A stage-structured predator-prey model with disturbing pulse and time delays. (English) Zbl 1167.34372

Summary: We propose a stage-structured predator-prey model with disturbing pulse and time delays and obtain the sufficient conditions for the global attractivity of predator-eradiation periodic solution and permanence of the system. We also show that time delay, pulse catching rate and the period of pulsing can affect the dynamics of the system.


34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
92D25 Population dynamics (general)
Full Text: DOI


[1] Hastings, A., Age-dependent predation is not a simple process, I. Continuous time models, Theor. Popul. Biol., 23, 47-62 (1983)
[2] Hastings, A., Delay in recruitment at different trophic levels, effects on stability, J. Math. Biol., 21, 35-44 (1984) · Zbl 0547.92014
[3] Wang, W.; Chen, L., A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33, 83-91 (1997)
[4] Gourley, S. A.; Kuang, Y., A stage structured predator-prey model and its dependence on through-stage delay and death rate, J. Math. Biol., 49, 188-200 (2004) · Zbl 1055.92043
[5] Liu, S.; Chen, L., Extinction and permanence in competitive stage-structured system with time-delay, Nonlinear Anal. TMA, 51, 1347-1361 (2002) · Zbl 1021.34065
[6] Liu, S.; Chen, L.; Luo, G.; Jiang, Y., Asymptotic behaviors of competitive Lotka-Volterra system with stage structure, J. Math. Anal. Appl., 271, 124-138 (2002) · Zbl 1022.34039
[7] Song, X.; Cui, J., The stage-structured predator-prey system with delay and harvesting, Appl. Anal., 81, 1127-1142 (2002) · Zbl 1049.34096
[8] Ou, L., The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay, J. Math. Anal. Appl., 283, 534-548 (2003) · Zbl 1035.34046
[9] Xiao, Y.; Chen, L.; ven den Bosch, F., Dynamical behavior for a stage-structured SIR infectious disease model, Nonlinear Anal. RWA, 3, 175-190 (2002) · Zbl 1007.92032
[10] Aiello, W. G.; Freedman, H. I., A time-delay model of single-species growth with stage structure, Math. Biosci., 101, 139-153 (1990) · Zbl 0719.92017
[11] Tang, S.; Chen, L., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44, 185-199 (2002) · Zbl 0990.92033
[12] D’Onofrio, A., Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model., 36, 473-489 (2002) · Zbl 1025.92011
[13] Liu, B.; Chen, L., The periodic competing Lotka-Volterra model with impulsive effect, IMA J. Math. Med. Biol., 21, 129-145 (2004) · Zbl 1055.92056
[14] Roberts, M. G.; Kao, R. R., The dynamics of an infectious disease in a population with birth pulse, Math. Biosci., 149, 23-36 (1998) · Zbl 0928.92027
[15] Tang, S.; Chen, L., The effect of seasonal harvesting on stage-structured population models, J. Math. Biol., 48, 357-374 (2004) · Zbl 1058.92051
[16] Liu, B.; Zhang, Y.; Chen, L., The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. RWA, 6, 227-243 (2005) · Zbl 1082.34039
[17] Zhang, S.; Tan, D.; Chen, L., Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations, Chaos Soliton Fract, 28, 367-376 (2006) · Zbl 1083.37537
[18] Yan, J., Stability for impulsive delay differential equations, Nonlinear Anal. TMA, 63, 66-80 (2005) · Zbl 1082.34069
[19] Leonid, B.; Elena, B., Linearized oscillation theory for a nonlinear delay impulsive equation, J. Comput. Appl. Math., 161, 477-495 (2003) · Zbl 1045.34039
[20] Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal. TMA, 53, 1041-1062 (2003) · Zbl 1037.34061
[21] Goh, B. S., Global stability in two species interactions, J. Math. Biol., 3, 313-318 (1976) · Zbl 0362.92013
[22] Hastings, A., Global stability in two species system, J. Math. Biol., 5, 399-403 (1978) · Zbl 0382.92008
[23] He, X., Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198, 355-370 (1996) · Zbl 0873.34062
[24] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence and stability of a stage-structured predator-prey model, J. Math. Anal. Appl., 262, 499-528 (2001) · Zbl 0997.34069
[25] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay, Comput. Math. Appl., 159, 863-880 (2004) · Zbl 1056.92063
[26] Wang, W.; Chen, L., A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33, 83-91 (1997)
[27] Hui, J.; Zhu, D., Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Solution Fract, 29, 233-251 (2006) · Zbl 1095.92067
[28] Bainov, D.; Simeonov, P., System with Impulse Effect, Stability, Theory and Applications (1989), John Wiley and Sons: John Wiley and Sons New York · Zbl 0676.34035
[29] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[30] Song, X.; Chen, L., Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170, 173-186 (2001) · Zbl 1028.34049
[31] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press Inc.: Academic Press Inc. San Diego, CA · Zbl 0777.34002
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