1. Introduction

The dynamic relationship between predator and its prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator-prey models have been studied extensively (e.g., see [1–10] and references cited therein), but they are questioned by several biologists. Thus, the Lotka-Volterra type predator-prey model with the Beddington-DeAngelis functional response has been proposed and has been well studied. The model can be expressed as follows:

(1.1)

The functional response in system (1.1) was introduced by Beddington [11] and DeAngelis et al. [12]. It is similar to the well-known Holling type II functional response but has an extra term in the denominator which models mutual interference between predators. It can be derived mechanistically from considerations of time utilization [11] or spatial limits on predation. But few scholars pay attention to this model. Hwang [6] showed that the system has no periodic solutions when the positive equilibrium is locally asymptotical stability by using the divergency criterion. Recently, Fan and Kuang [9] further considered the nonautonomous case of system (1.1), that is, they considered the following system:

(1.2)

For the general nonautonomous case, they addressed properties such as permanence, extinction, and globally asymptotic stability of the system. For the periodic (almost periodic) case, they established sufficient criteria for the existence, uniqueness, and stability of a positive periodic solution and a boundary periodic solution. At the end of their paper, numerical simulation results that complement their analytical findings were present.

However, we note that ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecosystem is the question of whether an ecosystem can withstand those unpredictable forces which persist for a finite period of time or not. In the language of control variables, we call the disturbance functions as control variables. In 1993, Gopalsamy and Weng [13] introduced a control variable into the delay logistic model and discussed the asymptotic behavior of solution in logistic models with feedback controls, in which the control variables satisfy certain differential equation. In recent years, the population dynamical systems with feedback controls have been studied in many papers, for example, see [13–22] and references cited therein.

It has been found that discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. It is reasonable to study discrete models governed by difference equations. Motivated by the above works, we focus our attention on the permanence and extinction of species for the following nonautonomous predator-prey model with time delay and feedback controls:

(1.3)

where , are the density of the prey species and the predator species at time , respectively. , are the feedback control variables. represent the intrinsic growth rate and density-dependent coefficient of the prey at time , respectively. denote the death rate and density-dependent coefficient of the predator at time , respectively. denotes the capturing rate of the predator; represents the rate of conversion of nutrients into the reproduction of the predator. Further, is a positive integer.

For the simplicity and convenience of exposition, we introduce the following notations. Let , and denote the set of integer satisfying We denote to be the space of all nonnegative and bounded discrete time functions. In addition, for any bounded sequence we denote ,

Given the biological sense, we only consider solutions of system (1.3) with the following initial condition:

(1.4)

It is not difficult to see that the solutions of system (1.3) with the above initial condition are well defined for all and satisfy

(1.5)

The main purpose of this paper is to establish a new general criterion for the permanence and extinction of system (1.3), which is dependent on feedback controls. This paper is organized as follows. In Section 2, we will give some assumptions and useful lemmas. In Section 3, some new sufficient conditions which guarantee the permanence of all positive solutions of system (1.3) are obtained. Moreover, under some suitable conditions, we show that the predator species will be driven to extinction.

2. Preliminaries

In this section, we present some useful assumptions and state several lemmas which will be useful in the proving of the main results.

Throughout this paper, we will have both of the following assumptions:

() , , , and are nonnegative bounded sequences of real numbers defined on such that

(2.1)

  , , and are nonnegative bounded sequences of real numbers defined on such that

(2.2)

Now, we state several lemmas which will be used to prove the main results in this paper.

First, we consider the following nonautonomous equation:

(2.3)

where functions , are bounded and continuous defined on with , . We have the following result which is given in [23].

Lemma 2.1.

Let be the positive solution of (2.3) with , then

  1. (a)

    there exists a positive constant such that

(2.4)

for any positive solution of (2.3);

  1. (b)

    for any two positive solutions and of (2.3).

Second, one considers the following nonautonomous linear equation:

(2.5)

where functions and are bounded and continuous defined on with and The following Lemma 2.2 is a direct corollary of Theorem of L. Wang and M. Q. Wang [24, page 125].

Lemma 2.2.

Let be the nonnegative solution of (2.5) with , then

  1. (a)

    for any positive solution of (2.5);

  2. (b)

    for any two positive solutions and of (2.5).

Further, considering the following:

(2.6)

where functions and are bounded and continuous defined on with , and The following Lemma 2.3 is a direct corollary of Lemma of Xu and Teng [25].

Lemma 2.3.

Let be the positive solution of (2.6) with , then for any constants and , there exist positive constants and such that for any and when one has

(2.7)

where is a positive solution of (2.5) with

Finally, one considers the following nonautonomous linear equation:

(2.8)

where functions are bounded and continuous defined on with and In [25], the following Lemma 2.4 has been proved.

Lemma 2.4.

