining which approximation method performs better
Summary
Invasion events are fundamental in population biology. In the study of disease there are interesting examples both at the cellular and wholeorganism level: while studying the onset or recovery of an infection, one might look at the probability that a single virion can infect a target cell and proliferate [9]; in epidemiology, the goal might be to estimate the probability that a newly introduced pathogen will become endemic in a na\{\i"}ve host population [4]. Suppose that the transition rate shape functions \lambda and \mu satisfy Assumptions 1.1 and 1.3 and that the leading order exponents satisfy \beta = \delta = 1.
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