The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments. (English) Zbl 1280.34078

Summary: In this paper, a delayed pest control model with stage-structure for pests by introducing a constant periodic pesticide input and harvesting prey (Crops) at two different fixed moments is proposed and analyzed. We assume only the pests are affected by pesticide. We prove that the conditions for global asymptotically attractive ‘predator-extinction’ periodic solution and permanence of the population of the model depend on time delay, pulse pesticide input, and pulse harvesting prey. By numerical analysis, we also show that constant maturation time delay, pulse pesticide input, and pulse harvesting prey can bring obvious effects on the dynamics of system, which also corroborates our theoretical results. We believe that the results will provide reliable tactic basis for the practical pest management. One of the features of present paper is to investigate the high-dimensional delayed system with impulsive effects at different fixed impulsive moments.


34K45 Functional-differential equations with impulses
92D25 Population dynamics (general)
34K13 Periodic solutions to functional-differential equations
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) · Zbl 0507.92016
[2] Hastings, A.: Delay in recruitment at different trophic levels, effects on stability. J. Math. Biol. 21, 35–44 (1984) · Zbl 0547.92014
[3] Qiu, Z., Wang, K.: The asymptotic behavior of a single population model with space-limited and stage-structure. Nonlinear Anal. Real World Appl. 6, 155–173 (2005) · Zbl 1088.34048
[4] Hui, J., Zhu, D.: Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos Solitons Fractals 29, 233–251 (2006) · Zbl 1095.92067
[5] Gourley, S.A., Kuang, Y.: A stage structured predator-prey model and its dependenceon through-stage delay and death rate. J. Math. Biol. 49, 188–200 (2004) · Zbl 1055.92043
[6] Wei, C., Chen, L.: Eco-epidemiology model with age structure and prey-dependent consumption for pest management. Appl. Math. Model. 33, 4354–4363 (2009) · Zbl 1179.34049
[7] Jiao, J., Chen, L.: Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators. Int. J. Biomath. 1, 197–208 (2008) · Zbl 1155.92355
[8] Song, X., Cui, J.: The stage-structured predator-prey system with delay and harvesting. Appl. Anal. 81, 1127–1142 (2002) · Zbl 1049.34096
[9] Ou, L., et al.: The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay. J. Math. Appl. 283, 534–548 (2003) · Zbl 1035.34046
[10] Shi, R., Chen, L.: The study of a ratio-dependent predator–prey model with stage structure in the prey. Nonlinear Dyn. 58, 443–451 (2009) · Zbl 1183.92083
[11] 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
[12] Zhang, H., Jiao, J., Chen, L.: Pest management through continuous and impulsive control strategies. Biosystems 90, 350–361 (2007)
[13] Meng, X., Li, Z., Wang, X.: Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects. Nonlinear Dyn. 59, 503–513 (2009) · Zbl 1183.92072
[14] 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
[15] 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
[16] Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009) · Zbl 1183.92003
[17] 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
[18] Zhang, S., Tan, D., Chen, L.: Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations. Chaos Solitons Fractals 28, 367–376 (2006) · Zbl 1083.37537
[19] Yan, J.: Stability for impulsive delay differential equations. Nonlinear Anal. 63, 66–80 (2005) · Zbl 1082.34069
[20] Leonid, B., Elena, B.: Linearized oscillation theory for a nonlinear delay impulsive equation. J. Comput. Appl. Math. 161, 477–495 (2003) · Zbl 1045.34039
[21] 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
[22] Gao, S., Chen, L., Teng, Z.: Impulsive vaccination of an SEIRS model with time delay and varying total population size. Bull. Math. Biol. 69, 731–745 (2007) · Zbl 1139.92314
[23] Meng, X., Jiao, J., Chen, L.: The dynamics of an age structured predator-prey model with disturbing pulse and time delays. Nonlinear Anal. Real World Appl. 9, 547–561 (2008) · Zbl 1142.34054
[24] Jiao, J., Long, W., Chen, L.: A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin. Nonlinear Anal. Real World Appl. 10(5), 3073–3081 (2009) · Zbl 1162.92330
[25] Bainov, D., Simeonov, P.: System with Impulsive Effect: Stability, Theory and Applications. Wiley, New York (1989) · Zbl 0676.34035
[26] Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) · Zbl 0719.34002
[27] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego, California (1993) · Zbl 0777.34002
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.