On stability of equations with an infinite distributed delay

For scalar equations of population dynamics with an infinite distributed delay (Formula presented), where f is the delayed production function, we consider asymptotic stability of the zero and a positive equilibrium K. It is assumed that the initial distribution is an arbitrary continuous function. Introducing conditions on the memory decay, we characterize functions f such that any solution with nonnegative nontrivial initial conditions tends to a positive equilibrium. The differences between finite and infinite delays are outlined, in particular, we present an example when the weak Allee effect (meaning that f′(0)=1 together with f(x)>x,x∈(0,K)) which has no effect in the finite delay case (all solutions are persistent) can lead to extinction in the case of an infinite delay.