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Abstract and Applied Analysis
Volume 2014, Article ID 867313, 19 pages
http://dx.doi.org/10.1155/2014/867313
Research Article

Dynamic Behaviors of a Discrete Lotka-Volterra Competition System with Infinite Delays and Single Feedback Control

1College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, China
2Department of Mathematics, Ningde Normal University, Fujian 352100, China

Received 21 May 2014; Revised 28 July 2014; Accepted 28 July 2014; Published 14 October 2014

Academic Editor: Ming Mei

Copyright © 2014 Liang Zhao 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 nonautonomous discrete two-species Lotka-Volterra competition system with infinite delays and single feedback control is considered in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that if the the discrete Lotka-Volterra competitive system with infinite delays and without feedback control is permanent, then, by choosing some suitable feedback control variable, the permanent species will be driven to extinction. That is, the feedback control variable, which represents the biological control or some harvesting procedure, is the unstable factor of the system. Such a finding overturns the previous scholars’ recognition on feedback control variables.