Congratulations to Becky Borchering! The research paper stemming from the second chapter of her dissertation was accepted today.

This paper has a fun working title that we always joke about in the Stochastics Lab. Originally, we expected this to be an easy project — almost too trivial to write up as a research article. I remember telling Becky, “Sometimes you just have to take a simple project, write out the results, and share them with people. Even if it’s just a ten-page paper, it can start a conversation that is much more substantial.” That was in November of 2014. Here we are and finally “10 Page Paper” is accepted … a mere three yearsa and *thirty-two* pages later.

Never underestimate an simple idea!!

Continuum Approximation of Invasion Probabilities ( pdf )

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.

Update! the paper is now published online here: https://epubs.siam.org/doi/abs/10.1137/17M1155259