When an individual with a novel trait is introduced in a new environment, we would like to understand what drives the likelihood that its lineage will persist. In deterministic population models, whether the invasive population “succeeds” often depends on whether the parameters of the system fall in a super- or sub-critical regime.  In stochastic population models, the parameters must be super-critical for there to be any chance of invasion, but even in the super-critical regime, chance alone allows for many invasive lineages to quickly go extinct.

In this talk, we consider multiple methods for estimating the probability of invasion for a class of continuous-time (discrete-state) Markov chain models. Because these probabilities can be expressed in terms of an associated finite-dimensional set of difference equations, an exact numerical solution is available. However, it is not possible to directly interpret such results in terms of the parameters of the system. Therefore, multiple approximation methods have been introduced. One approach, known as the diffusion approximation, results from recasting the discrete system as a continuous state-space stochastic differential equation and calculating the associated invasion probability.  Another approach ignores all non-linear interaction effects and reduces the CTMC to a continuous-time branching process for which the probability of persisting is known. Both of these have been shown to be inadequate in certain circumstances and, following Doering et al ‘07, we investigate a first order WKB approximation that has been shown to be effective in estimating other CTMC properties.