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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 270191, 12 pages
http://dx.doi.org/10.1155/2013/270191
Research Article

Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

Faculty of Science, Shaanxi University of Science and Technology, Xi’an, Shaanxi 710021, China

Received 2 August 2012; Revised 9 December 2012; Accepted 21 December 2012

Academic Editor: Lan Xu

Copyright © 2013 Zongmin Yue 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.

Abstract

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.