Abstract and Applied Analysis

A Predator-Prey Model with Functional Response and Stage Structure for Prey

Xiao-Ke Sun, Hai-Feng Huo, and Xiao-Bing Zhang

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Abstract

A predator-prey system with Holling type II functional response and stage structure for prey is presented. The local and global stability are studied by analyzing the associated characteristic transcendental equation and using comparison theorem. The existence of a Hopf bifurcation at the positive equilibrium is also studied. Some numerical simulations are also given to illustrate our results.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 628103, 19 pages.

Dates
First available in Project Euclid: 1 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364845174

Digital Object Identifier
doi:10.1155/2012/628103

Mathematical Reviews number (MathSciNet)
MR2889095

Zentralblatt MATH identifier
1239.34100

Citation

Sun, Xiao-Ke; Huo, Hai-Feng; Zhang, Xiao-Bing. A Predator-Prey Model with Functional Response and Stage Structure for Prey. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 628103, 19 pages. doi:10.1155/2012/628103. https://projecteuclid.org/euclid.aaa/1364845174


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References

  • E. Beretta and Y. Kuang, “Geometric stability switch criteria in delay differential systems with delay dependent parameters,” SIAM Journal on Mathematical Analysis, vol. 33, no. 5, pp. 1144–1165, 2002.
  • Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993.
  • X.-Y. Song and L.-S. Chen, “Optimal harvesting and stability for a predator-prey system with stage structure,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 3, pp. 423–430, 2002.
  • W. Wang and L. Chen, “A predator-prey system with stage-structure for predator,” Computers & Mathematics with Applications, vol. 33, no. 8, pp. 83–91, 1997.
  • S. A. Gourley and Y. Kuang, “A stage structured predator-prey model and its dependence on maturation delay and death rate,” Journal of Mathematical Biology, vol. 49, no. 2, pp. 188–200, 2004.
  • S. Liu and J. Zhang, “Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 446–460, 2008.
  • L. Ou, G. Luo, Y. Jiang, and Y. Li, “The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay,” Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 534–548, 2003.
  • W. G. Aiello, H. I. Freedman, and J. Wu, “Analysis of a model representing stage-structured population growth with state-dependent time delay,” SIAM Journal on Applied Mathematics, vol. 52, no. 3, pp. 855–869, 1992.
  • F. Y. Wang and G. P. Pang, “The global stability of a delayed predator-prey system with two stage-structure,” Chaos, Solitons & Fractals, vol. 10, pp. 1016–1023, 2007.
  • S. Gao, L. Chen, and L. Sun, “Optimal pulse fishing policy in stage-structured models with birth pulses,” Chaos, Solitons & Fractals, vol. 25, no. 5, pp. 1209–1219, 2005.
  • X. Y. Li and W. Wang, “A discrete epidemic model with stage structure,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 947–958, 2005.
  • X. K. Sun and H. F. Huo, “Permanence of a Holling type II predator-prey system with stagestructure,” in Proceedings of the 6th Conference of Biomathematics, vol. 2, pp. 598–602, Advanced Biomedical, Tai'an, China, July 2008.
  • W. Wang, P. Fergola, S. Lombardo, and G. Mulone, “Mathematical models of innovation diffusion with stage structure,” Applied Mathematical Modelling, vol. 30, no. 1, pp. 129–146, 2006.
  • S. Y. Tang and L. S. Chen, “Multiple attractors in stage-structured population models with birth pulses,” Bulletin of Math Biology, vol. 65, pp. 479–495, 2003.
  • Y. Xiao, D. Cheng, and S. Tang, “Dynamic complexities in predator-prey ecosystem models with age-structure for predator,” Chaos, Solitons & Fractals, vol. 14, no. 9, pp. 1403–1411, 2002.
  • X. Song, L. Cai, and A. U. Neumann, “Ratio-dependent predator-prey system with stage structure for prey,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 3, pp. 747–758, 2004.
  • F. Chen, “Permanence of periodic Holling type predator-prey system with stage structure for prey,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1849–1860, 2006.
  • F. Chen and M. You, “Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 207–221, 2008.
  • J. Cui and Y. Takeuchi, “A predator-prey system with a stage structure for the prey,” Mathematical and Computer Modelling, vol. 44, no. 11-12, pp. 1126–1132, 2006.
  • C.-Y. Huang, M. Zhao, and L.-C. Zhao, “Permanence of periodic predator-prey system with two predators and stage structure for prey,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 503–514, 2010.
  • H. Zhang, L. Chen, and R. Zhu, “Permanence and extinction of a periodic predator-prey delay system with functional response and stage structure for prey,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 931–944, 2007.
  • X.-K. Sun, H.-F. Huo, and H. Xiang, “Bifurcation and stability analysis in predator-prey model with a stage-structure for predator,” Nonlinear Dynamics, vol. 58, no. 3, pp. 497–513, 2009.