# zbMATH — the first resource for mathematics

Dynamic behaviors of the impulsive periodic multi-species predator-prey system. (English) Zbl 1165.34308
Summary: The dynamic behaviors of an impulsive periodic predator-prey model with $$n$$-preys and $$m$$-predators are studied in this paper. By constructing a suitable Lyapunov function and using the Comparison theorem of impulsive differential equation, sufficient conditions which ensure the permanence and global attractivity of the system are obtained. At the same time, a set of criteria which guarantee that some species in the system are permanent and globally attractive while the remaining species are driven to extinction is obtained. Our results show that, for the multi-species predator-prey community, impulsivity is one of the important reasons that can change the long time behaviors of species.

##### MSC:
 34A37 Ordinary differential equations with impulses 34D05 Asymptotic properties of solutions to ordinary differential equations 92D25 Population dynamics (general)
##### Keywords:
predator-prey; impulsive; extinction; permanence; global attractivity
Full Text:
##### References:
 [1] Ahmad, S.; Montes de Oca, F., Extinction in nonautonomous $$T$$-periodic competitive lotka – volterra system, Appl. math. comput., 90, 2-3, 155-166, (1998) · Zbl 0906.92024 [2] Ding, T.; Huang, H.; Zanolin, F., A priori bounds and periodic solution for a class of planar systems with applications to lotka – volterra equations, Discrete contin. dyn. syst., 1, 103-117, (1995) · Zbl 0877.34035 [3] Lisena, B., Global attractive periodic models of predator – prey type, Nonlinear anal. RWA, 6, 133-144, (2005) · Zbl 1097.34029 [4] Lopez-Gomez, J.; Ortega, R.; Tineo, A., The periodic predator – prey lotka – volterra model, Adv. differential equations., 1, 403-432, (1996) · Zbl 0849.34026 [5] Teng, Z., Uniform persistence of the periodic predator – prey lotka – volterra systems, Appl. anal., 72, 339-352, (1998) · Zbl 1031.34045 [6] Yang, P.; Xu, R., Global attractivity of the periodic lotka – volterra system, J. math. anal. appl., 233, 1, 221-232, (1999) · Zbl 0973.92039 [7] Zhao, J.D.; Chen, W.C., Global asymptotic stability of a periodic ecological model, Appl. math. comput., 147, 3, 881-892, (2004) · Zbl 1029.92026 [8] Xia, Y.H.; Chen, F.D.; Chen, A.P.; Cao, J.D., Existence and global attractivity of an almost periodic ecological model, Appl. math. comput., 157, 2, 449-475, (2004) · Zbl 1049.92038 [9] Zhao, J.D.; Jiang, J.F., Permanence in nonautonomous lotka – volterra system with predator – prey, Appl. math. comput., 152, 99-109, (2004) · Zbl 1047.92050 [10] Zhao, J.D.; Jiang, J.F.; Lazer, A.C., The permanence and global attractivity in a nonautonomous lotka – volterra system, Nonlinear anal. RWA, 5, 4, 265-276, (2004) · Zbl 1085.34040 [11] Chen, F.D., Permanence in nonautonomous multi-species predator – prey system with feedback controls, Appl. math. comput., 173, 2, 694-709, (2006) · Zbl 1087.92059 [12] Chen, F.D., Permanence and global stability of nonautonomous lotka – volterra system with predator – prey and deviating arguments, Appl. math. comput., 173, 2, 1082-1100, (2006) · Zbl 1121.34080 [13] Chen, F.D., On a periodic multi-species ecological model, Appl. math. comput., 171, 1, 492-510, (2005) · Zbl 1080.92059 [14] Chen, F.D., On a nonlinear nonautonomous predator – prey model with diffusion and distributed delay, J. comput. appl. math., 180, 1, 33-49, (2005) · Zbl 1061.92058 [15] Chen, F.D., Permanence and global attractivity of a discrete multispecies lotka – volterra competition predator – prey systems, Appl. math. comput., 182, 1, 3-12, (2006) · Zbl 1113.92061 [16] Chen, F.D.; Shi, C.L., Global attractivity in an almost periodic multi-species nonlinear ecological model, Appl. math. comput., 180, 1, 376-392, (2006) · Zbl 1099.92069 [17] Chen, F.D.; Xie, X.D.; Shi, J.L., Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. comput. appl. math., 194, 2, 368-387, (2006) · Zbl 1104.34050 [18] Chen, F.D.; Xie, X.D., Periodicity and stability of a nonlinear periodic integro-differential prey-competition model with infinite delays, Commun. nonlinear sci. numer. simul., 12, 6, 876-885, (2007) · Zbl 1122.45006 [19] 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 [20] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, (1993), Longman Scientific and Technical · Zbl 0793.34011 [21] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific · Zbl 0719.34002 [22] Ahmad, S.; Stamova, I.M., Asymptotic stability of an $$N$$-dimensional impulsive competitive systems, Nonlinear anal. RWA, 8, 2, 654-663, (2007) · Zbl 1152.34342 [23] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. comput. modelling, 26, 59-72, (1997) · Zbl 1185.34014 [24] 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 [25] 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 [26] Liu, B.; Teng, Z.D.; Liu, W.B., Dynamic behaviors of the periodic lotka – volterra competing system with impulsive perturbations, Chaos solitons fractals, 31, 356-370, (2007) · Zbl 1145.34029 [27] 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 [28] Liu, X.N.; Chen, L.S., Complex dynamics of Holling type II lotka – volterra predator – prey system with impulsive perturbations on the predator, Chaos solitons fractals, 16, 311-320, (2003) · Zbl 1085.34529 [29] Tang, S.Y.; Chen, L.S., The periodic predator – prey lotka – volterra model with impulsive effect, J. Mach. med. biol., 2, 267-296, (2002) [30] Zhang, S.W.; Dong, L.Z.; Chen, L.S., The study of predator – prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos solitons fractals, 23, 631-643, (2005) · Zbl 1081.34041 [31] Zhang, S.W.; Tan, D.J.; Chen, L.S., The periodic $$n$$-species gilpin – ayala competition system with impulsive effect, Chaos solitons fractals, 26, 507-517, (2005) · Zbl 1065.92065 [32] 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) · Zbl 1011.34031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.