Abstract
One dimensional difference equations are widely used in population biol-
ogy. These seemingly simple models can show a variety of behaviors from stability to
chaos. (see Cull, Yorke, May, Feigenbaum) We show how the enveloping technique
can be used to demonstrate global and semi-global stability. We discuss the issue of
whether local stability implies global stability. We give some examples of more com-
plicated behavior which can co-exist with local stability. We show that local stability
implies global stability even for models slightly more complicated than the usual mod-
els. We address the issue of how complicated a model must be to have local without
global stability, and we describe our candidates for the simplest such models.
