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The Permanence and Extinction of a Discrete PredatorPrey System with Time Delay and Feedback Controls
Advances in Difference Equations volume 2010, Article number: 738306 (2010)
Abstract
A discrete predatorprey system with time delay and feedback controls is studied. Sufficient conditions which guarantee the predator and the prey to be permanent are obtained. Moreover, under some suitable conditions, we show that the predator species y will be driven to extinction. The results indicate that one can choose suitable controls to make the species coexistence in a long term.
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 predatorprey models have been studied extensively (e.g., see [1–10] and references cited therein), but they are questioned by several biologists. Thus, the LotkaVolterra type predatorprey model with the BeddingtonDeAngelis functional response has been proposed and has been well studied. The model can be expressed as follows:
The functional response in system (1.1) was introduced by Beddington [11] and DeAngelis et al. [12]. It is similar to the wellknown 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:
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 predatorprey model with time delay and feedback controls:
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 densitydependent coefficient of the prey at time , respectively. denote the death rate and densitydependent 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:
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
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
, , and are nonnegative bounded sequences of real numbers defined on such that
Now, we state several lemmas which will be used to prove the main results in this paper.
First, we consider the following nonautonomous equation:
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

(a)
there exists a positive constant such that
for any positive solution of (2.3);

(b)
for any two positive solutions and of (2.3).
Second, one considers the following nonautonomous linear equation:
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

(a)
for any positive solution of (2.5);

(b)
for any two positive solutions and of (2.5).
Further, considering the following:
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
where is a positive solution of (2.5) with
Finally, one considers the following nonautonomous linear equation:
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
3. Main Results
Theorem 3.1.
Suppose that assumptions and hold, then there exists a constant such that
for any positive solution of system (1.3).
Proof.
Given any solution of system (1.3), we have
for all where is the initial time.
Consider the following auxiliary equation:
from assumptions and Lemma 2.2, there exists a constant such that
where is the solution of (3.3) with initial condition By the comparison theorem, we have
From this, we further have
Then, we obtain that for any constant there exists a constant such that
According to the first equation of system (1.3), we have
for all Considering the following auxiliary equation:
thus, as a direct corollary of Lemma 2.1, we get that there exists a positive constant such that
where is the solution of (3.9) with initial condition By the comparison theorem, we have
From this, we further have
Then, we obtain that for any constant there exists a constant such that
Hence, from the second equation of system (1.3), we obtain
for all Following a similar argument as above, we get that there exists a positive constant such that
By a similar argument of the above proof, we further obtain
From (3.6) and (3.12)–(3.16), we can choose the constant , such that
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:
Theorem 3.2.
Suppose that assumptions hold, then there exists a constant such that
for any positive solution of system (1.3).
Proof.
According to assumptions and we can choose positive constants and such that
Consider the following equation with parameter :
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
where is the solution of (3.21) with initial condition
Let from (3.20), we obtain that there exist and such that
for all
We first prove that
for any positive solution of system (1.3). In fact, if (3.24) is not true, then there exists a such that
where is the solution of system (1.3) with initial condition , So, there exists an such that
Hence, (3.26) together with the third equation of system (1.3) lead to
for Let be the solution of (3.21) with initial condition by the comparison theorem, we have
In (3.22), we choose and since then for given we have
for all Hence, from (3.28), we further have
From the second equation of system (1.3), we have
for all Obviously, we have as Therefore, we get that there exists an such that
for any Hence, by (3.26), (3.30), and (3.32), it follows that
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
On the other hand, by (3.24), we have
Hence, there are two positive integer sequences and satisfying
and such that
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
where Hence,
The above inequality implies that
So, we can choose a large enough such that
From the third equation of system (1.3) and (3.38), we have
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
In (3.22), we choose and , since and , then for all , we have
Equation (3.44) together with (3.45) lead to
for all , and .
From the second equation of system(1.3), we have
for , , and Therefore, we get that
for any Further, from the first equation of systems (1.3), (3.46), and (3.48), we obtain
for any , , and Hence,
In view of (3.37) and (3.38), we finally have
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 singlespecie system with feedback control:
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

(a)
there exists a constant such that
(3.53)for any positive solution of system (3.52).

(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
for any solution of system (3.52) and We make transformation and Hence, system (3.52) is equivalent to
According to , there exists a small enough, such that , Noticing that implies that lie between and Therefore, , It follows from (3.55) that
Let then . It follows easily from (3.56) that
Therefore, as and we can easily obtain that and The proof is completed.
Considering the following equations:
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
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
for any positive solution of system (1.3).
Proof.
According to assumption we can choose positive constants , , and , such that for all we have
Considering the following equation with parameter :
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
We choose if there exists a constant such that for all otherwise Obviously, there exists an , such that
Now, We prove that
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
where and Hence, for all one has
Therefore, from system (1.3), Lemmas 3.3 and 3.4, it follows that
for any solution of system (1.3). Therefore, for any small positive constant there exists an such that for all we have
From the fourth equation of system (1.3), one has
In (3.63), we choose and Since then for all , we have
Equations (3.69), (3.71) together with the second equation of system (1.3) lead to
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
where is the solution of system (1.3) with initial condition for all On the other hand, it follows from (3.65) that
Hence, there are two positive integer sequences and satisfying
and such that
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
where Hence,
The above inequality implies that
Choosing a large enough such that
then for we have
for all Therefore, it follows from system (1.3) that
for all Further, by Lemmas 3.3 and 3.4, we obtain that for any small positive constant we have
for any , , and For any , , and by the first equation of systems (1.3) and (3.77), it follows that
Assume that is the solution of (3.62) with the initial condition , then from comparison theorem and the above inequality, we have
In (3.63), we choose and Since and then we have
Equation (3.86) together with (3.87) lead to
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
Hence,
In view of (3.76) and (3.77), we finally have
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
holds, then
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
for First, we show that there exists an such that Otherwise, there exists an , such that
Hence, for all one has
Therefore, from Lemma 3.3 and comparison theorem, it follows that for the above there exists an , such that
Hence, for we have
So, which is a contradiction. Therefor, there exists an such that
Second, we show that
where
is bounded. Otherwise, there exists an such that Hence, there must exist an such that , , and for Let be a nonnegative integer, such that
It follows from (3.101) that
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.
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Acknowledgments
This work was supported by the National Sciences Foundation of China (no. 11071283) and the Sciences Foundation of Shanxi (no. 20090110053).
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Keywords
 Periodic Solution
 Feedback Control
 Comparison Theorem
 Nonnegative Solution
 Discrete Time Model