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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 745296, 12 pages
http://dx.doi.org/10.1155/2014/745296
Research Article

Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays

1Department of Science, Northeastern University, Shenyang, Liaoning 110819, China
2Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, China

Received 12 September 2013; Accepted 16 January 2014; Published 10 March 2014

Academic Editor: Shengqiang Liu

Copyright © 2014 Xue Zhang 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

A singular prey-predator model with time delays is formulated and analyzed. Allee effect is considered on the growth of the prey population. The singular prey-predator model is transformed into its normal form by using differential-algebraic system theory. We study its dynamics in terms of local analysis and Hopf bifurcation. The existence of periodic solutions via Hopf bifurcation with respect to two delays is established. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold argument. Finally, numerical simulations are included supporting the theoretical analysis and displaying the complex dynamical behavior of the model outside the domain of stability.