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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.

Abstract

We investigate positive solutions of a prey-predator model with predator saturation and competition under homogeneous Dirichlet boundary conditions. First, the existence of positive solutions and some sufficient and necessary conditions is established by using the standard fixed point index theory in cones. Second, the changes of solution branches, multiplicity, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory. Finally, the exact number and type of positive solutions are proved when or converges to infinity.