  We study delay-independent stability in nonlinear models with a distributed
delay which have a positive equilibrium. Such models frequently occur in
population dynamics and other applications. In particular, we construct a
relevant difference equation such that its stability implies stability of the
equation with a distributed delay and a finite memory. This result is,
generally speaking, incorrect for systems with infinite memory. If the relevant
difference equation is unstable, we describe the general delay-independent
attracting set and also demonstrate that the equation with a distributed delay
is stable for small enough delays.

Dynamical Systems