Persistence probabilities for stream populations.

Individuals in streams and rivers are constantly at risk of being washed downstream and thereby lost to their population. The possibility of diffusion-mediated persistence of populations in advective environments has been the focus of a multitude of recent modeling efforts. Most of these recent models are deterministic, and they predict the existence of a critical advection velocity, above which a population cannot persist. In this work, we present a stochastic approach to the persistence problem in streams and rivers. We use the dominant eigenvalue of the advection-diffusion operator to transition from a spatially explicit description to a spatially implicit birth-death process, in which individual washout from the domain appears as an additional death term. We find that the deterministic persistence threshold is replaced by a smooth transition from almost sure persistence to extinction as advection velocity increases. More interestingly, we explore how temporal variation in flow rate and other parameters affect the persistence probability. In line with general expectations, we find that temporal variation often decreases the persistence probability, and we focus on a few examples of how variation can increase population persistence.