Invariant curves in a discrete-time two-species system

Abstract

The dynamics of discrete-time two-species systems may not be simple even if they have no positive fixed points. To reveal the behaviour of such systems, we focus on a special class of discrete-time two-species systems whose degenerate cases have a line segment of positive fixed points. As a main result, we show that the line segment of positive fixed points persists as an invariant curve under a small perturbation of the degenerate case. The absence of a positive fixed point ensures that such an invariant curve consists of heteroclinic orbits connecting two axial fixed points. The general result is applied to a discrete-time competition model of Ricker type with reproductive delay. The application reveals the existence of heteroclinic orbits connecting two axial 2-cycles, not only heteroclinic orbits connecting two axial fixed points.