## Abstract and Applied Analysis

### Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

#### Abstract

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delays ${\tau }_{1}\ne {\tau }_{2}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 835310, 16 pages.

Dates
First available in Project Euclid: 3 October 2014

https://projecteuclid.org/euclid.aaa/1412360638

Digital Object Identifier
doi:10.1155/2014/835310

Mathematical Reviews number (MathSciNet)
MR3212451

Zentralblatt MATH identifier
07023162

#### Citation

Dai, Yunxian; Lin, Yiping; Zhao, Huitao. Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 835310, 16 pages. doi:10.1155/2014/835310. https://projecteuclid.org/euclid.aaa/1412360638

#### References

• R. Xu and M. A. J. Chaplain, “Persistence and global stability in a delayed predator-prey system with Michaelis-Menten type functional response,” Applied Mathematics and Computation, vol. 130, no. 2-3, pp. 441–455, 2002.
• Y. L. Song, Y. H. Peng, and J. J. Wei, “Bifurcations for a predator-prey system with two delays,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 466–479, 2008.
• G.-P. Hu, W.-T. Li, and X.-P. Yan, “Hopf bifurcations in a predator-prey system with multiple delays,” Chaos, Solitons & Fractals, vol. 42, no. 2, pp. 1273–1285, 2009.
• M. Liao, X. Tang, and C. Xu, “Bifurcation analysis for a three-species predator-prey system with two delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 183–194, 2012.
• C. Xu and P. Li, “Dynamical analysis in a delayed predator-prey model with two delays,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 652947, 22 pages, 2012.
• J.-F. Zhang, “Stability and bifurcation periodic solutions in a Lotka-Volterra competition system with multiple delays,” Nonlinear Dynamics, vol. 70, no. 1, pp. 849–860, 2012.
• J. Xia, Z. Yu, and R. Yuan, “Stability and Hopf bifurcation in a symmetric Lotka-Volterra predator-prey system with delays,” Electronic Journal of Differential Equations, vol. 2013, no. 9, pp. 1–16, 2013.
• H. Wan and J. Cui, “A malaria model with two delays,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 601265, 8 pages, 2013.
• M. Zhao, “Hopf bifurcation analysis for a semiratio-dependent predator-prey system with two delays,” Abstract and Applied Analysis, vol. 2013, Article ID 495072, 13 pages, 2013.
• S. Feyissa and S. Banerjee, “Delay-induced oscillatory dynamics in humoral mediated immune response with two time delays,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 35–52, 2013.
• G. Zhang, Y. Shen, and B. Chen, “Hopf bifurcation of a predator-prey system with predator harvesting and two delays,” Nonlinear Dynamics, vol. 73, no. 4, pp. 2119–2131, 2013.
• X.-Y. Meng, H.-F. Huo, and H. Xiang, “Hopf bifurcation in a three-species system with delays,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 635–661, 2011.
• J.-F. Zhang, “Global existence of bifurcated periodic solutions in a commensalism model with delays,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11688–11699, 2012.
• Y. Wang, W. Jiang, and H. Wang, “Stability and global Hopf bifurcation in toxic phytoplankton-zooplankton model with delay and selective harvesting,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 881–896, 2013.
• Y. Ma, “Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 370–375, 2012.
• X.-Y. Meng, H.-F. Huo, and X.-B. Zhang, “Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4335–4348, 2011.
• Y. Song, J. Wei, and M. Han, “Local and global Hopf bifurcation in a delayed hematopoiesis model,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 11, pp. 3909–3919, 2004.
• Y. Song and J. Wei, “Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 1–21, 2005.
• J. Wei and M. Y. Li, “Global existence of periodic solutions in a tri-neuron network model with delays,” Physica D, vol. 198, no. 1-2, pp. 106–119, 2004.
• T. Zhao, Y. Kuang, and H. L. Smith, “Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 28, no. 8, pp. 1373–1394, 1997.
• S. J. Gao, L. S. Chen, and Z. D. Teng, “Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 721–729, 2008.
• J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.
• B. Hassard, D. Kazarinoff, and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.
• J. H. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799–4838, 1998.
• V. Hutson, “The existence of an equilibrium for permanent systems,” The Rocky Mountain Journal of Mathematics, vol. 20, no. 4, pp. 1033–1040, 1990.
• S. G. Ruan and J. J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 863–874, 2003. \endinput