Population Models and Enveloping
One dimensional nonlinear difference equations are commonly used to model population growth. Although such models can display wild behavior including chaos, the common models have the interesting property that they are globally stable if they are locally stable. We show that a model with a single positive equilibrium is globally stable if it is enveloped by a self-inverse function. In particular, we show that the standard population models are enveloped by linear fractional functions which are self-inverse. Although enveloping by a linear fractional is sufficient for global stability, we show by example that such enveloping is not necessary. We extend our results by showing that enveloping implies global stabilty even when f(x) is a discontinuous multifunction, which may be a more reasonable description of real biological data. We also show that our techniques can be applied to situations which are not population models. Finally, we mention some extensions and open questions.