## Persistence and global stability in discrete models of Lotka–Volterra type.(English)Zbl 1124.39011

Consider $$N_{io}> 0$$, $$N_{ip}\geq 0$$ for $$1\leq i\leq n$$ and $$p\leq 0$$, $$a_i> 0$$, $$c_i\in\mathbb{R}$$, $$a_{ij}\in\mathbb{R}$$, $$a_i+ a_{ii}> 0$$, $$k_{ij}\geq 0$$ for $$1\leq i\leq n$$, $$1\leq j\leq n$$ and the persistence and global asymptotic stability of the discrete models of Lotka-Volterra type $\begin{gathered} N_i(p+1)= N_{ip}\exp\Biggl\{c_i- a_i N_i(p)- \sum^n_{j=1} a_{ij} N_j(p- k_{ij})\Biggr\},\;p\geq 0,\;1\leq i\leq n,\\ N_i(p)= N_{ip},\quad p\leq 0,\quad 1\leq i\leq n.\end{gathered}\tag{1}$ The author proves that, under some assumptions, all solutions $$N_i(p)$$, $$1\leq i\leq n$$, of (1) are positive and bounded above, the system is persistent and the positive equilibrium of (1) is globally asymptotically stable for any $$k_{ij}\geq 0$$, $$1\leq i\leq n$$, $$1\leq j\leq n$$.
Reviewer: D. M. Bors (Iaşi)

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 92D25 Population dynamics (general)
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### References:

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