Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission. (English) Zbl 1234.92059
Summary: According to biological and chemical control strategies for pest control, we investigate an \(SI\) model for pest management, concerning periodic spraying of microbial pesticides and releasing infected pests at different fixed moments. By using Floquet and comparison theorems, we prove that the pest-extinction periodic solution is globally asymptotically stable when the impulsive period \(T\) is less than the critical value \(T _{\max }\). Otherwise, the system can be permanent. Our results provide reliable tactic basis for practical pest management.
MSC:
| 92D30 | Epidemiology |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 92D40 | Ecology |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
References:
| [1] | Anderson, R., May, R.: Infectious Diseases of Human: Dynamics and Control. Oxford University Press, Oxford (1991) |
| [2] | Bainov, D.D., Simeonov, P.S.: System with Impulse Effect, Theory and Applications. Ellis Harwood Series in Mathematics and its Applications. Ellis Harwood, Chichester (1993) · Zbl 0815.34001 |
| [3] | Barclay, H.J.: Models for pest control using predator release, habitat management and pesticide release in combination. J. Appl. Ecol. 19, 337–348 (1982) · doi:10.2307/2403471 |
| [4] | Bean, S.: Management and Analysis of Biological Populations. Elsevier, Amsterdam (1980) |
| [5] | Grasman, J., Van Herwaarden, O.A., Hemerik, L., Van Lenteren, J.C.: A two-component model of host–parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 169, 207–216 (2001) · Zbl 0966.92026 · doi:10.1016/S0025-5564(00)00051-1 |
| [6] | Georgescu, P., Moroşanu, G.: Pest regulation by means of impulsive controls. Appl. Math. Comput. 190, 790–803 (2007) · Zbl 1117.93006 · doi:10.1016/j.amc.2007.01.079 |
| [7] | Huffaker, C.B.: New Technology of Pest Control. Wiley, New York (1980) |
| [8] | Hethcote, H.: The mathematics of infectious disease. SIAM Rev. 42, 599–653 (2002) · Zbl 0993.92033 · doi:10.1137/S0036144500371907 |
| [9] | Jiao, J.J., Chen, L.S.: A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators. Appl. Math. Comput. 189, 1448–1458 (2007) · Zbl 1117.92053 · doi:10.1016/j.amc.2006.12.043 |
| [10] | Jiao, J.J., Chen, L.S., Luo, G.L.: An appropriate pest management SI model with biological and chemical control concern. Appl. Math. Comput. 196(1), 285–293 (2008) · Zbl 1130.92046 · doi:10.1016/j.amc.2007.05.072 |
| [11] | Liu, W.M., Levin, S.A., Lwasa, Y.: Influence of nonlinear rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956 |
| [12] | Liu, W.M., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359–380 (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162 |
| [13] | Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. Singapore, World Scientific (1989) · Zbl 0719.34002 |
| [14] | Pang, G.P., Chen, L.S.: Dynamic analysis of a pest-epidemic model with impulsive control. Math. Comput. Simul. 79(1), 72–84 (2008) · Zbl 1144.92037 · doi:10.1016/j.matcom.2007.10.002 |
| [15] | Tang, S.Y., Xiao, Y.N., Chen, L.S., Cheke, R.A.: Integrated pest management models and their dynamical behavior. Bull. Math. Biol. 67, 115–135 (2005) · Zbl 1334.91058 · doi:10.1016/j.bulm.2004.06.005 |
| [16] | VanLenteren, J.C.: Integrated pest management in protected crops. In: Dent, D. (ed.) Integrated Pest Management, pp. 311–320. Chapman and Hall, London (1995) |
| [17] | Wang, X., Song, X.Y.: Mathematical models for the control of a pest population by infected pest. Comput. Math. Appl. 56, 266–278 (2008) · Zbl 1145.92339 · doi:10.1016/j.camwa.2007.12.015 |
| [18] | Wang, X., Song, X.Y.: Analysis of an impulsive pest management SEI model with nonlinear incidence rate. Comput. Appl. Math. 29(1), 1–17 (2010) · Zbl 1184.92045 |
| [19] | Zhang, X.A., Chen, L.S.: The periodic solution of a class of epidemic models. Comput. Math. Appl. 38, 61–71 (1999) · Zbl 0939.92031 · doi:10.1016/S0898-1221(99)00206-0 |
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