Abstract
We consider a discrete Nicholson’s blowflies model involving a linear harvesting term. Under appropriate assumptions, sufficient conditions are established for the existence and exponential convergence of positive almost periodic solutions of this model. To expose the effectiveness of the main theorems, we support our result by a numerical example.
MSC:39A11.
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1 Introduction
In [1], Gurney et al. proposed the following nonlinear autonomous delay equation:
to describe the population of the Australian sheep blowfly and to agree with the experimental data obtained by Nicholson in [2]. Here is the size of the population at time t, β is the maximum per capita daily egg production, is the size at which the blowfly population reproduces at its maximum rate, α is the per capita daily adult death rate, and τ is the generation time. Equation (1.1) is recognized in the literature as Nicholson’s blowflies model. The dynamical behavior of solutions of this model and its various modifications have been extensively studied by many authors during the last couple of decades. For more details, we suggest to the readers that they consult [3–18].
Biologists have proposed that the process of harvesting population is of great significance in the exploitation of biological resources, i.e., in fishery, forestry and wildlife management. This justification has attracted the attention of many mathematicians who are interested in studying the dynamic behavior of population models governed by differential or difference equations [19, 20]. In their recent paper [21], in particular, Berezansky et al. have put forward a question about the asymptotic behavior of the well-known Nicholson’s blowflies model involving a linear harvesting term of the form
Recently, there have appeared many results concerning the investigation of periodic and almost periodic behaviors of system (1.2) via employing several utilities such as fixed-point theorems and the coincidence degree theory [22–25, 27, 28].
The delay difference equation
is a discrete analogue of the delay differential equation (1.1). For the purpose of convenience, however, we shall consider a discrete Nicholson’s blowflies model involving a linear harvesting term of the form
where and are almost periodic sequences.
A primary purpose of this paper is to study the almost periodic dynamics of equation (1.3). We shall employ the contraction mapping principle to establish sufficient conditions for the existence of an almost periodic solution for (1.3). By constructing an appropriate Lyapunov functional, we also prove that the solutions of (1.3) converge exponentially to an almost periodic solution. We provide a numerical example to illustrate the effectiveness of the main theorems. To the best of authors’ knowledge, no paper regarding the investigation of almost periodicity of equation (1.3) has been published. Thus, our paper is different and presents a new approach.
2 Preliminaries
For a bounded sequence g defined on , we define and as follows:
In the sequel, we assume that
and
Let , and . For each , we define the norm of φ as . Denote for all . For any , it is easy to see that there is a unique solution of (1.3) with
and for all .
Definition 1 A sequence is said to be almost periodic if for , there is a constant such that in any interval of length there exists a number such that the inequality
is satisfied for all .
The following basic results are essential for proving the main results in Section 3. They were provided in [26] for arbitrary time scale . If , then these results can be formulated as follows.
Theorem 1 ([26])
Let be an almost periodic sequence, then is bounded on .
Theorem 2 ([26])
If are almost periodic sequences, then and fg are almost periodic.
Theorem 3 ([26])
If is an almost periodic sequence, then is almost periodic if and only if is bounded on where .
Theorem 4 ([26])
If is an almost periodic sequence and is defined on the value field of , then is almost periodic.
We assume that
A.1 There exist two constants and such that
The following result tells that every solution of (1.3) persists.
Lemma 1 Let A.1 hold. Then for , the solution of (1.3) and (2.3) satisfy
Proof Set . Let be an interval such that
We claim that
For the sake of contradiction, we assume that (2.6) is not true. Then, one can find such that
In view of (1.3), A.1, (2.7) and the fact that , we have
which is a contradiction, and this implies that (2.6) holds. Next, we show that
On the contrary, assume that there exists such that
In virtue of A.1 and (2.6), we obtain
In view of (1.3), A.1, (2.9), (2.10) and the fact that , we have
which is a contradiction, and this implies that (2.8) holds. In view of (2.6) and (2.8), it follows that relation (2.4) is true. Hence the proof is complete. □
3 The main results
Consider the linear difference system
where , and is a matrix sequence. In what follows, we denote by any convenient norm either of a vector or of a matrix.
Definition 2 The difference system (3.1) is said to possess an exponential dichotomy on if there exists a projection P, that is, an matrix P such that , and constants , such that
where is the fundamental solution matrix of (3.1) and .
Consider the following almost periodic difference system:
where is an almost periodic matrix sequence and is an almost periodic vector sequence.
Theorem 5 If the linear system (3.1) admits an exponential dichotomy, then system (3.2) has a bounded solution in the form
where is the fundamental solution matrix of (3.1).
