Journal of Applied Mathematics

Periodic Solutions and Homoclinic Bifurcations of Two Predator-Prey Systems with Nonmonotonic Functional Response and Impulsive Harvesting

Mingzhan Huang and Xinyu Song

Full-text: Open access

Abstract

Two predator-prey models with nonmonotonic functional response and state-dependent impulsive harvesting are formulated and analyzed. By using the geometry theory of semicontinuous dynamic system, we obtain the existence, uniqueness, and stability of the periodic solution and analyse the dynamic phenomenon of homoclinic bifurcation of the first system by choosing the harvesting rate β as control parameter. Besides, we also study the homoclinic bifurcation of the second system about parameter δ on the basis of the theory of rotated vector field. Finally, numerical simulations are presented to illustrate the results.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 803764, 11 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305639

Digital Object Identifier
doi:10.1155/2014/803764

Mathematical Reviews number (MathSciNet)
MR3191137

Zentralblatt MATH identifier
07010756

Citation

Huang, Mingzhan; Song, Xinyu. Periodic Solutions and Homoclinic Bifurcations of Two Predator-Prey Systems with Nonmonotonic Functional Response and Impulsive Harvesting. J. Appl. Math. 2014 (2014), Article ID 803764, 11 pages. doi:10.1155/2014/803764. https://projecteuclid.org/euclid.jam/1425305639


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References

  • J. S. Tener, Muskoxen, Queens Printer, Ottawa, Canada, 1965.
  • J. C. Holmes and W. M. Bethel, “Modification of intermediate host behavior by parasites,” Zoological Journal of the Linnean Society, vol. 51, pp. 123–149, 1972.
  • G. S. K. Wolkowicz, “Bifurcation analysis of a predator-prey system involving group defence,” SIAM Journal on Applied Mathematics, vol. 48, no. 3, pp. 592–606, 1988.
  • S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2001.
  • 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.
  • K. Mischaikow and G. Wolkowicz, “A predator-prey system involving group defense: a connection matrix approach,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 11, pp. 955–969, 1990.
  • J. B. Collings, “The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model,” Journal of Mathematical Biology, vol. 36, no. 2, pp. 149–168, 1997.
  • V. H. Edwards, “The influence of high substrate concentrations on microbial kinetics,” Biotechnology and Bioengineering, vol. 12, no. 5, pp. 679–712, 1970.
  • L. S. Chen, “Pest control and geometric theory of Semi-continuous dynamical system,” Journal of Beihua University, vol. 12, pp. 1–9, 2011.
  • P. Yongzhen, L. Changguo, and C. Lansun, “Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay,” Mathematics and Computers in Simulation, vol. 79, no. 10, pp. 2994–3008, 2009.
  • Z. Liu and R. 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.
  • K. Negi and S. Gakkhar, “Dynamics in a Beddington-DeAngelis prey-predator system with impulsive harvesting,” Ecological Modelling, vol. 206, no. 3-4, pp. 421–430, 2007.
  • 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.
  • 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.
  • 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.
  • M. Huang, S. Liu, X. Song, and L. Chen, “Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 815–826, 2013.
  • C. Dai, M. Zhao, and L. Chen, “Homoclinic bifurcation in semi-continuous dynamic systems,” International Journal of Biomathematics, vol. 5, no. 6, Article ID 1250059, 2012.
  • Y. Q. Ye, Limit Cycle Theory, Shanghai Science and Technology Press, Shanghai, China, 1984. \endinput