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).

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.
Comparing with Theorem A, it is easy to see that (  ) in Theorem B are weaker than (  ) in Theorem A ( = 1, 2) 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][4][5][6][7][8][9][10] and the references cited therein.
The remaining part of this paper is organized as follows.In Section 2, we will introduce several lemmas.The permanence of system (1) is then studied in Section 3. In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results.

Preliminaries
In this section, we will introduce several useful lemmas.

Permanence
In this section, we detail the proof of our main result by several lemmas.The following lemma is a direct conclusion of [1].
Lemma 5. Assume holds then there exist two positive constants  1 and  1 such that where  1 and  1 are defined in the proof.
Proof.According to Lemma 4, for any  > 0 small enough, there exists enough large  1 > 0, such that, for  ≥  1 , Thus, it follows from ( 16) and the first equation of system (1) that for  ≥  1 , where Thus From the third equation of system (1), we have where  1 = 1 −   1 and  1 () =   1 ().Then, for any  ≤ , according to Lemma 2,(19), and (20) Note that 0 <   1 < 1; hence 0 <  1 < 1.Therefore, Then, there exists a positive integer  2 >  1 such that, for any positive solution of system (1),   1 where 1 exp{− 1 ( + 1)}.Substituting (23) into the first equation of system (1), we can get where By applying Lemma 1 to (24), it immediately follows that lim inf Setting  → 0 in the above inequality, we obtain lim inf It follows from (26) that there exists large enough This together with the third equation of system (1) leads to Hence, By applying Lemma 3, it follows from (29) that lim inf This completes the proof of Lemma 5.

Lemma 6. Assume that
holds; then there exist two positive constants  2 and  2 such that lim inf where  2 and  2 are defined in the proof.

Example and Numerical Simulation
(48) Equation ( 48) means that all conditions of Theorem B are satisfied in system (47).Thus, the system (47) is permanent.
Our numerical simulation supports our result (see Figure 1).However, that is to say, ( 1 ) does not hold and we could not obtain the result of the permanence from Theorem A. Thus our results improve the main results in [1].