Population Models: Stability in One Dimension1

Abstract

Some of the simplest models of population growth are one dimensional nonlinear difference equations. While such models can display wild behavior including chaos, the standard biological models have the interesting property that they display global stability if they display local stability. Various researchers have sought a simple explanation for this agreement of local and global stability. Here, we show that enveloping by a linear fractional function is sufficient for global stability. We also show that for seven standard biological models local stability implies enveloping and hence global stability. We derive two methods to demonstrate enveloping and show that these methods can easily be applied to the seven example models. Although enveloping by a linear fractional is a sufficient for global stability, we show by example that such enveloping is not necessary. We extend our results by showing that enveloping implies global stability even when f(x) is a discontinuous multi-function which might be a more reasonable description of real bilogical data. We show that our techniques can be applied to situations which are not population models. Finally, we give examples of population models which have local stability but not global stability.