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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 358930, 15 pages
Complex Dynamical Behaviors in a Predator-Prey System with Generalized Group Defense and Impulsive Control Strategy
Department of Mathematics, Qiannan Normal College for Nationalities, Duyun, Guizhou 558000, China
Received 1 November 2012; Accepted 20 May 2013
Academic Editor: Patricia J. Y. Wong
Copyright © 2013 Shunyi Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 1, no. 45, pp. 1–60, 1965.
- H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monograph Textbooks Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
- V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Conn, USA, 1961.
- D. L. DeAngelis, R. A. Goldstein, and R. V. O'Neill, “A Model for trophic interaction,” Ecology, vol. 56, pp. 881–892, 1975.
- J. R. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331–340, 1975.
- D. A. Miller, J. B. Grand, T. F. Fondell, and M. Anthony, “Predator functional response and prey survival: direct and indirect interactions affecting a marked prey population,” Journal of Animal Ecology, vol. 75, no. 1, pp. 101–110, 2006.
- J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, pp. 707–723, 1968.
- H. I. Freedman and S. G. Ruan, “Hopf bifurcation in three-species food chain models with group defense,” Mathematical Biosciences, vol. 111, no. 1, pp. 73–87, 1992.
- H. I. Freedman and G. S. K. Wolkowicz, “Predator-prey systems with group defence: the paradox of enrichment revisited,” Bulletin of Mathematical Biology, vol. 48, no. 5-6, pp. 493–508, 1986.
- H. I. Freedman and H. S. Quan, “Interactions leading to persistence in predator-prey systems with group defence,” Bulletin of Mathematical Biology, vol. 50, no. 5, pp. 517–530, 1988.
- D. M. Xiao and S. G. Ruan, “Codimension two bifurcations in a predator-prey system with group defense,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 11, no. 8, pp. 2123–2131, 2001.
- H. Ddumba, J. Y. T. Mugisha, J. W. Gonsalves, and G. I. H. Kerley, “The role of predator fertility and prey threshold bounds on the global and local dynamics of a predator-prey model with a prey out-flux dilution effect,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9169–9186, 2012.
- S. W. Zhang, F. Y. Wang, and L. S. Chen, “A food chain model with impulsive perturbations and Holling IV functional response,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 855–866, 2005.
- Z. L. Xiong, Y. Xue, and S. Y. Li, “A food chain system with Holling IV functional responses and impulsive effect,” International Journal of Biomathematics, vol. 1, no. 3, pp. 361–375, 2008.
- H. Baek, “A food chain system with Holling type IV functional response and impulsive perturbations,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1152–1163, 2010.
- J. J. Jiao, X. Z. Meng, and L. S. Chen, “A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1448–1458, 2007.
- S. W. Zhang, D. J. Tan, and L. S. Chen, “Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 27, no. 4, pp. 980–990, 2006.
- Z. J. Liu and R. H. Tan, “Impulsive harvesting and stocking in a Monod-Haldane functional response predator-prey system,” Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 454–464, 2007.
- Q. Wang, B. X. Dai, and Y. M. Chen, “Multiple periodic solutions of an impulsive predator-prey model with Holling-type IV functional response,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1829–1836, 2009.
- H. Baek and Y. Do, “Stability for a Holling type IV food chain system with impulsive perturbations,” Kyungpook Mathematical Journal, vol. 48, no. 3, pp. 515–527, 2008.
- C.-Y. Huang, Y.-J. Li, and H.-F. Huo, “The dynamics of a stage-structured predator-prey system with impulsive effect and Holling mass defence,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 87–96, 2012.
- Y. Z. Wang and M. Zhao, “Dynamic analysis of an impulsively controlled predator-prey model with Holling type IV functional response,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 141272, 18 pages, 2012.
- H.-F. Huo and W.-T. Li, “Dynamics of a nonautonomous semiratio-dependent predator-prey system with nonmonotonic functional responses,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 70656, 19 pages, 2006.
- X. P. Li and W. S. Yang, “Permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay,” Abstract and Applied Analysis, vol. 2009, Article ID 960823, 6 pages, 2009.
- S. W. Zhang, L. Z. Dong, and L. S. Chen, “The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 631–643, 2005.
- Y. Z. Pei, C. G. Li, L. S. Chen, and C. Wang, “Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators,” Advances in Complex Systems, vol. 8, no. 4, pp. 483–495, 2005.
- S. Y. Li, Z. L. Xiong, and X. Wang, “The study of a predator-prey system with group defense and impulsive control strategy,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2546–2561, 2010.
- D. D. Bainov and D. D. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman UK, 1993.
- V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific Publishing, Teaneck, NJ, USA, 1989.
- A. Lakmeche and O. Arino, “Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 7, no. 2, pp. 265–287, 2000.