On the stability and uniform persistence of a discrete model of Nicholson's blowflies (Q1900650)

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 \textit{V. Lj. Kocic} and \textit{G. Ladas} [Appl. Anal. 38, No. 1/2, 21-31 (1990; Zbl 0715.39003)].