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Note on the Persistent Property of a Discrete LotkaVolterra Competitive System with Delays and Feedback Controls
Advances in Difference Equations volumeÂ 2010, ArticleÂ number:Â 249364 (2010)
Abstract
A nonautonomous species discrete LotkaVolterra competitive system with delays and feedback controls is considered in this work. Sufficient conditions on the coefficients are given to guarantee that all the species are permanent. It is shown that these conditions are weaker than those of Liao et al. 2008.
1. Introduction
Traditional LotkaVolterra competitive systems have been extensively studied by many authors [1â€“7].The autonomous model can be expressed as follows:
where , , , denoting the density of the i th species at time . Montes de Oca and Zeeman [6] investigated the general nonautonomous species LotkaVolterra competitive system
and obtained that if the coefficients are continuous and bounded above and below by positive constants, and if for each there exists an integer such that
then exponentially for and where is a certain solution of a logistic equation. Teng [8] and Ahmad and Stamova [9] also studied the coexistence on a nonautonomous LotkaVolterra competitive system. They obtained the necessary or sufficient conditions for the permanence and the extinction. For more works relevant to system (1.1), one could refer to [1â€“9] and the references cited therein.
However, to the best of the authors' knowledge, to this day, still less scholars consider the general nonautonomous discrete LotkaVolterra competitive system with delays and feedback controls. Recently, in [1] Liao et al. considered the following general nonautonomous discrete LotkaVolterra competitive system with delays and feedback controls:
where is the density of competitive species; is the control variable; ; bounded sequences , , , , and ; and are positive integer; denote the sets of all integers and all positive real numbers, respectively; is the firstorder forward difference operator ; .
In [1], Liao et al. obtained sufficient conditions for permanence of the system (1.4).
They obtained what follows.
Lemma 1.1.
Assume that
hold, then system (1.4) is permanent, where
Since
Hence, the above inequality (1.5) implies
That is
It was shown that in [1] Liao et al. considered system (1.4) where all coefficients , , , , , and were assumed to satisfy conditions (1.9).
In this work, we shall study system (1.4) and get the same results as [1] do under the weaker assumption that
Our main results are the following Theorem 1.2.
Theorem 1.2.
Assume that (1.10) holds, then system (1.4) is permanent.
Remark 1.3.
The inequality (1.9) implies (1.10), but not conversely, for
Therefore, we have improved the permanence conditions of [1] for system (1.4).
Theorem 1.2 will be proved in Section 2. In Section 3, an example will be given to illustrate that (1.10) does not imply (1.9); that is, the condition (1.10) is better than (1.9).
2. Proof of Theorem 1.2
The following lemma can be found in [10].
Lemma 2.1.
Assume that and , and further suppose that

(1)
(2.1)
Then for any integer
Especially, if and is bounded above with respect to , then
Then for any integer
Especially, if and is bounded below with respect to , then
Following comparison theorem of difference equation is Theorem of [11, page 241].
Lemma 2.2.
Let , For any fixed is a nondecreasing function with respect to , and for , following inequalities hold: , If , then for all .
Now let us consider the following single species discrete model:
where and are strictly positive sequences of real numbers defined for and , Similarly to the proof of Propositions and in [12], we can obtain the following.
Lemma 2.3.
Any solution of system (2.7) with initial condition satisfies
where
The following lemma is direct conclusion of [1].
Lemma 2.4.
Let denote any positive solution of system (1.4).Then there exist positive constants such that
where
Proposition 2.5.
Suppose assumption (1.10) holds, then there exist positive constant and such that
Proof.
We first prove .
By Lemma 2.4 and by the first equation of system (1.4), we have
for sufficiently large, then
Thus
where
From the second equation of system (1.4), we have
Then, Lemma 2.1 implies that for any ,
where
For any small positive constant , there exists a such that
From the first equation of system (1.4), (2.18), and (2.20), we have
By Lemmas 2.2 and 2.3, we have
Setting in (2.22) leads to
Thus,
where
Second, we prove . For enough small , from the second equation of system (1.4), we have
for sufficient large . Hence
Thus, we obtain
This completes the proof.
3. An Example
In this section, we give an example to illustrate that (1.10) does not imply (1.9). Consider the twospecies system with delays and feedback controls for
We have
So
Therefore (1.10) holds.
But
Thus (1.9) does not hold.
References
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Kong, X., Chen, L. & Yang, W. Note on the Persistent Property of a Discrete LotkaVolterra Competitive System with Delays and Feedback Controls. Adv Differ Equ 2010, 249364 (2010). https://doi.org/10.1155/2010/249364
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DOI: https://doi.org/10.1155/2010/249364