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On the extinction for non-autonomous food chain systems with delays. (English) Zbl 1089.92062

Summary: General \(n\)-species nonautonomous Lotka-Volterra-type food chain systems with pure delays are considered. New sufficient conditions for which part of the \(n\) species is driven to extinction and the surplus part remains permanent are established.

MSC:

92D40 Ecology
34K20 Stability theory of functional-differential equations
Full Text: DOI

References:

[1] Ahmad, S., Extinction of species in nonautonomous Lotka-Volterra systems, Proc. Amer. Math. Soc., 127, 2905-2910 (1999) · Zbl 0924.34040
[2] Ahmad, S.; Montes de Oca, F., Extinction in nonautonomous \(T\)-periodic competitive Lotka-Volterra systems, Appl. Math. Comput., 90, 155-166 (1998) · Zbl 0906.92024
[3] Ahmad, S.; Montes de Oca, F., Average growth and extinction in a two dimensional Lotka-Volterra system, Dyn. Contin. Discrete Impulsive Syst. Ser. A, 9, 177-186 (2002) · Zbl 1081.34513
[4] Duan, K.; Teng, Z., The stability and persistence of Lotka-Volterra food chains, J. Biomath., 6, 148-154 (1991), (in Chinese)
[5] Freedman, H. I.; So, J., Global stability and persistence of simple food chains, Math. Biosci., 76, 69-86 (1985) · Zbl 0572.92025
[6] Gard, T. C., Persistence in food chains with general interactions, Math. Biosci., 51, 165-174 (1980) · Zbl 0453.92017
[7] Gard, T. C., Top predator persistence in differential equation models of food chains: the effects of omnivory and external forcing of lower trophic levels, J. Math. Biol., 14, 285-299 (1982) · Zbl 0494.92022
[8] Gard, T. C.; Hallam, T. G., Persistence in food webs—I Lotka-Volterra food chains, Bull. Math. Biol., 41, 877-891 (1979) · Zbl 0422.92017
[9] Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer New York · Zbl 0352.34001
[10] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[11] Li, Z.; Teng, Z., Permanence for nonautonomous food chain systems with delay, J. Math. Anal. Appl., 286, 724-740 (2003) · Zbl 1054.34126
[12] Ma, Z.; Zong, W.; Luo, Z., The thresholds of survival for an \(n\)-dimensional food chain model in a polluted environment, J. Math. Anal. Appl., 210, 440-458 (1997) · Zbl 0905.92030
[13] Montes de Oca, F.; Zeeman, M. L., Balancing survival and extinction in nonautonomous competitive Lotka-Volterra system, J. Math. Anal. Appl., 192, 360-370 (1995) · Zbl 0830.34039
[14] Montes de Oca, F.; Zeeman, M. L., Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 124, 3677-3687 (1996) · Zbl 0866.34029
[15] Teng, Z., On the nonautonomous Lotka-Volterra \(N\)-species competing systems, Appl. Math. Comput., 114, 175-185 (2000) · Zbl 1016.92045
[16] Teng, Z., On the permanence and extinction in nonautonomous Lotka-Volterra competitive systems with delays, Acta Math. Sinica, 44, 293-306 (2001), (in Chinese) · Zbl 1033.34079
[17] Teng, Z.; Chen, L., Permanence and extinction of periodic predator-prey systems in a patchy environment with delay, Nonlinear Anal. Real World Appl., 4, 335-364 (2003) · Zbl 1018.92033
[18] Teng, Z.; Yu, Y., The extinction in nonautonomous prey-predator Lotka-Volterra systems, Acta Math. Appl. Sinica, 15, 401-408 (1999) · Zbl 1007.92031
[19] Teng, Z.; Yu, Y., Some new results of nonautonomous Lotka-Volterra competitive systems with delays, J. Math. Anal. Appl., 241, 254-275 (2000) · Zbl 0947.34066
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