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Dynamic complexity of an Ivlev-type prey-predator system with impulsive state feedback control. (English) Zbl 1256.34032

Summary: The dynamic complexities of an Ivlev-type prey-predator system with impulsive state feedback control are studied analytically and numerically. Using the analogue of the Poincaré criterion, sufficient conditions for the existence and the stability of semitrivial periodic solutions can be obtained. Furthermore, the bifurcation diagrams and phase diagrams are investigated by means of numerical simulations, which illustrate the feasibility of the main results presented here.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34A37 Ordinary differential equations with impulses
34A36 Discontinuous ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. · Zbl 0718.34011
[2] G. Jiang and Q. Lu, “Impulsive state feedback control of a predator-prey model,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 193-207, 2007. · Zbl 1134.49024 · doi:10.1016/j.cam.2005.12.013
[3] J. M. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82-95, 1977. · Zbl 0348.34031 · doi:10.1137/0132006
[4] 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
[5] R. Doutt and P. DeBach, Biological Control of Insect Pests and Weeds, Reinhold, New York, NY, USA, 1964.
[6] J. Grasman, O. A. Van Herwaarden, L. Hemerik, and J. C. Van Lenteren, “A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control,” Mathematical Biosciences, vol. 169, no. 2, pp. 207-216, 2001. · Zbl 0966.92026 · doi:10.1016/S0025-5564(00)00051-1
[7] H. Yu, S. Zhong, R. P. Agarwal, and S. K. Sen, “Three-species food web model with impulsive control strategy and chaos,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 1002-1013, 2011. · Zbl 1221.34039 · doi:10.1016/j.cnsns.2010.05.014
[8] H. Baek, “Dynamic complexities of a three-species Beddington-DeAngelis system with impulsive control strategy,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 23-38, 2010. · Zbl 1194.34087 · doi:10.1007/s10440-008-9378-0
[9] W. Z. Gong, Q. F. Zhang, and X. H. Tang, “Existence of subharmonic solutions for a class of second-order p-Laplacian systems with impulsive effects,” Journal of Applied Mathematics, vol. 2012, Article ID 434938, 18 pages, 2012. · Zbl 1230.34030 · doi:10.1155/2012/434938
[10] X. Liu and L. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 16, no. 2, pp. 311-320, 2003. · Zbl 1085.34529 · doi:10.1016/S0960-0779(02)00408-3
[11] Z. Liu, S. Zhong, C. Yin, and W. Chen, “On the dynamics of an impulsive reaction-diffusion predator-prey system with ratio-dependent functional response,” Acta Applicandae Mathematicae, vol. 115, no. 3, pp. 329-349, 2011. · Zbl 1229.35305 · doi:10.1007/s10440-011-9624-8
[12] H. Yu, S. Zhong, M. Ye, and W. Chen, “Mathematical and dynamic analysis of an ecological model with an impulsive control strategy and distributed time delay,” Mathematical and Computer Modelling, vol. 50, no. 11-12, pp. 1622-1635, 2009. · Zbl 1185.37203 · doi:10.1016/j.mcm.2009.10.008
[13] J. Hui and L. Chen, “A Single species model with impulsive diffusion,” Acta Mathematicae Applicatae Sinica, vol. 21, no. 1, pp. 43-48, 2005. · Zbl 1180.92072 · doi:10.1007/s10255-005-0213-3
[14] H. Yu, S. Zhong, and M. Ye, “Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay,” Mathematics and Computers in Simulation, vol. 80, no. 3, pp. 619-632, 2009. · Zbl 1178.92058 · doi:10.1016/j.matcom.2009.09.013
[15] S. A. Hadley and L. K. Forbes, “Dynamical systems analysis of a five-dimensional trophic food web model in the Southern oceans,” Journal of Applied Mathematics, vol. 2009, Article ID 575047, 17 pages, 2009. · Zbl 1196.37125 · doi:10.1155/2009/575047
[16] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, NY, USA, 1983. · Zbl 0515.34001
[17] L. Zhang and M. Zhao, “Dynamic complexities in a hyperparasitic system with prolonged diapause for host,” Chaos, Solitons & Fractals, vol. 42, no. 2, pp. 1136-1142, 2009. · Zbl 05813506 · doi:10.1016/j.chaos.2009.03.007
