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The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators. (English) Zbl 1184.92059
Summary: We introduce a mutual interference age structured predator-prey (natural enemy-pest) model with constant maturation time delay for the prey, and then propose a pest management strategy by constant periodic releasing for the predator. We show that there exists a global attractive pest-eradication periodic solution when the periodic releasing amounts μ 1 and μ 2 are lager than some critical value. Further, to obtain a more effective pest control strategy, we give conditions (involving the estimate of μ 1 and μ 2 ) under which the model is uniformly permanent and the pest population is under an economic threshold level. We believe that the results will provide reliable tactic basis for the practical pest management.
34K60Qualitative investigation and simulation of models
34K35Functional-differential equations connected with control problems
[1]Hastings, A.: Age-dependent predation is not a simple process. I. continuous time models, Theor. populat. Biol. 23, 347-362 (1983) · Zbl 0507.92016 · doi:10.1016/0040-5809(83)90023-0
[2]Hastings, A.: Delay in recruitment at different trophic levels, effects on stability, J. math. Biol 21, 35-44 (1984) · Zbl 0547.92014 · doi:10.1007/BF00275221
[3]Bainov, D.; Simeonv, P.: System with impulsive effect: stability, theory and applications, (1989) · Zbl 0683.34032
[4]Lakshmikantham, V.; Bainov, D.; Simeonov, P.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[5]Aiello, W. G.; Freedman, H. I.: A time delay model of single-species growth with stage structure, Math. biosci. 101, 139-152 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[6]Wang, W.; Freedman, Z. I.; Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. math. 52, 855-869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048
[7]Song, X. Y.; Chen, L. S.: Optimal harvesting and stability for a two competitive system with stage structure, Math. biosci. 170, 173-186 (2001) · Zbl 1028.34049 · doi:10.1016/S0025-5564(00)00068-7
[8]Hui, J.; Chen, L. S.: Impulsive vaccination of SIR epidemic models with nonlinear incidence rates, Discr. cont. Dyn. syst. B 3, 595-606 (2004) · Zbl 1100.92040 · doi:10.3934/dcdsb.2004.4.595
[9]Liu, B.; Chen, L. S.; Zhang, Y. J.: The dynamics of a prey-dependent consumption model concerning impulsive control strategy, Appl. math. Comput. 169, 305-320 (2005) · Zbl 1074.92042 · doi:10.1016/j.amc.2004.09.053
[10]Liu, X. N.; Chen, L. S.: Complex dynamics of Holling type II Lotka – Volterra predator – prey system with impulsive perturbations on the predator, Chaos soliton fract. 16, 311-320 (2003) · Zbl 1085.34529 · doi:10.1016/S0960-0779(02)00408-3
[11]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[12]Tang, S. Y.; Chen, L. S.: Density-dependent birth rate, birth pulse and their population dynamics consequences, J. math. Biol. 44, 185-199 (2002) · Zbl 0990.92033 · doi:10.1007/s002850100121
[13]Zhang, S. W.; Chen, L. S.: A study of predator – prey models with the beddington – daanglis functional response and impulsive effect, Chaos soliton fract. 27, 237-248 (2006) · Zbl 1102.34032 · doi:10.1016/j.chaos.2005.03.039
[14]Hui, J.; Zhu, D.: Dynamics complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos soliton fract. 29, 233-251 (2006) · Zbl 1095.92067 · doi:10.1016/j.chaos.2005.08.025
[15]Liu, B.; Zhang, Y.; Sun, L.: The dynamical behaviors of a Lotka – Volterra predator model concerning integrated pest management, Nonlinear anal. RWA 6, 227-243 (2005) · Zbl 1082.34039 · doi:10.1016/j.nonrwa.2004.08.001
[16]Gao, S. J.; Chen, L. S.: Pulse vaccination strategy in a delayed SIR epidemic model with vertical transmission, Discr. cont. Dyn. syst. Ser. B 7, 77-86 (2007) · Zbl 1191.34062
[17]Meng, X. Z.; Jiao, J. J.; Chen, L. S.: Two profitless delays for an SEIRS epidemic disease model with vertical transmission and pulse vaccination, Chaos soliton fract. 40, 2114-2125 (2009) · Zbl 1198.34185 · doi:10.1016/j.chaos.2007.09.096
[18]Freedman, H. I.; Agarwal, M. J.; Devi, S.: Analysis of stability and persistence in a ratio-dependent predator – prey resource model, Int. J. Biomath. 1, 107-118 (2009)
[19]Meng, X. Z.; Chen, L. S.: The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Appl. math. Comput. 197, 582-597 (2008) · Zbl 1131.92056 · doi:10.1016/j.amc.2007.07.083
[20]Freedman, H. J.: Graphical stability, enrichment, and pest control by a natural enemy, Math. biosci. 31, 207-225 (1976) · Zbl 0373.92023 · doi:10.1016/0025-5564(76)90080-8
[21]Ferron, P.: Pest control using the fungi beauveria and metarhizinm, Microbial control in pests and plant diseases (1981)
[22]Van Lenteren, J. C.: Measures of success in biological control of anthropoids by augmentation of natural enemies, Measures of success in biological control (2000)
[23]Caltagirone, L. E.; Doutt, R. L.: The history of the vedalia beetle importation to California and its impact on the development of biological control, Ann. rev. Entomol. 34, 1-16 (1989)
[24]Bainov, D.; Simeonov, P.: Impulsive differential equations: periodic solutions and applications, (1993)
[25]Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003