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Dynamic analysis of an impulsive predator-prey model with disease in prey and Ivlev-type functional response. (English) Zbl 1246.92028
Summary: A predator-prey model with disease in the prey, Ivlev-type functional response, and impulsive effects is proposed. By using Floquet theory and small amplitude perturbations skill, sufficient conditions for existence and global stability of susceptible pest-eradication periodic solutions are obtained. By the impulsive comparison theorem, conditions ensuring the permanence of the system are established. Examples and simulations are given to show the complex dynamics of the key parameters.

37N25Dynamical systems in biology
34A37Differential equations with impulses
65C20Models (numerical methods)
Full Text: DOI
[1] H. K. Baek, S. D. Kim, and P. Kim, “Permanence and stability of an Ivlev-type predator-prey system with impulsive control strategies,” Mathematical and Computer Modelling, vol. 50, no. 9-10, pp. 1385-1393, 2009. · Zbl 1185.34067 · doi:10.1016/j.mcm.2009.07.007
[2] J. J. Jiao, X. S. Yang, S. Cai, and L. S. Chen, “Dynamical analysis of a delayed predator-prey model with impulsive diffusion between two patches,” Mathematics and Computers in Simulation, vol. 80, no. 3, pp. 522-532, 2009. · Zbl 1190.34107 · doi:10.1016/j.matcom.2009.07.008
[3] K. Y. Liu, X. Z. Meng, and L. S. Chen, “A new stage structured predator-prey Gomportz model with time delay and impulsive perturbations on the prey,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 705-719, 2008. · Zbl 1131.92064 · doi:10.1016/j.amc.2007.07.020
[4] M. X. Liu, Z. Jin, and M. Haque, “An impulsive predator-prey model with communicable disease in the prey species only,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3098-3111, 2009. · Zbl 1162.92043 · doi:10.1016/j.nonrwa.2008.10.010
[5] X.-Z. Meng, L.-S. Chen, and Q.-X. Li, “The dynamics of an impulsive delay predator-prey model with variable coefficients,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 361-374, 2008. · Zbl 1133.92029 · doi:10.1016/j.amc.2007.08.075
[6] Y. F. Shao and B. X. Dai, “The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 3567-3576, 2010. · Zbl 1218.34099 · doi:10.1016/j.nonrwa.2010.01.004
[7] Y. F. Shao, B. X. Dai, and Z. G. Luo, “The dynamics of an impulsive one-prey multi-predators system with delay and Holling-type II functional response,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2414-2424, 2010. · Zbl 1200.92044 · doi:10.1016/j.amc.2010.07.042
[8] Y. F. Shao, “Analysis of a delayed predator-prey system with impulsive diffusion between two patches,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 120-127, 2010. · Zbl 1201.34086 · doi:10.1016/j.mcm.2010.01.021
[9] R. Q. Shi, X. Jiang, and L. S. Chen, “A predator-prey model with disease in the prey and two impulses for integrated pest management,” Applied Mathematical Modelling, vol. 33, no. 5, pp. 2248-2256, 2009. · Zbl 1185.34015 · doi:10.1016/j.apm.2008.06.001
[10] X. Y. Song and Z. Y. Xiang, “The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects,” Journal of Theoretical Biology, vol. 242, no. 3, pp. 683-698, 2006. · doi:10.1016/j.jtbi.2006.05.002
[11] H. Su, B. X. Dai, Y. M. Chen, and K. W. Li, “Dynamic complexities of a predator-prey model with generalized Holling type III functional response and impulsive effects,” Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1715-1725, 2008. · Zbl 1152.34309 · doi:10.1016/j.camwa.2008.04.001
[12] W. M. Wang, H. Wang, and Z. Li, “The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1772-1785, 2007. · Zbl 1195.92066 · doi:10.1016/j.chaos.2005.12.025
[13] X. Q. Wang, W. M. Wang, and X. L. Lin, “Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2392-2404, 2009. · Zbl 1198.37135 · doi:10.1016/j.chaos.2007.10.035
[14] W. M. Wang, X. Q. Wang, and Y. Z. Lin, “Complicated dynamics of a predator-prey system with Watt-type functional response and impulsive control strategy,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1427-1441, 2008. · Zbl 1142.34342 · doi:10.1016/j.chaos.2006.10.032
[15] M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls,” Advances in Difference Equations, vol. 2010, Article ID 281612, 42 pages, 2010. · Zbl 1219.34104 · doi:10.1155/2010/281612 · eudml:226182
[16] L. S. Chen, X. Y. Song, and Z. Lu, Mathematical Models and Methods in Ecology, Sichuan Technology Publishing Company, Chengdu, China, 2003.
[17] J. Cost, Comparing predator-prey models qualitatively and quantitatively with ecological time-series data [Ph.D. thesis], Institute National Agronomique, Paris, France, 1998.
[18] Z. E. Ma, Mathematical Modelling and Study of Species Ecology, Anhui Education Publishing Company, Hefei, China, 1996.
[19] J.-W. Feng and S.-H. Chen, “Global asymptotic behavior for the competing predators of the Ivlev types,” Mathematica Applicata, vol. 13, no. 4, pp. 85-88, 2000. · Zbl 1037.92029
[20] L. Ling and W. M. Wang, “Dynamics of a Ivlev-type predator-prey system with constant rate harvesting,” Chaos, Solitons and Fractals, vol. 41, no. 4, pp. 2139-2153, 2009. · Zbl 1198.34061 · doi:10.1016/j.chaos.2008.08.024
[21] X. J. Wu and W. T. Huang, “Dynamics analysis of a one-prey multi-predator impulsive system with Ivlev-type functional,” Ecological Modelling, vol. 220, pp. 774-783, 2009.
[22] H. D. Burges and N. Hussey, Microbial Control of Insect and Mites, Academic Press, New York, NY, USA, 1971.
[23] L. A. Falcon, “Problem associated with the use of arthropod viruses in pest control,” Annual Review of Entomology, vol. 21, pp. 305-324, 1976.
[24] P. Georgescu and H. Zhang, “An impulsively controlled predator-pest model with disease in the pest,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 270-287, 2010. · Zbl 1192.34057 · doi:10.1016/j.nonrwa.2008.10.060
[25] B. Liu, Y. Zhang, and L. Chen, “The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management,” Nonlinear Analysis. Real World Applications, vol. 6, no. 2, pp. 227-243, 2005. · Zbl 1082.34039 · doi:10.1016/j.nonrwa.2004.08.001
[26] R. L. Doutt and P. DeBach, Biological Control of Insect Pests and Weeds, Reinhold Publishing Corporation, New York, NY, USA, 1964.
[27] D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solution and Applications, Longman scientific Technical, Harlow, UK, 1993. · Zbl 0815.34001
[28] V. Laksbmikanham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publisher, Singapore, 1989.
[29] M. De la Sen, A. Ibeas, S. Alonso-Quesada, and R. Nistal, “On the equilibrium points, boundedness and positivity of a sveirs epidemic model under constant regular constrained vaccination,” Informatica, vol. 22, no. 3, pp. 339-370, 2011. · Zbl 1263.92029
[30] H.-F. Huo and Z.-P. Ma, “Dynamics of a delayed epidemic model with non-monotonic incidence rate,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 459-468, 2010. · Zbl 1221.34197 · doi:10.1016/j.cnsns.2009.04.018
[31] I. A. Dzhalladova, J. Ba\vstinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,” Abstract and Applied Analysis, vol. 2011, Article ID 920412, 14 pages, 2011. · Zbl 1217.93150 · doi:10.1155/2011/920412