On stability of equations with an infinite distributed delay

For scalar equations of population dynamics with an infinite distributed delay x'(t ) = r (t) [integral(t)(-infinity) f(x(s)) d(s)R (t, s ) - x (t)], x (t )=phi (t), t <= t(0), 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 is an element of (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.