Permanence of a general periodic single-species system with periodic impulsive perturbations. (English) Zbl 1116.92052

Liu, Xianning; Takeuchi, Yasuhiro

Japan J. Ind. Appl. Math. 24, No. 1, 57-65 (2007).

Summary: Sufficient conditions for permanence of a general periodic single-species system with periodic impulsive perturbations are obtained via comparison theory of impulsive differential equations. An application is given to a periodic impulsive logistic system.

MSC:

92D25	Population dynamics (general)
34A37	Ordinary differential equations with impulses
34C60	Qualitative investigation and simulation of ordinary differential equation models
34D99	Stability theory for ordinary differential equations

Keywords:

impulses; single-species; permanence; periodic perturbation

References:

[1]	D.D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications. Longman, England, 1993. · Zbl 0815.34001
[2]	E. Funasaki and M. Kot, Invasion and chaos in a periodically pulsed mass-action chemostat. Theor. Popul. Biol.,44 (1993), 203–224. · Zbl 0782.92020 · doi:10.1006/tpbi.1993.1026
[3]	S. Gao and L. Chen, The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses. Chaos, Solitons & Fractals,24 (2005), 1013–1023. · Zbl 1061.92059 · doi:10.1016/j.chaos.2004.09.091
[4]	A. Lakmeche, O. Arino, Birfurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynam. Contin. Discrete Impuls.,7 (2000), 265–287. · Zbl 1011.34031
[5]	V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989. · Zbl 0719.34002
[6]	W. Li and H. Huo, Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, J. Comput. Appl. Math.,174 (2005), 227–238. · Zbl 1070.34089 · doi:10.1016/j.cam.2004.04.010
[7]	W. Li and H. Huo, Existence and global attractivity of positive periodic solutions of functional differential equations with impulses. Nonlinear Analysis,59 (2004), 857–877. · Zbl 1061.34059
[8]	B. Liu, Y. Zhang and L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management. Nonlinear Anal. Real World Appl.,6 (2005), 227–243. · Zbl 1082.34039 · doi:10.1016/j.nonrwa.2004.08.001
[9]	X. Liu and L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations. J. Math. Anal. Appl.,289 (2004), 279–291. · Zbl 1054.34015 · doi:10.1016/j.jmaa.2003.09.058
[10]	X. Liu and L. Chen, Global attractivity of positive periodic solutions for nonlinear impulsive systems. Nonlinear Analysis,65 (2006), 1843–1857. · Zbl 1111.34010 · doi:10.1016/j.na.2005.10.041
[11]	M.G. Roberts and R.R. Kao, The dynamics of an infectious disease in a population with birth pulses. Math. Biosci.,149 (1998), 23–36. · Zbl 0928.92027 · doi:10.1016/S0025-5564(97)10016-5
[12]	B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol.,60 (1998), 1123–1148. · Zbl 0941.92026 · doi:10.1016/S0092-8240(98)90005-2
[13]	S. Tang and L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol.,44 (2002), 185–199. · Zbl 0990.92033 · doi:10.1007/s002850100121
[14]	S. Tang and L. Chen, Global attractivity in a ”food-limited” population model with impulsive effects. J. Math. Anal. Appl.,292 (2004), 211–221. · Zbl 1062.34055 · doi:10.1016/j.jmaa.2003.11.061
[15]	J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka–Volterra systems. Math. Comput. Model.,40 (2004), 509–518. · Zbl 1112.34052 · doi:10.1016/j.mcm.2003.12.011
[16]	B. Zhang and Y. Liu, Global attractivity for certain impulsive delay differential equations. Nonlinear Analysis,52 (2003), 725–736. · Zbl 1027.34086 · doi:10.1016/S0362-546X(02)00129-3
[17]	W. Zhu, D. Xu and Z. Yang, Global exponential stability of impulsive delay difference equation. Appl. Math. Comput.,181 (2006), 65–72. · Zbl 1148.39304 · doi:10.1016/j.amc.2006.01.015