Let be the nonnegative solution of (2.8) with , then, for any constants and , there exist positive constants and such that for any and when , one has

(2.9)

3. Main Results

Theorem 3.1.

Suppose that assumptions and hold, then there exists a constant such that

(3.1)

for any positive solution of system (1.3).

Proof.

Given any solution of system (1.3), we have

(3.2)

for all where is the initial time.

Consider the following auxiliary equation:

(3.3)

from assumptions and Lemma 2.2, there exists a constant such that

(3.4)

where is the solution of (3.3) with initial condition By the comparison theorem, we have

(3.5)

From this, we further have

(3.6)

Then, we obtain that for any constant there exists a constant such that

(3.7)

According to the first equation of system (1.3), we have

(3.8)

for all Considering the following auxiliary equation:

(3.9)

thus, as a direct corollary of Lemma 2.1, we get that there exists a positive constant such that

(3.10)

where is the solution of (3.9) with initial condition By the comparison theorem, we have

(3.11)

From this, we further have

(3.12)

Then, we obtain that for any constant there exists a constant such that

(3.13)

Hence, from the second equation of system (1.3), we obtain

(3.14)

for all Following a similar argument as above, we get that there exists a positive constant such that

(3.15)

By a similar argument of the above proof, we further obtain

(3.16)

From (3.6) and (3.12)–(3.16), we can choose the constant , such that

(3.17)

This completes the proof of Theorem 3.1.

In order to obtain the permanence of system (1.3), we assume that

() where is some positive solution of the following equation:

(3.18)

Theorem 3.2.

Suppose that assumptions hold, then there exists a constant such that

(3.19)

for any positive solution of system (1.3).

Proof.

According to assumptions and we can choose positive constants and such that

(3.20)

Consider the following equation with parameter :

(3.21)

Let be any positive solution of system (3.18) with initial value By assumptions and Lemma 2.2, we obtain that is globally asymptotically stable and converges to uniformly for Further, from Lemma 2.3, we obtain that, for any given and a positive constant ( is given in Theorem 3.1), there exist constants and such that for any and when , we have

(3.22)

where is the solution of (3.21) with initial condition

Let from (3.20), we obtain that there exist and such that

(3.23)

for all

We first prove that

(3.24)

for any positive solution of system (1.3). In fact, if (3.24) is not true, then there exists a such that

(3.25)

where is the solution of system (1.3) with initial condition , So, there exists an such that

(3.26)

Hence, (3.26) together with the third equation of system (1.3) lead to

(3.27)

for Let be the solution of (3.21) with initial condition by the comparison theorem, we have

(3.28)

In (3.22), we choose and since then for given we have

(3.29)

for all Hence, from (3.28), we further have

(3.30)

From the second equation of system (1.3), we have

(3.31)

for all Obviously, we have as Therefore, we get that there exists an such that

(3.32)

for any Hence, by (3.26), (3.30), and (3.32), it follows that

(3.33)

for any where Thus, from (3.23) and (3.33), we have which leads to a contradiction. Therefore, (3.24) holds.

Now, we prove the conclusion of Theorem 3.2. In fact, if it is not true, then there exists a sequence of initial functions such that

(3.34)

On the other hand, by (3.24), we have

(3.35)

Hence, there are two positive integer sequences and satisfying

(3.36)

and such that

(3.37)
(3.38)

By Theorem 3.1, for any given positive integer , there exists a such that , , , and for all Because of as there exists a positive integer such that and as Let for any , we have

(3.39)

where Hence,

(3.40)

The above inequality implies that

(3.41)

So, we can choose a large enough such that

(3.42)

From the third equation of system (1.3) and (3.38), we have

(3.43)

for any , , and Assume that is the solution of (3.21) with the initial condition , then from comparison theorem and the above inequality, we have

(3.44)

In (3.22), we choose and , since and , then for all , we have

(3.45)

Equation (3.44) together with (3.45) lead to

(3.46)

for all , and .

From the second equation of system(1.3), we have

(3.47)

for , , and Therefore, we get that

(3.48)

for any Further, from the first equation of systems (1.3), (3.46), and (3.48), we obtain

(3.49)

for any , , and Hence,

(3.50)

In view of (3.37) and (3.38), we finally have

(3.51)

which is a contradiction. Therefore, the conclusion of Theorem 3.2 holds. This completes the proof of Theorem 3.2.

In order to obtain the permanence of the component of system (1.3), we next consider the following single-specie system with feedback control:

(3.52)

For system (3.52), we further introduce the following assumption:

suppose , where , are given in the proof of Lemma 3.3.

For system(3.52), we have the following result.

Lemma 3.3.

Suppose that assumptions hold, then

  1. (a)

    there exists a constant such that

    (3.53)

    for any positive solution of system (3.52).

  2. (b)

    if assumption holds, then each fixed positive solution of system (3.52) is globally uniformly attractive on

Proof.

