Permanence for a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

A new set of sufficient conditions for the permanence of a ratio-dependent predator-prey system with Holling type III functional response and feedback controls are obtained. The result shows that feedback control variables have no influence on the persistent property of the system, thus improving and supplementing the main result of Yang (2008). 1. Introduction The aim of this paper is to investigate the permanent property of the following discrete ratio-dependent predator-prey system with Holling type III and feedback controls: where and are the densities of the prey population and predator population at time , respectively, for , and are all bounded nonnegative sequences such that Here, for any bounded sequence , , . By the biological meaning, we will focus our discussion on the positive solutions of system (1). So, we consider (1) together with the following initial conditions: It is not difficult to see that the solutions of (1)–(3) are well defined and satisfy Recently, Yang [1] proposed and studied the permanence of system (1). Set Using the comparison theorem of difference equation, Yang obtained the following result. Theorem A (see [1]). Assume that hold; then system (1) is permanent. Theorem A shows that feedback control variables play important roles in the persistent property of the system (1). But the question is whether or not the feedback control variables have influence on the permanence of the system. On the other hand, as was pointed out by Fan and Wang [2], “if we use the method of comparison theorem, then the additional condition, in some extent, is necessary. But for the system itself, this condition may not necessary.[sic]” In [2], by establishing a new difference inequality, Fan and Wang showed that feedback control has no influence on the permanence of a single species discrete model. Their success motivated us to consider the persistent property of system (1). Indeed, in this paper, we will apply the analysis technique of Fan and Wang [2] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system. We finally obtain the following main results. Theorem B. Assume that hold; then system (1) is permanent. Comparing with Theorem A, it is easy to see that in Theorem B are weaker than in Theorem A and feedback control variables have no influence on the permanent property of system (1), so our results improve the main results in [1]. For more works on this direction, one could refer to [3–10] and the references cited therein. The remaining part of this paper is organized