Persistence and convergence of ecosystems: an analysis of some second order difference equations.

Three second order difference equation models are analyzed and numerical solutions computed. It is shown that two concepts of ecosystem stability, the local property of convergence and the global property of persistence, do not coincide, and that the exislity are obtained and shown as parameter space diagrams. Examples of solution trajectories representative of different regions of this space are computed and discussed. A wide range of oscillatory behavior, as noted in recent papers by several authors, results. In addition, the erratic nature of regions of convergence to stable solutions is discussed.