On stability of equations with an infinite distributed delay

Abstract For scalar equations of population dynamics with an infinite distributed delay x ′ ( t ) = r ( t ) ∫ − ∞ t f ( x ( s ) ) d s R ( t , s ) − x ( t ) , x ( t ) = φ ( 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 ∈ ( 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.