Research Article Permanence and Global Attractivity of a Delayed Discrete Predator-Prey System with General Holling-Type Functional Response and Feedback Controls

This paper discusses a delayed discrete predator-prey system with general Holling-type functional response and feedback controls. Firstly, sufficient conditions are obtained for the permanence of the system. After that, under some additional conditions, we show that the periodic solution of the system is global stable.


Introduction
The following predator-prey system with Holling-type II functional response and delays ẋ1 t and some generalized systems of general Holling-type functional response have been studied by many scholars see 1-3 and the references cited therein .It has been found that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations.Discrete time models can also provide efficient computational models of continuous models for numerical simulations see 4-12 .In 4 , Yang considered the following delayed discrete predator-prey system with general Holling-type functional response: Sufficient conditions which guarantee the existence of at least one positive periodic solution are obtained by using the continuation theorem of coincidence degree theory.But Yang did not consider the permanence and globally attractivity of system 1.2 , which are two of the most important topics in the study of population dynamics.
On the other hand, as was pointed out by Huo and Li 13 , ecosystem in the real world is continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates.Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time.In the language of control variables, we call the disturbance functions as control variables for more discussion on this section, one could refer to 12-16 for more details .Though much works dealt with the continuous time model.However, to the best of the author's knowledge, up to this day, there are still no scholars that propose and study the system 1.2 with feedback control.Therefore, the main purpose of this paper is to study the following delayed discrete predator-prey system with general Holling-type functional response and feedback control: where 1, 2, are bounded nonnegative sequences and Here, for any bounded sequence {a k }, a M sup k∈N {a k }, and a L inf k∈N {a k }, where N {0, 1, 2, . ..}.This paper is organized as follows.In Section 2, we will introduce some definition and establish several useful lemma.The permanence of system 1.3 is then studied in Section 3. In Section 4, based on the permanence result, under the assumption that all the delays are equal to zero and the coefficients of the system are periodic sequences, we obtain a set of sufficient conditions which guarantee the existence and stability of a unique globally attractive positive periodic solution of the system.
By the biological meaning, we will focus our discussion on the positive solution of system 1.3 .So it is assumed that the initial conditions of 1.3 are of the form where τ max{τ 1 , τ 2 , τ 3 , τ 4 }.
One can easily show that the solutions of 1.3 with the initial condition 1.5 are defined and remain positive for all k ∈ N.

Preliminaries
In this section, we will introduce the definition of permanence and several useful lemmas.Definition 2.1.System 1.3 is said to be permanent if there exist positive constants x * i , u * i , x i * , u i * , which are independent of the solution of the system, such that for any positive solution x 1 k , x 2 k , u 1 k , u 2 k of system 1.3 satisfies Lemma 2.2.Assume that x k satisfies where {a k } and {b k } are positive sequences, x k 0 > 0, θ is a positive constant, and k 0 ∈ N. Then one has where {a k } and {b k } are positive sequences, x k 0 > 0, θ is a positive constant, and where Proof.The proofs of Lemmas 2.2 and 2.3 are very similar to those of 6, Propositions 2.1 and 2.2 , respectively.So we omit the detail here.
Lemma 2.4.Assume that x k satisfies where {a k } and {b k } are positive sequences, x k 0 > 0, θ and τ are positive constants, and where Proof.From the above equation, one has Sequently we can easily obtain that 2.9 So one has

2.10
By Lemma 2.2, we can complete the proof of Lemma 2.4.
Lemma 2.5.Assume that x k satisfies where {a k } and {b k } are positive sequences, x k 0 > 0, θ and τ are positive constants, and where Proof.From the above equation, one has Sequently we can easily obtain that

2.14
So one has

2.15
By Lemma 2.3, we can complete the proof of Lemma 2.5.
Lemma 2.6 is a direct corollary of 17, Theorem 6.2, page 125 by L. Wang and M. Q. Wang.

Lemma 2.6. Consider the following first-order difference equation:
where A, B are positive constants.Assuming A < 1, for any solution {y k } of the above system, one has The following comparison theorem for the difference equation is of 17, Theorem 2.1, page 241 by L. Wang and M. Q. Wang.Lemma 2.7.Let k ∈ {k 0 , k 0 1, . . ., k 0 l, . ..}, r ≥ 0. For fixed k, g k, r is a nondecreasing function with respect to r, and for k ≥ k 0 , the following inequalities hold:

2.18
If y k 0 ≤ u k 0 , then y k ≤ u k for all k ≥ k 0 .

Permanence
In this section, we establish a permanent result for system 1.3 .
Proposition 3.1.In addition to 1.4 , assume further that be any positive solution of system 1.3 , from the first equation of 1.3 , it follows that By applying Lemmas 2.4 and 2.7, we obtain lim sup where Similarly, from the second equation of 1.3 , it follows that Under the assumption H 1 , by applying Lemmas 2.4 and 2.7, we obtain lim sup where

3.9
For any positive constant ε small enough, it follows from 3.5 and 3.8 that there exists large enough K 1 > τ such that

3.10
Then the third equation of 1.3 leads to And so

3.12
By applying Lemmas 2.6 and 2.7, it follows from 3.12 that lim sup

3.13
Setting ε → 0 in the above inequality leads to lim sup Thus we complete the proof of Proposition 3.1.
Proposition 3.2.In addition to 1.4 , assume further that

