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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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