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An impulsive predator-prey system with modified Leslie-Gower and Holling type II schemes. (English) Zbl 1148.34034
Summary: An impulsive predator-prey system with modified Leslie-Gower and Holling-type II schemes is presented. By using the Floquet theory of impulsive equation and the small amplitude perturbation method, the global asymptotical stability of a prey-free positive periodic solution and the permanence of the system are discussed. The corresponding threshold conditions are obtained respectively. Finally, numerical simulations are given.

34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)
34D05Asymptotic stability of ODE
34C25Periodic solutions of ODE
34A37Differential equations with impulses
Full Text: DOI
[1] 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 · doi:10.1016/j.chaos.2004.05.044
[2] Holling, C. S.: The functional response of predator to prey density and its role in mimicry and population regulation, Mem ent sec can 45, 1-60 (1965)
[3] Cushing, J. M.: Periodic time-dependent predator -- prey system, SIAM J appl math 32, 82-95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006
[4] Leslie, P. H.: Some further notes on the use of matrices in population mathematics, Biometrica 35, 213-245 (1948) · Zbl 0034.23303
[5] Leslie, P. H.; Gower, J. C.: The properties of a stochastic model for the predator -- prey type of interaction between two species, Biometrica 47, 219-234 (1960) · Zbl 0103.12502
[6] Pielou, E. C.: An introduction to mathematical ecology, (1969) · Zbl 0259.92001
[7] Aziz-Alaoui, M. A.; Okiye, M. Daher: Boundedness and global stability for a predator -- prey model with modified Leslie -- gower and Holling-type II type schemes, Appl math lett 16, 1069-1075 (2003) · Zbl 1063.34044 · doi:10.1016/S0893-9659(03)90096-6
[8] 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 · doi:10.1016/S0960-0779(02)00408-3
[9] Zhang, S. W.; Chen, L. S.: A Holling II functional response food chain model with impulsive perturbations, Chaos, solitons & fractals 24, 1269-1278 (2005) · Zbl 1086.34043
[10] 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
[11] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[12] Zhang, Y. J.; Liu, B.; Chen, L. S.: Extinction and permanence of a two-prey one-predator system with impulsive effect, IMA J appl math 20, 1-17 (2003) · Zbl 1046.92051 · doi:10.1093/imammb/20.4.309
[13] Lu, Z. H.; Chi, X. B.; Chen, L. S.: Impulsive control strategies in biological control of pesticide, Theor pop biol 64, 39-47 (2003) · Zbl 1100.92071 · doi:10.1016/S0040-5809(03)00048-0
[14] Smith, H. L.: Cooperative systems of differential equations with concave nonlinearities, Nonlinear anal TMA 10, 1037-1052 (1986) · Zbl 0612.34035 · doi:10.1016/0362-546X(86)90087-8
[15] Zhang, S. W.; Chen, L. S.: Chaos in three species food chain system with impulsive perturbations, Chaos, solitons & fractals 24, 73-83 (2005) · Zbl 1066.92060