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Existence of positive periodic solutions for a generalized predator-prey model with diffusion feedback controls
Advances in Difference Equations volume 2013, Article number: 52 (2013)
Abstract
By employing the continuation theorem of coincidence degree theory, we derive a sufficient condition for the existence and attractivity of at least a positive periodic solution of the generalized predator-prey model with exploited term.
MSC:92D25, 34C25, 34K15.
1 Introduction
In recent years, the existence of positive periodic solutions of the prey-predator model has been widely studied [1–3]. The qualitative analysis of predator-prey systems is an interesting mathematical problem and has attracted a great attention of many mathematicians and biologists [4, 5]. Recently, Xu and Chen [6] investigated the two-species ratio-dependent predator-prey different model with time delay. Since a realistic model requires the inclusion of the effect of changing environment, recently, Shihua and Feng [7] have considered the following model:
where (), (), , , , and are strictly positive continuous w-periodic functions.
In the paper, we will study the following model:
where (), , , are the same as in model (1.1). Some assumptions on the above functions on () will be given in next section.
Our aim in this paper is to establish a sufficient condition for the existence and attractivity of at least a positive w-periodic solution of model (1.2).
2 Main result
To obtain the existence of positive periodic solutions of system (1.2), we summarize some concepts and results from [5] that will be basic for this section.
Let X, Z be Banach spaces, let be a linear mapping, and let be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if and ImL is closed in Z. If L is a Fredholm mapping of index zero, there exist continuous projectors and such that and . It follows that is invertible. We denote the inverse of that map by Kp. If Ω is an open-bounded subset of X, the mapping N will be called L-compact on if is bounded and is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism .
In the proof of our existence theorem, we will use the continuation theorem of Gaines and Mawhin [8].
Lemma 2.1 [8]
Let L be a Fredholm mapping of index zero and let N be L-compact on . Suppose the following:
-
(i)
for each , every solution x of is such that ;
-
(ii)
for each ;
-
(iii)
.
The has at least one solution in .
For convenience, we introduce the notations
where f is a continuous w-periodic function.
Our main result on the global existence of a positive periodic solution of system (1.2) is stated in the following theorem.
Theorem 2.1 Assume that
() there exists a constant A such that for , , when ,
() there exists a constant B such that for , when ,
() there exists a constant C () such that for , , when ,
() there exists a constant D () such that for , , when ,
() .
Then system (1.1) has at least one positive w-periodic solution.
Proof
Consider the system
Let , , then system (1.2) changes into system (2.1). Hence it is easy to see that system (2.1) has a w-periodic solution , then is a positive w-periodic solution of system (1.2). Therefore, for (1.2) to have at least one positive w-periodic solution, it is sufficient that (2.1) has at least one w-periodic solution. In order to apply Lemma 2.1 to system (2.1), we take
and
for any (or Z). Then X and Z are Banach spaces with the norm . Let
Then it follows that
and P, Q are continuous projectors such that
Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) is given by
Thus
and
where
and
Obviously, QN and are continuous. It is not difficult to show that is compact for any open bounded by using the Arzela-Ascoli theorem. Moreover, is clearly bounded. Thus, N is L-compact on with any open bounded set .
Now we reach the point where we search for an appropriate open bounded subset Ω for the application of the continuation theorem (Lemma 2.1). Corresponding to the operator equation , , we have
Assume that is a solution of system (2.2) for a certain .
Because of , there exist such that
It is clear that
From this and system (2.2), we obtain
There are two cases to consider for (2.3) and (2.4).
Case 1. Assume that , then .
From this and (2.3), we have
which, together with condition () in Theorem 2.1, gives
Thus
Case 2. Assume that , then .
From this and (2.4), we have
which, together with condition () in Theorem 2.1, gives
Thus
From Case 1 and Case 2, we obtain
From (2.5), we get
Thus
There are two cases to consider for (2.6) and (2.7).
