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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 101386, 17 pages
http://dx.doi.org/10.1155/2012/101386
Research Article

The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

Department of Mathematics Education, Catholic University of Daegu, Gyeongsan 712-702, Republic of Korea

Received 26 August 2011; Accepted 28 October 2011

Academic Editor: Agacik Zafer

Copyright © 2012 Hunki Baek. 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 Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations.