Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016, Article ID 7920482, 7 pages
http://dx.doi.org/10.1155/2016/7920482
Research Article

A Novel Method to Determine the Local Stability of the n-Species Lotka-Volterra System with Multiple Delays

1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2Department of Mathematics, Taizhou University, Taizhou 225300, China

Received 18 May 2016; Revised 4 November 2016; Accepted 20 November 2016

Academic Editor: Leonid Shaikhet

Copyright © 2016 Xiao-Ping Chen and Hao Liu. 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. H. I. Friedman, Deterministic Mathematical Models in Population Ecology, Dekker, New York, NY, USA, 1998.
  2. J. D. Murray, Mathematical biology, vol. 19 of Biomathematics, Springer, Berlin, Second edition, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. K. Golpalsamy, “Globally asymptotic stability in a periodic Lotka-Volterra system,” Journal of the Australian Mathematical Society, Series B, vol. 24, pp. 160–170, 1982. View at Google Scholar
  4. Y. Kuang and H. L. Smith, “Global stability for infinite delay Lotka-Volterra type systems,” Journal of Differential Equations, vol. 103, no. 2, pp. 221–246, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. X.-Z. Li, C.-L. Tang, and X.-H. Ji, “The criteria for globally stable equilibrium in n-dimensional Lotka-Volterra systems,” Journal of Mathematical Analysis and Applications, vol. 240, no. 2, pp. 600–606, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. Xiao and W. Li, “Limit cycles for the competitive three dimensional Lotka-Volterra system,” Journal of Differential Equations, vol. 164, no. 1, pp. 1–15, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. L. Zhang and Z. Teng, “N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations,” Nonlinear Analysis. Real World Applications, vol. 12, no. 6, pp. 3152–3169, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. J. Zhen and Z.-E. Ma, “Stability for a competitive Lotka-Volterra system with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 51, no. 7, pp. 1131–1142, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. H. Park, “Stability for a competitive Lotka-Volterra system with delays: LMI optimization approach,” Applied Mathematics Letters, vol. 18, no. 6, pp. 689–694, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Y. G. Sun and F. W. Meng, “LMI approach to stability for a competitive Lotka-Volterra system with time-varying delays,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 291–297, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. J. Qiu and J. Cao, “Exponential stability of a competitive Lotka-Volterra system with delays,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 819–829, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. G. Qin, “Stability and hopf bifurcations of a three-species symbiosis model with delays,” in Proceedings of the International Workshop on Chaos-Fractals Theories and Applications (IWCFTA '09), pp. 272–276, Shenyang City, China, November 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. J. Ma, Q. Zhang, and Q. Gao, “Stability of a three-species symbiosis model with delays,” Nonlinear Dynamics, vol. 67, no. 1, pp. 567–572, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. Hofbauer and J. W. So, “Diagonal dominance and harmless off-diagonal delays,” Proceedings of the American Mathematical Society, vol. 128, no. 9, pp. 2675–2682, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. A. Campbell, “Delay independent stability for additive neural networks,” Differential Equations and Dynamical Systems. An International Journal for Theory, Applications, and Computer Simulations, vol. 9, pp. 115–138, 2001. View at Google Scholar · View at MathSciNet
  16. K. Gopalsamy, “Global asymptotic stability in Volterra's population systems,” Journal of Mathematical Biology, vol. 19, no. 2, pp. 157–168, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. W. Michiels and S. I. Niculescu, Stability and Stabilization of Time-Delay Systems, An Eigenvalue Based Approach, SIAM, 2007.
  18. X.-P. Chen and H. Dai, “Stability analysis of time-delay systems using a contour integral method,” Applied Mathematics and Computation, vol. 273, pp. 390–397, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. 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