Note on the Persistent Property of a Discrete Lotka-Volterra Competitive System with Delays and Feedback Controls
© Xiangzeng Kong et al. 2010
Received: 26 June 2010
Accepted: 12 September 2010
Published: 14 September 2010
A nonautonomous -species discrete Lotka-Volterra 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.
then exponentially for and where is a certain solution of a logistic equation. Teng  and Ahmad and Stamova  also studied the coexistence on a nonautonomous Lotka-Volterra 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 [1–9] and the references cited therein.
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 first-order forward difference operator ; .
In , Liao et al. obtained sufficient conditions for permanence of the system (1.4).
They obtained what follows.
It was shown that in  Liao et al. considered system (1.4) where all coefficients , , , , , and were assumed to satisfy conditions (1.9).
Our main results are the following Theorem 1.2.
Assume that (1.10) holds, then system (1.4) is permanent.
Therefore, we have improved the permanence conditions of  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 .
Following comparison theorem of difference equation is Theorem of [11, page 241].
Let , For any fixed is a nondecreasing function with respect to , and for , following inequalities hold: , If , then for all .
where and are strictly positive sequences of real numbers defined for and , Similarly to the proof of Propositions and in , we can obtain the following.
The following lemma is direct conclusion of .
We first prove .
This completes the proof.
3. An Example
Therefore (1.10) holds.
Thus (1.9) does not hold.
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