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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 627419, 30 pages
http://dx.doi.org/10.1155/2012/627419
Research Article

Uniqueness and Multiplicity of a Prey-Predator Model with Predator Saturation and Competition

1College of Science, Xi'an Technological University, Xi'an, Shaanxi 710032, China
2College of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi 710062, China

Received 15 September 2012; Revised 26 November 2012; Accepted 10 December 2012

Academic Editor: Junjie Wei

Copyright © 2012 Xiaozhou Feng and Lifeng Li. 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. A. D. Bazykin, Nonlinear Dynamics of interacting Populations, World Scientific, Singapore, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  2. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, Md, USA, 1925.
  3. V. Volterra, “Variations and fluctuations of the number of individuals in animal species living together,” in Animal Ecology, R. N. Chapman, Ed., pp. 409–448, 1931. View at Google Scholar
  4. H. Nie and J. Wu, “Multiplicity and stability of a predator-prey model with non-monotonic conversion rate,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 10, no. 1, pp. 154–171, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Wu and G. Wei, “Coexistence states for cooperative model with diffusion,” Computers & Mathematics with Applications, vol. 43, no. 10-11, pp. 1277–1290, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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 · View at MathSciNet
  7. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992. View at MathSciNet
  8. 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 · View at MathSciNet
  9. J. López-Gómez and R. Pardo, “Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case,” Differential and Integral Equations, vol. 6, no. 5, pp. 1025–1031, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. E. N. Dancer, “On uniqueness and stability for solutions of singularly perturbed predator-prey type equations with diffusion,” Journal of Differential Equations, vol. 102, no. 1, pp. 1–32, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. N. Lakos, “Existence of steady-state solutions for a one-predator–two-prey system,” SIAM Journal on Mathematical Analysis, vol. 21, no. 3, pp. 647–659, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Blat and K. J. Brown, “Global bifurcation of positive solutions in some systems of elliptic equations,” SIAM Journal on Mathematical Analysis, vol. 17, no. 6, pp. 1339–1353, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. H. 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 · View at MathSciNet
  14. Y. H. Du and Y. Lou, “Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation,” Proceedings of the Royal Society of Edinburgh A, vol. 131, no. 2, pp. 321–349, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. H. Wu, “Global bifurcation of coexistence state for the competition model in the chemostat,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal A, vol. 39, no. 7, pp. 817–835, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Nie and J. Wu, “Positive solutions of a competition model for two resources in the unstirred chemostat,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 231–242, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. H. Nie and J. Wu, “Asymptotic behavior of an unstirred chemostat model with internal inhibitor,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 889–908, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. H. 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 · View at MathSciNet
  19. M. X. Wang and Q. Wu, “Positive solutions of a prey-predator model with predator saturation and competition,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 708–718, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. 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 · View at MathSciNet
  21. M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” Journal of Functional Analysis, vol. 8, no. 2, pp. 321–340, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. G. Crandall and P. H. Rabinowitz, “Bifurcation, perturbation of simple eigenvalues and linearized stability,” Archive for Rational Mechanics and Analysis, vol. 52, pp. 161–180, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, China, 1990. View at MathSciNet
  24. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, NY, USA, 1966. View at MathSciNet
  25. K. Wonlyul and R. Kimun, “Coexistence states of a predator-prey system with non-monotonic functional response,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 8, no. 3, pp. 769–786, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet