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Abstract and Applied Analysis
Volume 2011, Article ID 547060, 16 pages
http://dx.doi.org/10.1155/2011/547060
Research Article

Positive Solutions for a General Gause-Type Predator-Prey Model with Monotonic Functional Response

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 30 April 2011; Revised 12 June 2011; Accepted 20 June 2011

Academic Editor: Gabriella Tarantello

Copyright © 2011 Guohong Zhang and Xiaoli Wang. 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. R. Peng and J. Shi, “Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case,” Journal of Differential Equations, vol. 247, no. 3, pp. 866–886, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. F. Yi, J. Wei, and J. Shi, “Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,” Journal of Differential Equations, vol. 246, no. 5, pp. 1944–1977, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. W. Ko and K. Ryu, “A qualitative study on general Gause-type predator-prey models with non-monotonic functional response,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2558–2573, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. W. Ko and K. Ryu, “Coexistence states of a predator-prey system with non-monotonic functional response,” Nonlinear Analysis: Real World Applications, vol. 8, no. 3, pp. 769–786, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. W. Ko and K. Ryu, “A qualitative study on general Gause-type predator-prey models with constant diffusion rates,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 217–230, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003. View at Publisher · View at Google Scholar
  7. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
  8. T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, Hackensack, NJ, USA, 2005. View at Publisher · View at Google Scholar
  9. E. N. Dancer, “On positive solutions of some pairs of differential equations,” Transactions of the American Mathematical Society, vol. 284, no. 2, pp. 729–743, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. E. N. Dancer, “On positive solutions of some pairs of differential equations. II,” Journal of Differential Equations, vol. 60, no. 2, pp. 236–258, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. E. N. Dancer, “On the indices of fixed points of mappings in cones and applications,” Journal of Mathematical Analysis and Applications, vol. 91, no. 1, pp. 131–151, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. E. N. Dancer, “Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,” The Bulletin of the London Mathematical Society, vol. 34, no. 5, pp. 533–538, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. L. Li, “Coexistence theorems of steady states for predator-prey interacting systems,” Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 143–166, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. P. Y. H. Pang and M. Wang, “Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion,” Proceedings of the London Mathematical Society, vol. 88, no. 1, pp. 135–157, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. R. Peng and M. Wang, “On multiplicity and stability of positive solutions of a diffusive prey-predator model,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 256–268, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. X. Zeng, “A ratio-dependent predator-prey model with diffusion,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1062–1078, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. 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
  18. Y. Du and Y. Lou, “Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 131, no. 2, pp. 321–349, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Y. Du and Y. Lou, “Some uniqueness and exact multiplicity results for a predator-prey model,” Transactions of the American Mathematical Society, vol. 349, no. 6, pp. 2443–2475, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Y. Du and Y. Lou, “S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model,” Journal of Differential Equations, vol. 144, no. 2, pp. 390–440, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” Journal of Functional Analysis, vol. 8, pp. 321–340, 1971. View at Google Scholar · View at Zentralblatt MATH
  22. J. López-Gómez and M. Molina-Meyer, “Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas,” Journal of Differential Equations, vol. 209, no. 2, pp. 416–441, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. S. Cano-Casanova, “Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 49, no. 3, pp. 361–430, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. S. Cano-Casanova and J. López-Gómez, “Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems,” Journal of Differential Equations, vol. 178, no. 1, pp. 123–211, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, vol. 426 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2001. View at Publisher · View at Google Scholar
  26. J. López-Gómez, “The steady states of a non-cooperative model of nuclear reactors,” Journal of Differential Equations, vol. 246, no. 1, pp. 358–372, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,” Journal of Functional Analysis, vol. 7, pp. 487–513, 1971. View at Google Scholar · View at Zentralblatt MATH