3.18
Proof.Let x 1 k , x 2 k , u 1 k , u 2 k be any positive solution of system 1.3 .From H 2 and H 3 , there exists a small enough positive constant ε such that Also, according to Proposition 3.1, for the above ε, there exists

3.21
Then from the first equation of 1.3 , one has ε , so the above inequality follows that 3.24 Here we use the fact that exp From 3.19 and 3.23 , by Lemmas 2.5 and 2.7, one has lim inf

3.25
Setting ε → 0 in the above inequality leads to lim inf where

3.27
and Similarly, from the second equation of 1.3 , one has ε , so the above inequality leads to

3.30
Here we use the fact that exp{ α From 3.20 and 3.29 , by Lemmas 2.5 and 2.7, one has lim inf

3.31
Setting ε → 0 in the above inequality leads to lim inf where

3.33
Then the third equation of 1.3 leads to

3.34
And so

3.35
By applying Lemmas 2.6 and 2.7, it follows from 3.35 that lim sup

3.36
Setting ε → 0 in the above inequality leads to lim sup where where u 2 * q L 2 x 2 * /η M 2 .Thus we complete the proof of Proposition 3.2.
Theorem 3.3.In addition to 1.4 , assume further that H 1 , H 2 , and H 3 hold, then system 1.3 is permanent.
It should be noticed that, from the proofs of Propositions 3.1 and 3.2, we know that under the conditions of Theorem 3.3, the set is an invariant set of system 1.3 .

Existence and stability of a periodic solution
In this section, we consider the stability property of system 1.3 under the assumption τ i 0 i 1, 2, 3, 4 , that is, we consider the following system: which are similar to system 1.3 but do not include delays.In this section, we always assume that {r i k }, {b i k }, {α 1 k }, {e i k }, {η i k }, {q i k } are bounded nonnegative periodic sequences with a common period ω and satisfy Also it is assumed that the initial conditions of 4.1 are of the form Using a similar way, under some conditions, we can obtain the permanence of system 4.1 .
As above, still let x * i and u * i , i 1, 2, be the upper bound of {x i k } and {u i k }, x i * and let u i * , i 1, 2, be the lower bound of {x i k } and {u i k }, where x * i , u * i , x i * , and u i * are independent of the solution of system 4.1 .Our first result concerns with the existence of a periodic solution.
Theorem 4.1.In addition to 4.2 , assume further that H 1 , H 2 , and H 3 hold, then system 4.1 has a periodic solution denoted by {x , Ω is an invariant set of system 4.1 .Thus, we can define a mapping F on Ω by Obviously, F depends continuously on x 1 0 , x 2 0 , u 1 0 , u 2 0 .Thus F is continuous and maps a compact set Ω into itself.Therefore, F has a fixed point x 1 , x 2 , u 1 , u 2 .It is easy to see that the solution {x 1 k , x 2 k , u 1 k , u 2 k } passing through x 1 , x 2 , u 1 , u 2 is a periodic solution of system 4.1 .This completes the proof.Now, we study the globally stability property of the periodic solution obtained in Theorem 4.1.Theorem 4.2.In addition to the conditions of Theorem 4.1, if system 4.1 satisfies where the definition of W i , i 1, 2, 3 can be seen in the following proof, then the ω-periodic solution To complete the proof, it suffices to show that

4.10
Since Similarly, we get where Because of the boundedness of {x 1 k }, {x 2 k }, {x 1 k }, {x 2 k }, g ξ 6 k is bounded, where g x x p / 1 mx p and g means the derivation of the function g x .Let |g ξ 6 k | < W 3 .Also, one has

4.15
In view of 4.5 -4.8 , we can choose a ε > 0 such that Also, from Propositions 3.1 and 3.2, there exist K 3 > K 2 such that 4.17 Then from 4.11 , for k > K 3 , one has

4.18
So from 4.13 , for k > K 3 , one has

4.19
Also, for k > K 3 , one has In view of 4.18 -4.21 , one has

4.24
This completes the proof.
1 k is the density of prey species at kth generation, x 2 k is the density of predator species at kth generation, u 1 k and u 2 k are control variables.Also, r 1 k , b 1 k denote the intrinsic growth rate and density-dependent coefficient of the prey, respectively, r 2 k , b 2 k denote the death rate and density-dependent coefficient of the predator, α 1 k denote the capturing rate of the predator, α 2 k /α 1 k represent the rate of conversion of nutrients into the reproduction of the predator.Further, τ i i 1, 2, 3, 4 are nonnegative constants and m, p are positive constants.In this paper, we always assume that {r ∈ 0, 1 for i 1, 2, 3, 4, 5.Because of the boundedness of{x 1 k }, {x 2 k }, {x 1 k }, {x 2 k }, |f 1 ξ 2 k , x 2 k |, |f 2 x 1 k , ξ 3 k | are bounded,where f 1 and f 2 mean the partial derivation of the function f x, y .Let |f 1

Figure 1 :
Dynamics behaviors of system 1.3 with initial condition x 1 p , x 2 p , u 1 p , u 2 p Inequalities 5.3 -5.5 show that H 1 -H 3 are fulfilled.From Theorem 3.3, system 1.3 is permanent.Figure1is the numeric simulation of the solution of system 1.3 with initial condition x 1 p , x 2 p , u 1 p , u 2 p 1.4, 0.8, 0.3, 0.03 P −1, 0 .