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Complexity
Volume 2017, Article ID 2578043, 12 pages
https://doi.org/10.1155/2017/2578043
Research Article

Delay-Induced Oscillations in a Competitor-Competitor-Mutualist Lotka-Volterra Model

Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China

Correspondence should be addressed to Changjin Xu; moc.621@304jcx

Received 14 January 2017; Accepted 30 March 2017; Published 26 April 2017

Academic Editor: Alicia Cordero

Copyright © 2017 Changjin Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Aggelis, D. V. Vayenas, V. Tsagou, and S. Pavlou, “Prey-predator dynamics with predator switching regulated by a catabolic repression control mode,” Ecological Modelling, vol. 183, no. 4, pp. 451–462, 2005. View at Publisher · View at Google Scholar · View at Scopus
  2. N. Bairagi and D. Jana, “On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 35, no. 7, pp. 3255–3267, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. K. Chakraborty, M. Chakraborty, and T. K. Kar, “Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 4, pp. 613–625, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. pada Das, K. Kundu, and J. Chattopadhyay, “A predator-prey mathematical model with both the populations affected by diseases,” Ecological Complexity, vol. 8, no. 1, pp. 68–80, 2011. View at Publisher · View at Google Scholar · View at Scopus
  5. C.-H. Hsu, C.-R. Yang, T.-H. Yang, and T.-S. Yang, “Existence of traveling wave solutions for diffusive predator-prey type systems,” Journal of Differential Equations, vol. 252, no. 4, pp. 3040–3075, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. Jiao, S. Cai, and L. Chen, “Analysis of a stage-structured predatory-prey system with birth pulse and impulsive harvesting at different moments,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2232–2244, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. T. K. Kar and A. Ghorai, “Dynamic behaviour of a delayed predator-prey model with harvesting,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9085–9104, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. W. Ko and K. Ryu, “Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 1109–1115, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Q. M. Liu and R. Xu, “Periodic solution for a delayed three-species food-chain system with Holling type-II functional response,” International Journal of Mathematics and Mathematical Sciences, vol. 64, pp. 4057–4070, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Y. Saito, J. Sugie, and Y.-H. Lee, “Global asymptotic stability for predator-prey models with environmental time-variations,” Applied Mathematics Letters, vol. 24, no. 12, pp. 1973–1980, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. X. Lv, P. Yan, and S. Lu, “Existence and global attractivity of positive periodic solutions of competitor-competitor-mutualist Lotka-Volterra systems with deviating arguments,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 823–832, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. M. Xiao and J.-D. Cao, “Stability and Hopf bifurcation in a delayed competitive web sites model,” Physics Letters A, vol. 353, no. 2-3, pp. 138–150, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Xu, X. Tang, and M. Liao, “Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2920–2936, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. C. Xu, X. Tang, M. Liao, and X. He, “Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays,” Nonlinear Dynamics, vol. 66, no. 1-2, pp. 169–183, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  16. R. Xu and Z. E. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  17. X.-P. Yan and W.-T. Li, “Bifurcation and global periodic solutions in a delayed facultative mutualism system,” Physica D. Nonlinear Phenomena, vol. 227, no. 1, pp. 51–69, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  19. C. B. Yu and J. J. Wei, “Stability and bifurcation analysis in a basic model of the immune response with delays,” Chaos, Solitons & Fractals, vol. 41, no. 3, pp. 1223–1234, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. Yuan and F. Zhang, “Stability and global Hopf bifurcation in a delayed predator-prey system,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 959–977, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  21. M. L. Zeeman, “Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,” Dynamics and Stability of Systems, vol. 8, no. 3, pp. 189–217, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  22. X. Zhou, X. Shi, and X. Song, “Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 129–136, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. 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: Mathematical Analysis, vol. 10, no. 6, pp. 863–874, 2003. View at Google Scholar · View at MathSciNet · View at Scopus
  24. B. Hassard, D. Kazarino, and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet