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Abstract and Applied Analysis
Volume 2011, Article ID 934569, 16 pages
Research Article

A Three-Species Food Chain System with Two Types of Functional Responses

1Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
2Department of Mathematics Education, Catholic University of Daegu, Gyeongsan-si, Gyeongbuk 712-702, Republic of Korea

Received 20 July 2010; Revised 11 December 2010; Accepted 19 January 2011

Academic Editor: H. B. Thompson

Copyright © 2011 Younghae Do 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.


In recent decades, many researchers have investigated the ecological models with three and more species to understand complex dynamical behaviors of ecological systems in nature. However, when they studied the models with three species, they have just considered the functional responses between prey and mid-predator and between mid-predator and top predator as the same type. However, in the paper, in order to describe more realistic ecological world, a three-species food chain system with two types of functional response, Holling type and Beddington-DeAngelis type, is considered. It is shown that this system is dissipative. Also, the local and global stability of equilibrium points of the system is established. In addition, conditions for the persistence of the system are found according to the existence of limit cycles. Some numerical examples are given to substantiate our theoretical results. Moreover, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the largest Lyapunov exponent.