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Stable periodic solution of a discrete periodic Lotka-Volterra competition system. (English) Zbl 1019.39004
Consider the following discrete Lotka-Volterra competition system \[ \begin{cases} x(n+1)= x(n)\exp \biggl[r_1(n) \bigl(1-x(n)/K_1(n)-\mu_2(n) y(n) \bigr) \biggr],\\ y(n+1)=y(n) \exp\biggl[r_2(n) \bigl(1-\mu_1(n)x(n)-y(n) /K_2 (n)\bigr) \biggr] \end{cases} \] where \(K_i(n)\), \(r_i(n)\) and \(\mu_i(n)\), \(i=1,2\) are bounded non-negative sequences. Sufficient conditions are given for the persistence of the system, i.e. the existence of a compact subset \(E\subset \mathbb{R}^2_+\) such that each solution will eventually enter and remain in \(E\). The existence and stability of periodic solution is established, too.

MSC:
39A11 Stability of difference equations (MSC2000)
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