Abstract
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.
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Hastings, A.: Age-dependent predation is not a simple process, I. Continuous time models. Theor. Popul. Biol. 23, 47–62 (1983)
Hastings, A.: Delay in recruitment at different trophic levels, effects on stability. J. Math. Biol. 21, 35–44 (1984)
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)
Hui, J., Zhu, D.: Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos Solitons Fractals 29, 233–251 (2006)
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)
Wei, C., Chen, L.: Eco-epidemiology model with age structure and prey-dependent consumption for pest management. Appl. Math. Model. 33, 4354–4363 (2009)
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)
Song, X., Cui, J.: The stage-structured predator-prey system with delay and harvesting. Appl. Anal. 81, 1127–1142 (2002)
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)
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)
Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139–153 (1990)
Zhang, H., Jiao, J., Chen, L.: Pest management through continuous and impulsive control strategies. Biosystems 90, 350–361 (2007)
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)
Liu, B., Chen, L.: The periodic competing Lotka–Volterra model with impulsive effect. IMA J. Math. Med. Biol. 21, 129–145 (2004)
Roberts, M.G., Kao, R.R.: The dynamics of an infectious disease in a population with birth pulse. Math. Biosci. 149, 23–36 (1998)
Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009)
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)
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)
Yan, J.: Stability for impulsive delay differential equations. Nonlinear Anal. 63, 66–80 (2005)
Leonid, B., Elena, B.: Linearized oscillation theory for a nonlinear delay impulsive equation. J. Comput. Appl. Math. 161, 477–495 (2003)
Liu, X., Ballinger, G.: Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal. 53, 1041–1062 (2003)
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)
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)
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)
Bainov, D., Simeonov, P.: System with Impulsive Effect: Stability, Theory and Applications. Wiley, New York (1989)
Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego, California (1993)
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The work of X. Meng was supported by Sweden postdoctoral fellowships by the Wenner-Gren Foundations, the work of T. Zhang was supported by Soft Science Research Program of Shandong Province of China (2009RKB153) and Qingdao Science and Technology Development Program (KZJ-46).
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Zhang, T., Meng, X. & Song, Y. The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments. Nonlinear Dyn 64, 1–12 (2011). https://doi.org/10.1007/s11071-010-9840-1
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DOI: https://doi.org/10.1007/s11071-010-9840-1