Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 409264, 7 pages
http://dx.doi.org/10.1155/2014/409264
Research Article

Traveling Wave Solution in a Diffusive Predator-Prey System with Holling Type-IV Functional Response

1Polytechnic Institute of Jiangxi Science and Technology Normal University, Nanchang 330038, China
2Department of Mathematics, Chongqing Normal University, Chongqing 400030, China
3School of Mathematics and Physics, University of South China, Hengyang 421001, China

Received 13 November 2013; Accepted 27 January 2014; Published 6 March 2014

Academic Editor: Chaudry Masood Khalique

Copyright © 2014 Deniu Yang 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. J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, pp. 707–723, 1968. View at Publisher · View at Google Scholar
  2. W. Sugie and J. A. Howell, “Kinetics of phenol by washed cell,” Biotechnology and Bioengineering, vol. 23, pp. 2039–2049, 1980. View at Google Scholar
  3. A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, pp. 1530–1535, 1992. View at Publisher · View at Google Scholar
  4. Y.-H. Fan and W.-T. Li, “Global asymptotic stability of a ratio-dependent predator-prey system with diffusion,” Journal of Computational and Applied Mathematics, vol. 188, no. 2, pp. 205–227, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. A. Gourley and N. F. Britton, “A predator-prey reaction-diffusion system with nonlocal effects,” Journal of Mathematical Biology, vol. 34, no. 3, pp. 297–333, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. L.-L. Wang and W.-T. Li, “Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response,” Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp. 341–357, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet
  8. R. A. Gardner, “Existence of travelling wave solutions of predator-prey systems via the connection index,” SIAM Journal on Applied Mathematics, vol. 44, no. 1, pp. 56–79, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1974.
  10. A. Okubo, Diffusion and Ecological Problems: Mathematical Models, vol. 10 of Lectures in Biomathematics, Springer, Berlin, Germany, 1980. View at MathSciNet
  11. M. R. Owen and M. A. Lewis, “How predation can slow, stop or reverse a prey invasion,” Bulletin of Mathematical Biology, vol. 63, pp. 655–684, 2001. View at Publisher · View at Google Scholar
  12. J. Huang, G. Lu, and S. Ruan, “Existence of traveling wave solutions in a diffusive predator-prey model,” Journal of Mathematical Biology, vol. 46, no. 2, pp. 132–152, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. R. Dunbar, “Travelling wave solutions of diffusive 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
  14. S. R. Dunbar, “Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits,” SIAM Journal on Applied Mathematics, vol. 46, no. 6, pp. 1057–1078, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W.-T. Li and S.-L. Wu, “Traveling waves in a diffusive predator-prey model with Holling type-III functional response,” Chaos, Solitons & Fractals, vol. 37, no. 2, pp. 476–486, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. View at MathSciNet
  17. J. P. LaSalle, “Stability theory for ordinary differential equations,” Journal of Differential Equations, vol. 4, pp. 57–65, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer, Berlin, Germany, 1994. View at MathSciNet
  19. J. D. Murray, Mathematical Biology, vol. 19, Springer, Berlin, Germany, 2nd edition, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. Mischaikow and J. F. Reineck, “Travelling waves in predator-prey systems,” SIAM Journal on Mathematical Analysis, vol. 24, no. 5, pp. 1179–1214, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, vol. 140, American Mathematical Society, Providence, RI, USA, 1994. View at MathSciNet
  22. S. R. Dunbar, “Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in 𝐑4,” Transactions of the American Mathematical Society, vol. 286, no. 2, pp. 557–594, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. L. Chow and W. C. Tam, “Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion,” Bulletin of Mathematical Biology, vol. 38, no. 6, pp. 643–658, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. C. Conley, Isolated Invariant Sets and the Morse Index, vol. 38 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1978. View at MathSciNet