Noise-induced Extinction in a Bistable System

A considerable amount of analytical work has been done in recent years to understand extinction in finite discrete population systems in the presence of demographic noise. This has been made possible by advances in employing the WKB method for solving a wide variety of stochastic single population systems. Previous works have focussed on extinction in logistic/Verhulst-type models with a single stable fixed point, Allee-type models which has an additional unstable nonzero fixed point and switching behaviour of bistable finite population systems. However, the mean extinction time in bistable systems with an absorbing state hasn’t received much focus in these studies. In this work, we study the mean extinction time (MTE) in a phenomenological single population model exhibiting bistability using the WKB method. The quasi-stationary distribution (QSD) is derived in the limit of a large parameter, N , of the model, and by matching the QSD in different regimes of stability, an analytical expression for the mean extinction time has been derived. The MTE is exponentially large in N but shows a significant reduction in the extinction time due to the presence of bistable fixed points as compared to models that have only single stable states such as the Verhulst model. It is demonstrated that the analytical results compare well with that derived from direct numerical calculations.