Based on assumptions , conclusion (a) can be proved by a similar argument as in Theorems 3.1 and 3.2.

Here, we prove conclusion (b). Letting be some solution of system (3.52), by conclusion (a), there exist constants , , and , such that

(3.54)

for any solution of system (3.52) and We make transformation and Hence, system (3.52) is equivalent to

(3.55)

According to , there exists a small enough, such that , Noticing that implies that lie between and Therefore, , It follows from (3.55) that

(3.56)

Let then . It follows easily from (3.56) that

(3.57)

Therefore, as and we can easily obtain that and The proof is completed.

Considering the following equations:

(3.58)

then we have the following result.

Lemma 3.4.

Suppose that assumptions hold, then there exists a positive constant such that for any positive solution of system (3.58), one has

(3.59)

where is the solution of system (3.52) with and

The proof of Lemma 3.4 is similar to Lemma 3.3, one omits it here.

Let be a fixed solution of system (3.52) defined on one assumes that

  

Theorem 3.5.

Suppose that assumptions hold, then there exists a constant such that

(3.60)

for any positive solution of system (1.3).

Proof.

According to assumption we can choose positive constants , , and , such that for all we have

(3.61)

Considering the following equation with parameter :

(3.62)

by Lemma 2.4, for given and ( is given in Theorem 3.1.), there exist constants and , such that for any and when we have

(3.63)

We choose if there exists a constant such that for all otherwise Obviously, there exists an , such that

(3.64)

Now, We prove that

(3.65)

for any positive solution of system (1.3). In fact, if (3.65) is not true, then for , there exist a and such that for all

(3.66)

where and Hence, for all one has

(3.67)

Therefore, from system (1.3), Lemmas 3.3 and 3.4, it follows that

(3.68)

for any solution of system (1.3). Therefore, for any small positive constant there exists an such that for all we have

(3.69)

From the fourth equation of system (1.3), one has

(3.70)

In (3.63), we choose and Since then for all , we have

(3.71)

Equations (3.69), (3.71) together with the second equation of system (1.3) lead to

(3.72)

for all where Obviously, we have as which is contradictory to the boundedness of solution of system (1.3). Therefore, (3.65) holds.

Now, we prove the conclusion of Theorem 3.5. In fact, if it is not true, then there exists a sequence of initial functions, such that

(3.73)

where is the solution of system (1.3) with initial condition for all On the other hand, it follows from (3.65) that

(3.74)

Hence, there are two positive integer sequences and satisfying

(3.75)

and such that

(3.76)
(3.77)

By Theorem 3.1, for given positive integer , there exists a such that , , , and for all Because that as there is a positive integer such that and as Let for any , we have

(3.78)

where Hence,

(3.79)

The above inequality implies that

(3.80)

Choosing a large enough such that

(3.81)

then for we have

(3.82)

for all Therefore, it follows from system (1.3) that

(3.83)

for all Further, by Lemmas 3.3 and 3.4, we obtain that for any small positive constant we have

(3.84)

for any , , and For any , , and by the first equation of systems (1.3) and (3.77), it follows that

(3.85)

Assume that is the solution of (3.62) with the initial condition , then from comparison theorem and the above inequality, we have

(3.86)

In (3.63), we choose and Since and then we have

(3.87)

Equation (3.86) together with (3.87) lead to

(3.88)

for all , , and .

So, for any , , and from the second equation of systems (1.3), (3.61), (3.77), (3.84), and (3.88), it follows that

(3.89)

Hence,

(3.90)

In view of (3.76) and (3.77), we finally have

(3.91)

which is a contradiction. Therefore, the conclusion of Theorem 3.5 holds.

Remark 3.6.

In Theorems 3.2 and 3.5, we note that are decided by system(1.3), which is dependent on the feedback control . So, the control variable has impact on the permanence of system (1.3). That is, there is the permanence of the species as long as feedback controls should be kept beyond the range. If not, we have the following result.

Theorem 3.7.

Suppose that assumption

(3.92)

holds, then

(3.93)

for any positive solution of system (1.3).

Proof.

By the condition, for any positive constant ( where is given in Theorem 3.5), there exist constants and such that

(3.94)

for First, we show that there exists an such that Otherwise, there exists an , such that

(3.95)

Hence, for all one has

(3.96)

Therefore, from Lemma 3.3 and comparison theorem, it follows that for the above there exists an , such that

(3.97)

Hence, for we have

(3.98)

So, which is a contradiction. Therefor, there exists an such that

Second, we show that

(3.99)

where

(3.100)

is bounded. Otherwise, there exists an such that Hence, there must exist an such that , , and for Let be a nonnegative integer, such that

(3.101)

It follows from (3.101) that

(3.102)

which leads to a contradiction. This shows that (3.99) holds. By the arbitrariness of it immediately follows that as This completes the proof of Theorem 3.7.