A dynamic dichotomy for a system of hierarchical difference equations

A system of difference equations that arises in population dynamics is studied. Criteria are given for the existence of equilibria lying in the positive cone and for the existence of periodic cycles lying on the boundary of the cone. These equilibria and cycles arise from a bifurcation that occurs as a fundamental parameter R ₀ increases through the value 1. Under monotone conditions on the nonlinearities and for R ₀ near 1, we derive criteria for the stability of the equilibria and we determine the global dynamics on the boundary of the cone. We show that boundary orbits tend to periodic cycles (all of which we classify into four types). A dynamic dichotomy is established between the equilibria and the cycles, which asserts that one is stable and the other is unstable. We also establish a dynamic dichotomy between the equilibria and the boundary of the cone.