Absolute and delay-dependent stability of equations with a distributed delay

Summary: "We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Models with a unique positive equilibrium 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 lower and upper solution bounds and also demonstrate that the equation with a distributed delay is stable for small enough delays."