Barrier Transitions Driven by Fluctuations, with Applications to Ecology and Evolution

Abstract Questions that involve "barrier crossings" arise in both ecology and evolution. When two species compete, and the population dynamics have multiple equilibria, a barrier crossing corresponds to fluctuations driving the populations across a deterministic separatrix. Similarly, fluctuation may drive quantitative traits across valleys in the surface of individual fitness, particularly when the fluctuations interact with a deterministic force such as a correlated selection reponse. In this paper, I show how to formulate and solve the barrier crossing problem. Two classes of models are considered. In models based on stochastic differential or difference equations, fluctuations are superimposed upon a system of deterministic differential or difference equations. In the case of a Markov chain, there may be no underlying deterministic system, so that the system is only characterized by Markovian transition probabilities. The validity of the diffusion approximation and the origin of macroscopic population dynamics from a underlying purely probabilistic system are investigated.