Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014 (2014), Article ID 107565, 11 pages
http://dx.doi.org/10.1155/2014/107565
Research Article

Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay

Department of Science, Northeastern University, No. 3-11, Wenhua Road, Heping District, Shenyang, Liaoning 110004, China

Received 3 August 2013; Accepted 22 January 2014; Published 3 March 2014

Academic Editor: Junjie Wei

Copyright © 2014 Xue Zhang and Qing-ling Zhang. 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

This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.