Nonautonomous Semiflows

Abstract

There are many nonautonomous models that describe the population dynamics in a fluctuating environment. Solutions of these systems can generate nonautonomous semiflows on phase spaces. The purpose of this chapter is to develop the theory of nonautonomous semiflows. It is well known that the existence and stability of periodic solutions of a periodic differential system are equivalent to those of fixed points of its associated Poincaré map (see, e.g., [106]) . In Section 3.1 we introduce the concept of periodic semiflows and prove that uniform persistence of a periodic semiflow also reduces to that of its associated Poincaré map under a general abstract setting. To illustrate the applications of the theory of monotone discrete dynamical systems to periodic problems, we then discuss periodic cooperative ordinary differential systems and scalar parabolic equations. In particular, we establish threshold dynamics in terms of principal multipliers and eigenvalues, and show how to obtain corresponding results for autonomous cases of these systems. Two practical examples are also provided.