Case 1. Assume that , then . From this and (2.5), we have
which, together with condition () in Theorem 2.1, gives
Hence
Case 2. Assume that , then . From this and (2.7), we have
which, together with condition () in Theorem 2.1, gives
Hence
From Case 1 and Case 2, we have
From Theorem 2.1(), we get
From (2.11)-(2.22), we obtain, for ,
and
Clearly, () are independent of λ. Denote , here is taken sufficiently large such that each solution of the system
satisfies , provided that system (2.23) has a solution or a number of solutions, and that
where will appear in below.
Now we take . This satisfies condition (i) of Lemma 2.1. When , u is a constant vector in with . If system (2.23) has one or more solutions, then
where are one constant.
If system (2.23) does not have a solution, then naturally
This shows that condition (ii) of Lemma 2.1 is satisfied finally. We will prove that condition (iii) of Lemma 2.1 is satisfied. We only prove that when , . When , u is a constant vector in with . Our proof will be broken into three steps as follows.
Step 1. We prove
To this end, we define the mapping by
where is a parameter, when , u is a constant vector in with . We will show that when , , if the conclusion is not true, i.e., the constant vector u with satisfies , then from
it follows the arguments of (2.11)-(2.22) that
Thus
which contradicts the fact that .
According to topological degree theory, we have
Step 2. We prove
where , are two chosen positive constants such that
To this end, we define the mapping by
where is a parameter. We will prove that when , . When , u is a constant vector in with . Now we consider two possible cases:
-
(i)
When , from condition (iii) in Theorem 2.1, we have . Moreover, , thus . Therefore, .
-
(ii)
When , if , from condition () in Theorem 2.1, we have . However, . Therefore, . If , we also consider two possible cases: (a) ; (b) . (a) When , from condition () in Theorem 2.1, we have
Therefore . (b) When , if , then from condition () in Theorem 2.1, we obtain . Consequently, . If , we can claim when , . Otherwise, from
we have
and
i.e.,
Thus
and
Therefore
which contradicts the fact that . Based on the above discussion, for any , we have . According to topological degree theory, we obtain
Step 3. We prove
To this end, we define the mapping by
where is a parameter and , are two chosen positive constants such that . We will prove that when , . If it is not true, then the constant vector u satisfies with . Thus we have
(2.24) implies
We claim that ; otherwise, if , then from condition () in Theorem 2.1, we have
Consequently,
which contradicts (2.23). We also claim that . If , then . However, .
Thus
which contradicts (2.24). (2.26) gives
that is,
Thus
and
Therefore
which leads to a contradiction. Therefore, by means of topological degree theory, we have
From the proof of the three steps above, we obtain
Because of condition () in Theorem 2.1, the system of algebraic equations
has a unique solution which satisfies
Thus
Therefore, from (2.20), we have
This completes the proof of Theorem 2.1. □
3 An example
Consider the system
where is a positive constant, all the parameters are positive continuous w-periodic functions with periodic .
In Theorem 2.1, , . It is easily shown that if , then and if , then . We also can show if
and if , then .
() ,
() .
By Theorem 2.1, we have the following theorem.
Theorem 3.1 If () and () hold, the system (3.1) has at least one positive w-periodic solution. Consider the system
where is a positive constant, all the parameters are positive continuous w-periodic functions with periodic .
In Theorem 2.1, , . It is easily shown that if , then and if , then . We also can show if , and if , then .
() ,
() .
By Theorem 2.1, we have the following theorem.
Theorem 3.2 If () and () hold, system (3.2) has at least one positive w-periodic solution.
References
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Acknowledgements
This research is partially supported by the National Natural Science Foundation of China (11071222, 11101126), the Hunan provincial Natural Science Foundation of China (12JJ3008) and the Research Fund of Hunan provincial Education Department (11B113).
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Each of the authors, CD and YW, contributed to each part of this study equally and read and approved the final version of the manuscript.
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Du, C., Wu, Y. Existence of positive periodic solutions for a generalized predator-prey model with diffusion feedback controls. Adv Differ Equ 2013, 52 (2013). https://doi.org/10.1186/1687-1847-2013-52
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DOI: https://doi.org/10.1186/1687-1847-2013-52