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The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls
Advances in Difference Equations volume 2010, Article number: 738306 (2010)
Abstract
A discrete predator-prey 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 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:
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:
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:
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:
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
such that
for any positive solution 
of (2.3);
-
(b)
for any two positive solutions 
and 
of (2.3).
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).
for any positive solution 
of (2.5);
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 single-specie 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 
such that
of system (3.52).
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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This work was supported by the National Sciences Foundation of China (no. 11071283) and the Sciences Foundation of Shanxi (no. 2009011005-3).
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Li, Q., Liu, H. & Zhang, F. The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls. Adv Differ Equ 2010, 738306 (2010). https://doi.org/10.1155/2010/738306
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DOI: https://doi.org/10.1155/2010/738306