Journal of Applied Mathematics

Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge

Lingshu Wang and Guanghui Feng

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Abstract

A ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, when τ = τ 0 . By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 978758, 10 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305624

Digital Object Identifier
doi:10.1155/2014/978758

Mathematical Reviews number (MathSciNet)
MR3191142

Zentralblatt MATH identifier
07010815

Citation

Wang, Lingshu; Feng, Guanghui. Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge. J. Appl. Math. 2014 (2014), Article ID 978758, 10 pages. doi:10.1155/2014/978758. https://projecteuclid.org/euclid.jam/1425305624


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References

  • W. O. Kermack and A. G. Mckendrick, “A contribution on the mathematical theory of epidemics,” Proceedings of the Royal Society A, vol. 115, pp. 700–721, 1927.
  • R. M. Anderso and R. M. May, Intections Disease of Humans Dynamics and Control, Oxford University Press, Oxford, UK, 1991.
  • A. K. Pal and G. P. Samanta, “Stability analysis of an eco-epidemiological model incorporating a prey refuge,” Nonlinear Analysis: Modelling and Control, vol. 15, no. 4, pp. 473–491, 2010.
  • S. Wang, The research of eco-epidemiological of models incorporating prey refuges [Ph.D. thesis], Lanzhou University, 2012.
  • Y. Xiao and L. Chen, “A ratio-dependent predator-prey model with disease in the prey,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 397–414, 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.
  • Y. N. Xiao and L. S. Chen, “Global stability of a predator-prey system with stage structure for the predator,” Acta Mathematica Sinica, vol. 20, no. 1, pp. 63–70, 2004.
  • R. Xu and Z. Ma, “Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1444–1460, 2008.
  • R. Xu and Z. Ma, “Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 669–684, 2008.
  • W. G. Aiello and H. I. Freedman, “A time-delay model of single-species growth with stage structure,” Mathematical Biosciences, vol. 101, no. 2, pp. 139–153, 1990.
  • Y. Kuang and J. W.-H. So, “Analysis of a delayed two-stage population model with space-limited recruitment,” SIAM Journal on Applied Mathematics, vol. 55, no. 6, pp. 1675–1696, 1995.
  • J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.
  • Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, New York, NY, USA, 1993. \endinput