## Journal of Applied Mathematics

### Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

#### Abstract

We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay $\tau$ as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay $\tau$ passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 201936, 10 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808185

Digital Object Identifier
doi:10.1155/2013/201936

Mathematical Reviews number (MathSciNet)
MR3124603

Zentralblatt MATH identifier
06950554

#### Citation

Shi, Ruiqing; Qi, Junmei; Tang, Sanyi. Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays. J. Appl. Math. 2013 (2013), Article ID 201936, 10 pages. doi:10.1155/2013/201936. https://projecteuclid.org/euclid.jam/1394808185

#### References

• A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, Md, USA, 1925.
• V. Volterra, “Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,” Memorie dell'accademia dei Lincei, vol. 2, pp. 31–113, 1926.
• B. Liu, Z. Teng, and L. Chen, “Analysis of a predator-prey modelwith Holling II functional response concerning impulsive control strategy,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 347–362, 2006.
• G. Jiang and Q. Lu, “Impulsive state feedback control of a predator-prey model,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 193–207, 2007.
• S. Zhang, D. Tan, and L. Chen, “Chaotic behavior of a chemostat model with Beddington-DeAngelis functional response and periodically impulsive invasion,” Chaos, Solitons and Fractals, vol. 29, no. 2, pp. 474–482, 2006.
• A. Leung, “Periodic solutions for a prey-predator differential delay equation,” Journal of Differential Equations, vol. 26, no. 3, pp. 391–403, 1977.
• K. Gopalsamy, “Time lags and global stability in two-species competition,” Bulletin of Mathematical Biology, vol. 42, no. 5, pp. 729–737, 1980.
• X. Wen and Z. Wang, “The existence of periodic solutions for some models with delay,” Nonlinear Analysis: Real World Applications, vol. 3, no. 4, pp. 567–581, 2002.
• X. Chen, “Periodicity in a nonlinear discrete predator-prey system with state dependent delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 2, pp. 435–446, 2007.
• X.-P. Yan and C.-H. Zhang, “Hopf bifurcation in a delayed Lokta-Volterra predator-prey system,” Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 114–127, 2008.
• S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159–173, 2001.
• L.-L. Wang and W.-T. Li, “Existence and global stability of positive periodic solutions of a predator-prey system with delays,” Applied Mathematics and Computation, vol. 146, no. 1, pp. 167–185, 2003.
• X.-P. Yan and Y.-D. Chu, “Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 198–210, 2006.
• L. Zhou, Y. Tang, and S. Hussein, “Stability and Hopf bifurcation for a delay competition diffusion system,” Chaos, Solitons & Fractals, vol. 14, no. 8, pp. 1201–1225, 2002.
• Z. Liu and R. Yuan, “Stability and bifurcation in a harvested one-predator–two-prey model with delays,” Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1395–1407, 2006.
• X. Liu and D. Xiao, “Complex dynamic behaviors of a discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 80–94, 2007.
• F. Wang and G. Zeng, “Chaos in a Lotka-Volterra predator-prey system with periodically impulsive ratio-harvesting the prey and time delays,” Chaos, Solitons and Fractals, vol. 32, no. 4, pp. 1499–1512, 2007.
• C. Sun, M. Han, Y. Lin, and Y. Chen, “Global qualitative analysis for a predator-prey system with delay,” Chaos, Solitons & Fractals, vol. 32, no. 4, pp. 1582–1596, 2007.
• C. Çelik, “The stability and Hopf bifurcation for a predator-prey system with time delay,” Chaos, Solitons and Fractals, vol. 37, no. 1, pp. 87–99, 2008.
• C. Çelik, “Hopf bifurcation of a ratio-dependent predator-prey system with time delay,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1474–1484, 2009.
• W. Ma and Y. Takeuchi, “Stability analysis on a predator-prey system with distributed delays,” Journal of Computational and Applied Mathematics, vol. 88, no. 1, pp. 79–94, 1998.
• Z. Teng and M. Rehim, “Persistence in nonautonomous predator-prey systems with infinite delays,” Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 302–321, 2006.
• X. Liao and G. Chen, “Hopf bifurcation and chaos analysis of Chen's system with distributed delays,” Chaos, Solitons & Fractals, vol. 25, no. 1, pp. 197–220, 2005.
• C. Çelik, “Dynamical behavior of a ratio dependent predator-prey system with distributed delay,” Discrete and Continuous Dynamical Systems B, vol. 16, no. 3, pp. 719–738, 2011.
• Y. Song and Y. Peng, “Stability and bifurcation analysis on a logistic model with discrete and distributed delays,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1745–1757, 2006.
• Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, Mass, USA, 1993.
• 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.
• R. Xu and Z. Wang, “Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 70–86, 2006.
• J. Jiao, L. Chen, and S. Cai, “A delayed stage-structured Holling II predator-prey model with mutual interference and impulsive perturbations on predator,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1946–1955, 2009.
• C. Wei and L. Chen, “Eco-epidemiology model with age structure and prey-dependent consumption for pest management,” Applied Mathematical Modelling, vol. 33, no. 12, pp. 4354–4363, 2009.
• X. Meng, J. Jiao, and L. Chen, “The dynamics of an age structured predator-prey model with disturbing pulse and time delays,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 547–561, 2008.
• H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714–1726, 2008.
• R. Shi and L. Chen, “Staged-structured Lotka-Volterra predator-prey models for pest management,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 258–265, 2008.
• J. M. Cushing, Integro-Differential Equations and Delay Models in Population Dynamics, Springer, Berlin, Germany, 1977.
• B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.