[18] I. M. Cabrera, H. L. B. Díaz, and M. Martínez, “Dynamical system and nonlinear regression for estimate host-parasitoid relationship,” Journal of Applied Mathematics, vol. 2010, Article ID 851037, 10 pages, 2010. · Zbl 1211.37103 · doi:10.1155/2010/851037
[19] M. Zhao, L. Zhang, and J. Zhu, “Dynamics of a host-parasitoid model with prolonged diapause for parasitoid,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 455-462, 2011. · Zbl 1221.37208 · doi:10.1016/j.cnsns.2010.03.011
[20] F. D. Chen, “Periodicity in a ratio-dependent predator-prey system with stage structure for predator,” Journal of Applied Mathematics, vol. 2005, no. 2, pp. 153-169, 2005. · Zbl 1103.34060 · doi:10.1155/JAM.2005.153
[21] A. Lakmeche and O. Arino, “Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment,” Dynamics of Continuous, Discrete & Impulsive Systems B, vol. 7, no. 2, pp. 265-287, 2000. · Zbl 1011.34031
[22] S. Tang and L. Chen, “Density-dependent birth rate, birth pulses and their population dynamic consequences,” Journal of Mathematical Biology, vol. 44, no. 2, pp. 185-199, 2002. · Zbl 0990.92033 · doi:10.1007/s002850100121
[23] L. Nie, Z. Teng, L. Hu, and J. Peng, “The dynamics of a Lotka-Volterra predator-prey model with state dependent impulsive harvest for predator,” BioSystems, vol. 98, no. 2, pp. 67-72, 2009. · doi:10.1016/j.biosystems.2009.06.001
[24] G. Zeng, L. Chen, and L. Sun, “Existence of periodic solution of order one of planar impulsive autonomous system,” Journal of Computational and Applied Mathematics, vol. 186, no. 2, pp. 466-481, 2006. · Zbl 1088.34040 · doi:10.1016/j.cam.2005.03.003
[25] G. Jiang, Q. Lu, and L. Qian, “Complex dynamics of a Holling type II prey-predator system with state feedback control,” Chaos, Solitons & Fractals, vol. 31, no. 2, pp. 448-461, 2007. · Zbl 1203.34071 · doi:10.1016/j.chaos.2005.09.077
[26] L. Nie, J. Peng, Z. Teng, and L. Hu, “Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 544-555, 2009. · Zbl 1162.34007 · doi:10.1016/j.cam.2008.05.041
[27] L. Qian, Q. Lu, Q. Meng, and Z. Feng, “Dynamical behaviors of a prey-predator system with impulsive control,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 345-356, 2010. · Zbl 1187.34060 · doi:10.1016/j.jmaa.2009.08.048
[28] S. Tang and L. Chen, “Modelling and analysis of integrated pest management strategy,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 3, pp. 761-770, 2004. · Zbl 1114.92074 · doi:10.3934/dcdsb.2004.4.759
[29] S. Tang and R. A. Cheke, “State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences,” Journal of Mathematical Biology, vol. 50, no. 3, pp. 257-292, 2005. · Zbl 1080.92067 · doi:10.1007/s00285-004-0290-6
[30] V. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Conn, USA, 1961.
[31] R. E. Kooij and A. Zegeling, “A predator-prey model with Ivlev’s functional response,” Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 473-489, 1996. · Zbl 0851.34030 · doi:10.1006/jmaa.1996.0093
[32] J. Sugie, “Two-parameter bifurcation in a predator-prey system of Ivlev type,” Journal of Mathematical Analysis and Applications, vol. 217, no. 2, pp. 349-371, 1998. · Zbl 0894.34025 · doi:10.1006/jmaa.1997.5700
[33] J. Feng and S. Chen, “Global asympotic behavior for the competing predators of the Ivlev types,” Mathematica Applicata, vol. 13, no. 4, pp. 85-88, 2000. · Zbl 1037.92029
[34] V. L. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Conn, USA, 1955.
[35] H. Wang and W. Wang, “The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect,” Chaos, Solitons & Fractals, vol. 38, no. 4, pp. 1168-1176, 2008. · Zbl 1152.34310 · doi:10.1016/j.chaos.2007.02.008
[36] H. Wang, “Dispersal permanence of periodic predator-prey model with Ivlev-type functional response and impulsive effects,” Applied Mathematical Modelling, vol. 34, no. 12, pp. 3713-3725, 2010. · Zbl 1201.34077 · doi:10.1016/j.apm.2010.02.009
[37] P. Simeonov and D. Bainov, “Orbital stability of periodic solutions of autonomous systems with impulse effect,” International Journal of Systems Science, vol. 19, no. 12, pp. 2562-2585, 1988. · Zbl 0669.34044 · doi:10.1080/00207728808547133
[38] S. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley & Sons, New York, NY, USA, 1990. · Zbl 0691.58004
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