Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 613648, 7 pages
http://dx.doi.org/10.1155/2014/613648
Research Article

Travelling Waves of an n-Species Food Chain Model with Spatial Diffusion and Time Delays

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 21 February 2014; Accepted 28 March 2014; Published 20 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Fei Hu et al. 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. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1993. View at MathSciNet
  2. R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solution for athree-species Lotka-Volterra food-chain model with time delays,” Mathematical and Computer Modelling, vol. 40, no. 7-8, pp. 823–837, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. C. V. Pao, “Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 186–204, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. C. V. Pao, “Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays,” Nonlinear Analysis: Real World Applications, vol. 5, no. 1, pp. 91–104, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. C. V. Pao, “The global attractor of a competitor-competitor-mutualist reaction-diffusion system with time delays,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 9, pp. 2623–2631, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. Y. Tang and L. Zhou, “Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1290–1307, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. K. W. Schaaf, “Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,” Transactions of the American Mathematical Society, vol. 302, no. 2, pp. 587–615, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. A. Gardner, “Existence and stability of travelling wave solutions of competition models: a degree theoretic approach,” Journal of Differential Equations, vol. 44, no. 3, pp. 343–364, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer, New York, NY, USA, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. R. Dunbar, “Travelling wave solutions of diffuse Lotka-Volterra equations,” Journal of Mathematical Biology, vol. 17, no. 1, pp. 11–32, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. P. Bates and F. Chen, “Periodic traveling waves for a nonlocal integro-differential model,” Electronic Journal of Differential Equations, vol. 1999, pp. 1–19, 1999. View at Google Scholar · View at Scopus
  13. S. Ma and Y. Duan, “Asymptotic stability of traveling waves in a discrete convolution model for phase transitions,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 240–256, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. J. D. Murray, Mathematical Biology, Springer, Berlin, Germany, 1989. View at MathSciNet
  15. J. Li and Z. Liu, “Smooth and non-smooth traveling waves in a nonlinearly dispersive equation,” Applied Mathematical Modelling, vol. 25, no. 1, pp. 41–56, 2000. View at Publisher · View at Google Scholar · View at Scopus
  16. K. Wang and W. Wang, “Propagation of HBV with spatial dependence,” Mathematical Biosciences, vol. 210, no. 1, pp. 78–95, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. J. Wu and X. Zou, “Traveling wave fronts of reaction-diffusion systems with delay,” Journal of Dynamics and Differential Equations, vol. 13, no. 3, pp. 651–687, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X. Zou and J. Wu, “Local existence and stability of periodic traveling waves of lattice functional-differential equations,” The Canadian Applied Mathematics Quarterly, vol. 6, no. 4, pp. 397–418, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Huang and X. Zou, “Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays,” Journal of Mathematical Analysis and Applications, vol. 271, no. 2, pp. 455–466, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. S. Ma, “Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,” Journal of Differential Equations, vol. 171, no. 2, pp. 294–314, 2001. View at Publisher · View at Google Scholar · View at Scopus
  21. Q. Gan, R. Xu, X. Zhang, and P. Yang, “Travelling waves of a three-species Lotka-Volterra food-chain model with spatial diffusion and time delays,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2817–2832, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. E. Zeidler, Nonlinear Functional Analysis and its Applications: I. Fixed-Point Theorems, Springer, New York, NY, USA, 1986. View at MathSciNet