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Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses. (English) Zbl 1255.34056
Summary: A periodic n-species Gilpin-Ayala competition system with impulses is studied. By constructing a suitable Lyapunov function and using the comparison theorem of impulsive differential equations, a set of sufficient conditions which guarantee that some species in the system are permanent and globally attractive while the remaining species are driven to extinction are obtained. Our results show that the dynamic behaviors of the system we considered are quite different from the corresponding system without impulses.
MSC:
34D20Stability of ODE
93C15Control systems governed by ODE
34D05Asymptotic stability of ODE
92D25Population dynamics (general)
References:
[1]Ahmad, S.: On the nonautonomous Volterra–Lotka competition equations, Proc. amer. Math. soc. 117, 199-204 (1993) · Zbl 0848.34033 · doi:10.2307/2159717
[2]Ahmad, S.; De Oca, F. Montes: Extinction in nonautonomous T-periodic competitive Lotka–Volterra system, Appl. math. Comput. 90, No. 2–3, 155-166 (1998) · Zbl 0906.92024 · doi:10.1016/S0096-3003(97)00396-2
[3]Teng, Z. D.: On the nonautonomous Lotka–Volterra N-species competing systems, Appl. math. Comput. 114, 175-185 (2000) · Zbl 1016.92045 · doi:10.1016/S0096-3003(99)00110-1
[4]Zhao, J. D.; Jiang, J. F.: Average conditions for permanence and extinction in nonautonomous Lotka–Volterra system, J. math. Anal. appl. 299, 663-675 (2004) · Zbl 1066.34050 · doi:10.1016/j.jmaa.2004.06.019
[5]Ayala, F. J.; Gilpin, M. E.; Eherenfeld, J. G.: Competition between species: theoretical models and experimental tests, Theor. popul. Biol. 4, 331-356 (1973)
[6]Gilpin, M. E.; Ayala, F. J.: Global models of growth and competition, Proc. natl. Acad. sci. USA 70, 3590-3593 (1973) · Zbl 0272.92016 · doi:10.1073/pnas.70.12.3590
[7]Goh, B. S.; Agnew, T. T.: Stability in gilpin and ayala’s model of competition, J. math. Biol. 4, 275-279 (1977) · Zbl 0379.92017 · doi:10.1007/BF00280977
[8]Liao, X. X.; Li, J.: Stability in gilpin–ayala competition models with diffusion, Nonlinear anal. TMA 28, 1751-1758 (1997) · Zbl 0872.35054 · doi:10.1016/0362-546X(95)00242-N
[9]Chen, F. D.: Average conditions for permanence and extinction in nonautonomous gilpin–ayala competition model, Nonlinear anal. RWA 7, No. 4, 895-915 (2006) · Zbl 1119.34038 · doi:10.1016/j.nonrwa.2005.04.007
[10]Chen, F. D.: Permanence of a delayed non-autonomous gilpin–ayala competition model, Appl. math. Comput. 179, No. 1, 55-66 (2006) · Zbl 1096.92041 · doi:10.1016/j.amc.2005.11.079
[11]Chen, F. D.: Some new results on the permanence and extinction of nonautonomous gilpin–ayala type competition model with delays, Nonlinear anal. RWA 7, No. 5, 1205-1222 (2006) · Zbl 1120.34062 · doi:10.1016/j.nonrwa.2005.11.003
[12]Fan, M.; Wang, K.: Global periodic solutions of a generalized n-species gilpin–ayala competition model, Comput. math. Appl. 40, 1141-1151 (2000) · Zbl 0954.92027 · doi:10.1016/S0898-1221(00)00228-5
[13]Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[14]Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications, (1993)
[15]Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects, Math. comput. Modell. 26, 59-72 (1997) · Zbl 1185.34014 · doi:10.1016/S0895-7177(97)00240-9
[16]Jin, Z.; Ma, Z. E.; Han, M. A.: The existence of periodic solutions of the n-species Lotka–Volterra competition systems with impulsive, Chaos solitons fractals 22, 181-188 (2004) · Zbl 1058.92046 · doi:10.1016/j.chaos.2004.01.007
[17]Liu, X. N.; Chen, L. S.: Global dynamics of the periodic logistic system with periodic impulsive perturbations, J. math. Anal. appl. 289, 279-291 (2004) · Zbl 1054.34015 · doi:10.1016/j.jmaa.2003.09.058
[18]Liu, X. N.; Chen, L. S.: Global behaviors of a generalized periodic impulsive logistic system with nonlinear density dependence, Commun. nonlinear sci. Numer. simul. 10, 329-340 (2005) · Zbl 1070.34067 · doi:10.1016/j.cnsns.2003.03.001
[19]Tang, S. Y.; Chen, L. S.: The periodic predator-prey Lotka–Volterra model with impulsive effect, J. Mach med. Biol. 2, 267-296 (2002)
[20]Lakmeche, A.; Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. contin. Discrete impuls. Syst. 7, 265-287 (2000)
[21]Panetta, J. C.: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. math. Biol. 58, 425-447 (1996) · Zbl 0859.92014 · doi:10.1007/BF02460591
[22]Jin, Z.; Han, M. A.; Li, G. H.: The persistence in a Lotka–Volterra competition systems with impulsive, Chaos solitons fractals 24, 1105-1117 (2005) · Zbl 1081.34045 · doi:10.1016/j.chaos.2004.09.065
[23]Ahmad, S.; Stamova, I. M.: Asymptotic stability of an N-dimensional impulsive competitive systems, Nonlinear anal. RWA 8, No. 2, 654-663 (2007) · Zbl 1152.34342 · doi:10.1016/j.nonrwa.2006.02.004
[24]Zhang, S. W.; Tan, D. J.; Chen, L. S.: The periodic n-species gilpin–ayala competition system with impulsive effect, Chaos solitons fractals 26, No. 2, 507-517 (2005) · Zbl 1065.92065 · doi:10.1016/j.chaos.2005.01.020