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Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system. (English) Zbl 1066.34050
Authors’ abstract: An n-species nonautonomous competitive Lotka-Volterra system is considered. The average conditions on the coefficients are given to guarantee that all but one species are driven to extinction. The generalization for the result is presented, i.e., for each $r\leq n$ the average conditions on the coefficients are presented to guarantee that $r$ of the species of the system are permanent while the remaining $n-r$ species are driven to extinction. It is shown that these average conditions are improvement of those of {\it S. Ahmad} and {\it F. Montes de Oca} [Appl. Math. Comp. 90, 155--166 (1998; Zbl 0906.92024)] and {\it F. Montes de Oca} and {\it M. L. Zeeman} [Proc. Am. Math.Soc. 124, 3677--3687 (1996; Zbl 0866.34029) and J. Math. Anal. Appl. 192, 360--370 (1995; Zbl 0830.34039)].

##### MSC:
 34D05 Asymptotic stability of ODE 92D25 Population dynamics (general) 34C29 Averaging method
##### Keywords:
Lotka-Volterra system; permanence; averaging
Full Text:
##### References:
 [1] Ahmad, S.; Lazer, A. C.: Average conditions for global asymptotic stability in a nonautonomous Lotka -- Volterra system. Nonlinear anal. 40, 37-49 (2000) · Zbl 0955.34041 [2] De Oca, F. Montes; Zeeman, M. L.: Extinction in nonautonomous competitive Lotka -- Volterra systems. Proc. amer. Math. soc. 124, 3677-3687 (1996) · Zbl 0866.34029 [3] Ahmad, S.; De Oca, F. Montes: Extinction in nonautonomous T-periodic competitive Lotka -- Volterra system. Appl. math. Comput. 90, 155-166 (1998) · Zbl 0906.92024 [4] De Oca, F. Montes; Zeeman, M. L.: Balancing survival and extinction in nonautonomous competitive Lotka -- Volterra systems. J. math. Anal. appl. 192, 360-370 (1995) · Zbl 0830.34039 [5] Zeeman, M. L.: Extinction in competitive Lotka -- Volterra systems. Proc. amer. Math. soc. 123, 87-96 (1995) · Zbl 0815.34039 [6] Ahmad, S.: On the nonautonomous Volterra -- Lotka competition equations. Proc. amer. Math. soc. 117, 199-204 (1993) · Zbl 0848.34033 [7] Ahmad, S.: Extinction of species in nonautonomous Lotka -- Volterra systems. Proc. amer. Math. soc. 127, 2905-2910 (1999) · Zbl 0924.34040 [8] Teng, Z.: On the non-autonomous Lotka -- Volterra N-species competing systems. Appl. math. Comput. 114, 175-185 (2000) · Zbl 1016.92045 [9] Zhao, J. D.; Jiang, J. F.; Ahmad, S.: The permanence and global attractivity in a nonautonomous Lotka -- Volterra system. Nonlinear anal. RWA 5, 265-276 (2004) · Zbl 1085.34040 [10] Tineo, A.: On the asymptotic behavior of some population models. J. math. Anal. appl. 167, 516-529 (1992) · Zbl 0778.92018