The dynamics of pest control pollution model with age structure and time delay. (English) Zbl 1191.92074

Summary: By using a pollution model and impulsive delay differential equations, we investigate the dynamics of a pest control model with age structure for pest by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moment. We assume only the pests are affected by pesticides. We show that there exists a global attractive pest-extinction periodic solution when the periodic natural enemies release amount \(\mu _{1}\) and pesticide input amount \(\mu _{2}\) are larger than some critical value. Further, a condition for the permanence of the system is also given. By numerical analyses, we also show that constant maturation time delay, pulse pesticide input and pulse releasing of the natural enemies can bring obvious effects on the dynamics of system. We believe that the results will provide reliable tactic basis for the practical pest management.


92D40 Ecology
34K45 Functional-differential equations with impulses
34K13 Periodic solutions to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI


[1] Liu, B.; Teng, Z.D.; Chen, L.S., The effect of impulsive spraying pesticide on stage-structured population models with birth pulse, J. biol. syst., 13, 31-44, (2005) · Zbl 1125.92320
[2] Lu, Z.H.; Dai, B.X.; Chen, L.S., Impulsive control strategies in biological control of pesticide, Theor. popul. biol., 64, 39-47, (2003) · Zbl 1100.92071
[3] 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
[4] Bainov, D.; Simeonov, P., Impulsive differential equations: periodic solution and applications, Pitman monographs surveys in pure appl. math., 66, (1993) · Zbl 0815.34001
[5] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[6] Gao, S.J.; Chen, L.S.; Nieto, J.J.; Torres, A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045, (2006)
[7] Hui, J.; Zhu, D., Dynamics complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos solitons fract., 29, 233-251, (2006) · Zbl 1095.92067
[8] Liu, B.; Chen, L.S.; Zhang, Y.J., The effects of impulsive toxicant input on a population in a polluted environment, J. biol. syst., 11, 3, 265-274, (2003) · Zbl 1041.92044
[9] Liu, B.; Zhang, Y.J.; Chen, L.S., Dynamic complexities of a Holling I predator – prey model concerning periodic biological and chemical control, Chaos solitons fract., 22, 123-134, (2004) · Zbl 1058.92047
[10] Liu, B.; Chen, L.S., Dynamic complexities in lotka – volterra predator – prey system concerning impulsive control strategy, Int. J. bifur. chaos appl. sci. eng., 15, 517-531, (2005) · Zbl 1080.34026
[11] Liu, B.; Teng, Z.D.; Liu, W.B., Dynamic behaviors of the periodic lotka – volterra competing system with impulsive perturbations, Chaos solitons fract., 31, 356-370, (2007) · Zbl 1145.34029
[12] Gao, S.J.; Chen, L.S., Pulse vaccination strategy in a delayed SIR epidemic model with vertical transmission, Discrete contin. dyn. syst. ser. B, 7, 77-86, (2007) · Zbl 1191.34062
[13] 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
[14] Zhao, Z.; Chen, L.S.; Song, X.Y., Extinction and permanence of chemostat model with pulsed input in a polluted environment, Commun. nonlinear sci. numer. simul., 14, 1737-1745, (2009) · Zbl 1221.37209
[15] Zhao, Z.; Chen, L.S.; Song, X.Y., Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate, Math. comput. simul., 79, 500C510, (2008)
[16] Meng, X.Z.; Chen, L.S., Permanence and global stability in an impulsive lotka – volterra N-species competitive system with both discrete delays and continuous delays, Int. J. biomath., 1, 179-196, (2008) · Zbl 1155.92356
[17] Sun, S.; Chen, L.S., Permanence and complexity of the eco-epidemiological model with impulsive perturbation, Int. J. biomath., 1, 121-132, (2008) · Zbl 1166.92039
[18] Tang, S.Y.; Chen, L.S., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 44, 185-199, (2002) · Zbl 0990.92033
[19] Zhang, H.; Georgescu, P.; Chen, L.S., An impulsive predator – prey system with beddington – deangelis functional response and time delay, Int. J. biomath., 1, 1-17, (2008) · Zbl 1155.92045
[20] Nie, L.F.; Teng, Z.D.; Hu, L.; Peng, J.G., Existence and stability of periodic solution of a predator – prey model with state dependent impulsive effects, Math. comput. simul., 79, 2122-2134, (2009) · Zbl 1185.34123
[21] Meng, X.Z.; Chen, L.S.; Chen, H.D., Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. math. comput., 186, 516-529, (2008) · Zbl 1111.92049
[22] Meng, X.Z.; Chen, L.S., Permanence and global stability in an impulsive lotka – volterra N-species competitive system with both discrete delays and continuous delays, Int. J. biomath., 1, 2, 179-196, (2008) · Zbl 1155.92356
[23] Jin, Z.; Haque, M.; LIU, Q.X., Pulse vaccination in the periodic infection rate SIR epidemic model, Int. J. biomath., 1, 4, 409-432, (2008) · Zbl 1156.92028
[24] Shi, Ruiqing; Chen, L.S., Staged-structured lotka – volterra predator – prey models for pest management, Appl. math. comput., 203, 258-265, (2008) · Zbl 1152.92029
[25] Liu, Z.J.; Chen, L.S., Periodic solution of a two-species competitive system with toxicant and birth pulse, Chaos solitons fract., 32, 1703-1712, (2007) · Zbl 1137.34017
[26] Jiao, J.J.; Chen, L.S., Global attractivity of a stage-structure variable coefficients predator – prey system with time delay and impulsive perturbations on predators, Int. J. biomath., 1, 2, 197-208, (2008) · Zbl 1155.92355
[27] Song, X.Y.; Xiang, Z.Y., The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects, J. theor. biol., 242, 3, 683-698, (2006)
[28] Song, X.Y.; Hao, M.Y.; Meng, X.Z., A stage-structured predator – prey model with disturbing pulse and time delays, Appl. math. modell., 33, 1, 211-223, (2009) · Zbl 1167.34372
[29] Liu, X.N.; Chen, L.S., Complex dynamics of Holling type II lotka – volterra predator – prey system with impulsive perturbations on the predator, Chaos solitons fract., 16, 311-320, (2003) · Zbl 1085.34529
[30] 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
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