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The qualitative analysis of \(n\)-species Lotka-Volterra periodic competition systems. (English) Zbl 0756.34048

The author considers the \(n\)-species Lotka-Volterra system \[ dx_ i(t)/dt=x(t)\left(b_ i(t)-\sum^ n_{j=1}a_{ij}(t)x_ j(t)\right),\quad x_ i\geq 0,\quad i=1,\dots,n, \tag{1} \] where \(b_ i(t)\), \(a_{ij}(t)\) are continuous \(\omega\)-periodic functions with \(\int^ \omega_ 0b_ i(t)dt>0\) and \(a_{ij}(t)>0\). He proves that all solutions of (1) with positive initial values are ultimately bounded, and he obtains sufficient conditions for the existence and global attractivity of a positive periodic solution. His results include and generalize those of K. Gopalsamy [J. Aust. Math. Soc., Ser. B. 27, 66-72 (1985; Zbl 0588.92019); ibid. 24, 160-170 (1982; Zbl 0498.92016)] and J. M. Cushing [J. Math. Biol. 10, 385-400 (1980; Zbl 0455.92012)].
Reviewer: W.Müller (Berlin)

MSC:

34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
34D40 Ultimate boundedness (MSC2000)
34C11 Growth and boundedness of solutions to ordinary differential equations
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References:

[1] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities, Volume 1 & 2 (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[2] Gopalsamy, K., Global asymptotic stability in a periodic Lotka-Volterra system, J. Australian Math. Soc., Series B, 27, 66-72 (1985) · Zbl 0588.92019
[3] Cushing, J. M., Two species competition in a periodic environment, J. Math. Biology, 10, 4, 385-400 (1980) · Zbl 0455.92012
[4] Gopalsamy, K., Exchange of equilibria in two species Lotka-Volterra competition models, J. Australian Math. Soc., Series B, 24, 160-170 (1982) · Zbl 0498.92016
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