On the Stability and Uniform Persistence of a Discrete Model of Nicholson′s Blowflies

Consider the delay difference equation Nn+1 − Nn = −δ Nn + pNn−ke−aNn −k, n = 0, 1, 2, …, (1) which is a discrete analogue of the delay differential equation Ṅ(t) = −δN(t) + pN(t − τ)e−aN(t−τ), t ≥ 0 (cf. [8]). We show that when p ≤ δ, the zero solution of (1) is uniformly asymptotically stable and every non-negative solution of(1) tends to zero as n → ∞, while for p > δ, (1) is uniformly persistent. Moreover, if in addition to p > δ we also have [(1 − δ)−k−1 − 1] ln(p/δ) ≤ 1, then every positive solution of (1) tends to the positive equilibrium N* = (1/a) ln(p/δ).