# American Institute of Mathematical Sciences

June  2013, 6(3): 723-729. doi: 10.3934/dcdss.2013.6.723

## Dynamic behaviour of a periodic competitive system with pulses

 1 Dipartimento di Matematica, Universitá degli studi di Bari, 70125 Bari, Italy

Received  March 2010 Revised  December 2010 Published  December 2012

In this article we consider an $n$-dimensional competitive Lotka-Volterra system with periodic coefficients and impulses. We provide sufficient conditions for the existence and global attractivity of a positive periodic solution.
Citation: Benedetta Lisena. Dynamic behaviour of a periodic competitive system with pulses. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 723-729. doi: 10.3934/dcdss.2013.6.723
##### References:
 [1] S. Ahmad and I. M.Stamova, Asymptotic stability of an N-dimensional impulsive competitive system, Nonlinear Anal. Real World Appl., 8 (2007), 654-663. doi: 10.1016/j.nonrwa.2006.02.004. [2] S. Ahmad and A. C. Lazer, Average growth and extinction in a competitive Lotka-Volterra system, Nonlinear Anal., 62 (2005), 545-557. doi: 10.1016/j.na.2005.03.069. [3] S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000), 37-49. doi: 10.1016/S0362-546X(00)85003-8. [4] D. D. Bainov and P. S. Simeonov, "Systems with Impulse Effect," Elles Horwood Limited, Chichester, 1989 [5] D. D. Bainov and P. S. Simeonov, "Impulsive Differential Equations: Periodic Solutions and Applications," Longmann Scientific and Technical, New York, 1993. [6] B. Lisena, Stability and periodicity in competitive systems with impulses, Mediterr. J. Math., 6 (2009), 291-302. doi: 10.1007/s00009-009-0009-4. [7] B. Lisena, Coexistence and extinction in competitive systems with impulses, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 17 (2010), 619-637. [8] B. Lisena, Global stability in periodic competitive systems, Nonlinear Anal. Real World Appl., 5 (2004), 613-627. doi: 10.1016/j.nonrwa.2004.01.002. [9] B. Liu, Z. Teng and W. Liu, Dynamic behaviors of the periodic Lotka-Volterra competing systems with impulsive perturbations, Chaos Solitons Fractals, 31 (2007), 356-370. doi: 10.1016/j.chaos.2005.09.059. [10] I. Stamova, "Stability Analysis of Impulsive Functional Differential Equations," Walter de Gruyter, Berlin, New York, 2009. doi: 10.1515/9783110221824. [11] J. Zhen, M. Han and G. Li, The persistence in a Lotka-Volterra competition system with impulsive, Chaos Solitons Fractals, 24 (2005), 1105-1117. doi: 10.1016/j.chaos.2004.09.065.

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##### References:
 [1] S. Ahmad and I. M.Stamova, Asymptotic stability of an N-dimensional impulsive competitive system, Nonlinear Anal. Real World Appl., 8 (2007), 654-663. doi: 10.1016/j.nonrwa.2006.02.004. [2] S. Ahmad and A. C. Lazer, Average growth and extinction in a competitive Lotka-Volterra system, Nonlinear Anal., 62 (2005), 545-557. doi: 10.1016/j.na.2005.03.069. [3] S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000), 37-49. doi: 10.1016/S0362-546X(00)85003-8. [4] D. D. Bainov and P. S. Simeonov, "Systems with Impulse Effect," Elles Horwood Limited, Chichester, 1989 [5] D. D. Bainov and P. S. Simeonov, "Impulsive Differential Equations: Periodic Solutions and Applications," Longmann Scientific and Technical, New York, 1993. [6] B. Lisena, Stability and periodicity in competitive systems with impulses, Mediterr. J. Math., 6 (2009), 291-302. doi: 10.1007/s00009-009-0009-4. [7] B. Lisena, Coexistence and extinction in competitive systems with impulses, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 17 (2010), 619-637. [8] B. Lisena, Global stability in periodic competitive systems, Nonlinear Anal. Real World Appl., 5 (2004), 613-627. doi: 10.1016/j.nonrwa.2004.01.002. [9] B. Liu, Z. Teng and W. Liu, Dynamic behaviors of the periodic Lotka-Volterra competing systems with impulsive perturbations, Chaos Solitons Fractals, 31 (2007), 356-370. doi: 10.1016/j.chaos.2005.09.059. [10] I. Stamova, "Stability Analysis of Impulsive Functional Differential Equations," Walter de Gruyter, Berlin, New York, 2009. doi: 10.1515/9783110221824. [11] J. Zhen, M. Han and G. Li, The persistence in a Lotka-Volterra competition system with impulsive, Chaos Solitons Fractals, 24 (2005), 1105-1117. doi: 10.1016/j.chaos.2004.09.065.
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