Positive Steady States

Abstract

When a population can persist in a given region, we want to know what the long-term dynamics of the population are. Is there a positive steady state? Is it asymptotically stable? And what is the spatial distribution? From a management point of view, a unique stable steady state is the simplest: Even if the population is perturbed somewhat in one generation, it will return to its steady state over time. But many other scenarios can arise. For example, a positive state can be unstable and the population could cycle between different states. How does spatial dispersal affect these dynamics? Can it stabilize or destabilize a steady state? What are the spatial distributions throughout a cycle? In the case of an Allee effect, there could be a positive steady state even if the trivial state is locally stable. How does spatial dispersal affect the ability of a population to persist in that case? In this chapter, we present some analytical methods and numerical results that explore these questions. The effects are inherently nonlinear and therefore much harder to study completely.