Periodic systems of population models and enveloping functions

Abstract

We show that P. Cull’s concept of enveloping functions can be applied to periodic systems of population models to ensure the existence of globally asymptotically stable attractors for such systems without ever considering the compositions of the maps involved. We give conditions to ensure that a period-n system of population models sharing a fixed point and enveloping function has a globally asymptotically stable trivial geometric cycle. We also give conditions on two-periodic systems of population models that do not share a fixed point or an enveloping function that ensures the existence of a globally attracting geometric 2-cycle as well as providing bounds for the location of the attractor. A technique is outlined to establish the existence of a period-n attractor for a period-n system of population models that do not share a fixed point using the theorems for the period-2 case. Furthermore, we give examples that show enveloping provides sufficient but not necessary conditions for the existence of globally asymptotically stable geometric cycles in such systems.