The dynamics of an age structured predator-prey model with disturbing pulse and time delays. (English) Zbl 1142.34054

Summary: We formulate a general and robust prey-dependent consumption predator-prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. In this paper, the main feature is that we introduce time delay and pulse into the predator-prey (natural enemy-pest) model with age structure, exhibit a new modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
92D25 Population dynamics (general)
34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] 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
[2] Bainov, D.; Simeonov, P., System with Impulsive Effect: Stability, Theory and Applications (1989), Wiley: Wiley New York · Zbl 0683.34032
[3] 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
[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] Hastings, A., Age-dependent predation is not a simple process, I, Continuous time models, Theor. Popul. Biol., 23, 47-62 (1983)
[6] Hastings, A., Delay in recruitment at different trophic levels, effects on stability, J. Math. Biol., 21, 35-44 (1984) · Zbl 0547.92014
[7] Hui, J.; Zhu, D., Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos, Solitons and Fractals, 29, 233-251 (2006) · Zbl 1095.92067
[8] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press San Diego, CA · Zbl 0777.34002
[9] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[10] Leonid, B.; Elena, B., Linearized oscillation theory for a nonlinear delay impulsive equation, J. Comput. Appl. Math., 161, 477-495 (2003) · Zbl 1045.34039
[11] 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
[12] Liu, B.; Zhang, Y.; Chen, L., The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. Real World Appl., 6, 227-243 (2005) · Zbl 1082.34039
[13] Liu, S.; Chen, L., Extinction and permanence in competitive stage-structured system with time-delay, Nonlinear Anal., 51, 1347-1361 (2002) · Zbl 1021.34065
[14] 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
[15] Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal., 53, 1041-1062 (2003) · Zbl 1037.34061
[16] Ou, L., The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay, J. Math. Appl., 283, 534-548 (2003) · Zbl 1035.34046
[17] 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
[18] Song, X.; Cui, J., The stage-structured predator-prey system with delay and harvesting, Appl. Anal., 81, 1127-1142 (2002) · Zbl 1049.34096
[19] 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
[20] Tang, S.; Chen, L., The effect of seasonal harvesting on stage-structured population models, J. Math. Biol., 48, 357-374 (2004) · Zbl 1058.92051
[21] Wang, W.; Chen, L., A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33, 83-91 (1997)
[22] Xiao, Y.; Chen, L.; ven den Bosch, F., Dynamical behavior for a stage-structured SIR infectious disease model, Nonlinear Anal. Real World Appl., 3, 175-190 (2002) · Zbl 1007.92032
[23] Yan, J., Stability for impulsive delay differential equations, Nonlinear Anal., 63, 66-80 (2005) · Zbl 1082.34069
[24] Zhang, S.; Tan, D.; Chen, L., Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations, Chaos, Solitons and Fractals, 28, 367-376 (2006) · Zbl 1083.37537
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.