| Contents |
| One dimensional nonlinear difference equations are commonly
used to model population growth. Although such models can display
wild behavior including chaos, the standard population models
have the interesting property that they are globally stable
if they are locally stable. We show that models with a single
positive equilibrium are globally stable if they are
{\em 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
{\em enveloping implies global stability} 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 give examples of population models which have
local stability but not global stability. |
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