Qualitative theory of a third-order nonlinear system with examples in population dynamics and chemical kinetics

Abstract

A general autonomous system of three differential equations with specified sign constraints on the flow is studied. Examples of this system arise in numerous applications, such as genetic feedback, the Belousov-Zhabotinskii reaction, enzyme kinetics, predator-prey interactions, and interpopulation competition. In the case where there is a unique equilibrium with a two-dimensional unstable manifold, solutions near the equilibrium but not approaching it are considered. Under broad assumptions either all such solutions approach the same stable or semistable periodic orbit, or else all become unbounded.