Limiting the complexity of limit sets in self-regulating systems

Abstract

In the analysis of models of ecosystems one seeks to discover conditions that limit the complexity of the possible behavior of solutions since “intuitively” one feels that the full range of possible complex behaviors in systems of order three or more ought not to occur for simple ecological interactions. It has been shown by Hirsch [6,7] that the solutions of competitive and cooperative systems have limit sets which cannot be more complicated than invariant sets of systems of one lower dimension. In particular, autonomous 2-dimensional systems of these types have only “trivial” dynamics in the sense that all bounded solutions approach equilibrium asymptotically. In planar systems, for example, the absence of limit cycles makes the dynamics trivial in the sense that bounded solutions can have only critical points or orbits connecting critical points in their omega limit sets [3]. In this note we prove a theorem which appears to be useful in limiting the complexity of limit sets for a class of biologically important equations.