Forwards Attractor Structures in a Planar Cooperative Non-autonomous Lotka–Volterra System

Abstract

The global attractor of a dissipative dynamical system provides the necessary information to understand the asymptotic dynamics of all the system’s solutions. A crucial question consists in finding the structure of this set. In this paper we provide a full characterization of the structure of attractors for a planar non-autonomous Lotka–Volterra cooperative system. We show sufficient conditions for the existence of forward attractors and give a full description of them by proving the existence of such bounded global solutions that all bounded global solutions join them, i.e. converge towards them when time tends to plus and minus infinity. These results generalize those known in an autonomous framework. The case of particular interest in our work is the situation where globally forward-stable non-autonomous solutions have both coordinates strictly positive. We study this case in detail and obtain sufficient conditions that the problem parameters must satisfy in order to obtain various structures of non-autonomous attractors. This allows us to understand different paths of the solutions towards the unique globally stable one.