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On the stability and uniform persistence of a discrete model of Nicholson’s blowflies. (English) Zbl 0834.39009
The authors study the behaviour of nonnegative solutions to the difference equation $N_{k+ 1}- N_k= - \delta N_k+ pN_{k- i} e^{- aN_{k- i}},\tag{1}$ where $$k\in \mathbb{N}\cup 0$$, $$N_i\geq 0$$, $$\delta$$, $$a$$, $$p$$ are positive constants. This equation is the discrete analogue of some delay differential equation used to model the dynamics of Nicholson blowflies. The authors show that (1) has a unique global attractor in the cases 1) $$p\leq \delta$$ and 2) $$p> \delta$$, $$[(1- \delta)^{- i- 1}- 1]\ln (p/\delta)\leq 1$$, $$N_i>0\;\forall i$$. Moreover, in the first case the attractor (zero solution) is uniformly asymptotically stable, while for $$p> \delta$$, nonzero nonnegative solutions are uniformly separated from zero at $$+\infty$$. Note that the condition 2) is weaker than the analogous one of V. Lj. Kocic and G. Ladas [Appl. Anal. 38, No. 1/2, 21-31 (1990; Zbl 0715.39003)].

##### MSC:
 39A12 Discrete version of topics in analysis 39A10 Additive difference equations 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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