The dynamics of some discrete population models

IN THIS paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation A II+1 = AA, + F(A,_,) (E) converge as n + 00. In (E), A is a constant real number with 0 0 sufficiently small. Thus, formally, equation (E) comes from x(t + h) x(t) h = -yx(t) + D(x(t r)) for small h. If we set ~,z(f) := x(ht), then the preceding equation for t = n and r/h = m becomes yh@ + 1) = (1 ?‘h)Y, (n) + hD(Y,(n m)), which is of the form (E), where A := 1 yh and F(u) := hD(u), u 2 0. Clearly h > 0 is small if and only if A E (0, 1). 1069 @h)