One dimensional nonlinear difference equations have been 
used to model a variety of biological phenomena including 
population growth.  The standard biological models have the 
interesting characteristic 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.