Abstract
The permanence of a single-species population discrete model with feedback control is considered. We found that if we use the method of comparison theorem, then the additional condition, to some extent, is necessary. But for the system itself, this condition may not be necessary. Here, we use new methods instead of the comparison theorem to get the permanence of the system in consideration; the additional condition in Chen's paper (2007) is deleted.
1. Introduction
In [1], Li and Zhu investigated the single-species population discrete model with feedback control:which is a difference form of the single model with feedback control:
In [1, Theorem 2.1], under the assumptions that are all -periodic sequences, is a nonnegative integer, which may be zero, and is the first forward difference operator: They obtained what follows.
Lemma 1.1. System (1.1) has at least one positive -periodic solution.
The following notations and definition will be useful to our discussion.
Let denote the set of all bounded sequences , and let be the set of all such that . Given denoteAlso for , set
Definition 1.2. System (1.1) is said to be permanent if there exist two positive constants , such thatfor any solution of (1.1).
Recently, by using the comparison theorem, Chen [2] investigated the permanence of system (1.1) under the basic assumptions that
(H) are all bounded sequences.
He obtained what follows.
Lemma 1.3. Assume that (H) andhold, then system (1.1) is permanent, where
In [3, 4], we studied the discrete predator-prey system which takes the formUsing the method of comparison theorem, we obtained the following lemma.
Lemma 1.4. Assume
that the following conditions hold:
(H1)(H2)
Furthermore, assume thatholds. Then, system (1.8) is permanent.
We should point out that conditions (H1) and (H2) are sufficient for the permanence of system (1.8) (the reader can refer to [5]); that is, condition (1.9) is not necessary for its permanence.
We found that if we use the method of comparison theorem, then the additional condition, to some extent, is necessary. But for the system itself, this condition may not be necessary. Motivated by the above problem, we discuss the permanence of system (1.1) again; our investigation shows that condition (1.6) is also not necessary.
2. Main Results
In the remainder of this paper, for biological reasons, we only consider the solution of system (1.1) with initial condition
One can easily show that any solution of system (1.1) with initial condition (2.1) remains positive.
First, we state our main result below.
Theorem 2.1. Under the basic assumptions (H), the system (1.1) is permanent.
In order to prove our main result, firstly we give some lemmas which will be useful for the following discussion.
Lemma 2.2. Assume that and and further suppose that
(1)Then for any integerEspecially, if andis bounded above with respect to,
then(2)Then for any integerEspecially, ifand is bounded below with respect to,
then
Proof. Since the proof of (2) is similar to that of (1), we only need to prove (1). For any integer thenwhich implies thatFrom the above inequality, (2.4) is obvious and the proof is complete.
The following lemma can be found in [1, 6].
Lemma 2.3. Assume that satisfies and
(1)for whereis a positive constant and is a positive integer. Then,(2)forwhereis a positive constant and is a positive integer. Further assume that and Then,
The following two lemmas are direct conclusions of [2].
Lemma 2.4. There exists a positive constant such thatIn fact, one can choose
Lemma 2.5. There exists a positive constant such thatSimilarly, one can choose
Lemma 2.6. There exists a positive constant such that
Proof. By Lemmas 2.4 and 2.5
and by the first equation of system (1.1), we havefor sufficiently large; thenNotice that the sequence is bounded below; thusFor simplicity, we set thenFrom the second equation of
system (1.1), we haveThen, Lemma 2.2 implies that for
any Note thatHence, there exists a
positive integer such that for any solution of system (1.1), as In fact, we can choose then we getfor
Since is bounded above, set thenConsidering the first equation
of system (1.1), we have
In order to prove this lemma, by Lemma 2.2(2), for the
rest we only need to proveNotice thatHere, we use the Bernoulli
inequality for
Thus,where can be chosen as
This completes the proof.
Lemma 2.7. There exists a positive constant such that
Proof. Without loss of generality, for any there exists a positive integer such thatThen, the second equation
implies that
By Lemma 2.2(2), we havewhere can be chosen as
The proof is complete.
Proof of Theorem 2.1. From Lemmas 2.4–2.7 and the definition of permanence, the conclusion is obvious.
3. Discussion
In many situations, as our investigations show, the permanence and the existence of periodic solutions of a system are closely related with each other. But sometimes when we want to get the permanence of the system under the precondition that the periodic solution exists, we need some additional conditions. This mainly charges upon the method we used, especially when we use the comparison theorem. To make the comparison theorem holds, some additional conditions must be given, while to the system itself, these conditions may not be necessary. Just as in [2, 3], the conditions for the permanence of the system include some additional conditions besides the conditions for the existence of periodic solutions, while in this text and in [5], the conditions for the permanence of the system are exactly the same as the conditions for the existence of periodic solutions. This is because in this text and in [5] we use new methods instead of the comparison theorem.
Acknowledgments
This work is supported by the NSF of Ludong University (24070301, 24070302, 24200301) and by the Program for Innovative Research Team at Ludong University.