Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka–Volterra Models with Diffusion

Abstract

Lotka–Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis, or prey-predator behavior involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extended to allow explicit time dependence, generally in a periodic, quasiperiodic, or almost periodic fashion. The presence of more general nonautonomous terms in the equations leads to nontrivial difficulties which have stalled the development of the theory in this direction. However, the theory of nonautonomous dynamical systems has received much attention in the last decade, and this has opened new possibilities in the analysis of classical models with general nonautonomous terms. In this paper we use the recent theory of attractors for nonautonomous PDEs to obtain new results on the permanence and the existence of forwards and pullback asymptotically stable global solutions associated to nonautonomous Lotka–Volterra systems describing competition, symbiosis, or prey-predator phenomena. We note in particular that our results are valid for prey-predator models, which are not order-preserving: even in the “simple” autonomous case the uniqueness and global attractivity of the positive equilibrium (which follows from the more general results here) is new.