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Stable positive periodic solution of time dependent Lotka-Volterra periodic mutualistic system. (English) Zbl 0808.34055

The main result of this paper is: Consider a Lotka-Volterra mutualistic system in a periodic environment, \(dx_ i/dt = x_ i\) \((r_ i(t) + \sum^ n_{j = 1} a_{ij} (t)x_ j)\) \((i = 1, \dots, n)\), with \(a_{ii} (t) < 0\) and \(a_{ij} (t) \geq 0\) \((i \neq j)\) and with \(\omega\)-periodic \((0 < \omega < + \infty)\) functions \(r_ i (t)\) and \(a_{ij} (t)\). Assume \(\int^ \omega_ 0 r_ i(t) dt > 0\) and let \(\overline a_{ij} = \max_{0 \leq t \leq \omega} a_{ij} (t)\). If the leading principal minors of the matrix \((\overline a_{ij})\) alternate in sign, then the system has a unique and globally asymptotically stable positive \(\omega\)-periodic solution. This result extends a somewhat similar result for time-independent Lotka-Volterra mutualistic systems. The paper implicitly assumes continuity of the \(a_{ij} (t)\)’s. It ends with the “Note: The proof of theorem 2 has some errors, but it can be corrected easily”. The corrections should have been indicated and were not.

MSC:

34D20 Stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
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