Proof By direct substitution, we obtain
It follows that
Moreover, we have
where . By Theorem 1, is a bounded solution of system (3.2). The proof is complete. □
Theorem 6 ([26])
Let be an almost periodic sequence on and
then the linear system
admits an exponential dichotomy on .
Set
If we define the norm =, for any , then one can easily deduce that is a Banach space.
We assume that
A.2 .
Theorem 7 Let A.1 and A.2 hold. Then, there exists a unique positive almost periodic solution of (1.3) in .
Proof For any , we consider an auxiliary equation
Since , it follows from Theorem 6 that the linear system
admits an exponential dichotomy on . By Theorem 5 and Theorem 6, we deduce that system (3.6) has a bounded solution of the form
In virtue of Theorem 2, Theorem 3, Theorem 4, and using the almost periodicity of and the fact that the uniform limit of almost periodic sequences is also almost periodic, we deduce that is also almost periodic.
Define a mapping by setting
It is easy to see that is a closed subset of . For any , we have
By the fact that , we obtain
or
Using that , we end up with
On the other hand, we have
By virtue of the fact that , we obtain
This tells that the mapping T is a self-mapping from to .
Let . Then
In virtue of the fact that , we observe that
By A.1, we get
Therefore, by (1.3), (2.1) and (3.12), we have
Therefore, we end up with
which implies by A.2 that the mapping T is contractive on . Therefore, the mapping T possesses a unique fixed point such that . Thus, is an almost periodic solution of (1.3) in the . The proof is complete. □
We assume that
A.3 .
Theorem 8 Let A.1 and A.3 hold. Further, assume is a positive almost periodic solution of (1.3) in the set . Then, the solution of (1.3) with converges exponentially to as .
Proof Set and , where . Then
The result of Lemma 1 tells that is positive and bounded on and
Define a function by setting
It is clear that Φ is continuous on . Then, by A.3 we have
which implies that there exist two constants and such that
We consider the discrete Lyapunov functional
Calculating the difference of along the solution of (3.13), we have
for all .
Let
Then, we claim that
Assume, on the contrary, that there exists such that
which implies that
In virtue of (3.12), (3.18) and (3.20), we obtain
Thus,
which contradicts (3.16). Hence (3.19) holds. It follows that for all . The proof is complete. □
4 An example and concluding remark
In this section, we construct an example to demonstrate the results obtained in the previous sections. A concluding remark is also reported.
Example 1 Consider the following Nicholson’s blowflies model with a linear harvesting term of the form
where
and
It is clear that
and
Thus, conditions (2.1) and (2.2) hold. Let and . Then
and this shows that condition A.1 is satisfied. It remains to check conditions A.2 and A.3. However, one can see the validity of these conditions since
Therefore, we conclude that all assumptions of Theorem 7 and Theorem 8 are fulfilled. Hence, system (4.1) has a positive almost periodic solution in . Moreover, if , then converges exponentially to as .
Remark 1 It is well known that the optimal management of renewable resources has direct relationship to the sustainable development of exploitation of population. One way to handle this is to study population models involving harvesting, dispersal or competition. Assuming that harvesting is a function of the delayed estimate of the true population, Nicholson’s blowflies model involving a linear harvesting term has been the object of recent research.
Following this trend, we study the almost periodic behavior of a discrete analogue of Nicholson’s blowflies model involving a linear harvesting term of form (1.3). It is worth mentioning here that most of the discrete analogues of Nicholson’s models investigated in the literature have involved a linear part of form . In this paper, however, we consider Nicholson’s model of form (1.3) to guarantee the convergence of the series appears in the solution representation (3.8).
A result concerning the persistence of the solutions is provided prior to proving the main theorems. Under the assumptions A.1-A.3, sufficient conditions are established for the existence and exponential convergence of positive almost periodic solutions of (1.3). Our approach is based on the contraction mapping principle as well as on the construction of an appropriate Lyapunov functional.
The results of this paper could be generalized to Nicholson’s model involving multiple delays and multiple linear harvesting terms. As Nicholson’s model under consideration is involving a linear harvesting term, one can easily figure out that some of the results reported in the literature might be no longer applicable for proving the existence and exponential convergence of almost periodic solutions of (1.3). This implies that the main theorems of this paper improve and extend some of previously obtained results.
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The authors would like to express their sincere thanks to the editor Prof. Dr. Elena Braverman for handling our paper during the reviewing process and to the referees for suggesting some corrections that help making the content of the paper more accurate.
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Alzabut, J., Bolat, Y. & Abdeljawad, T. Almost periodic dynamics of a discrete Nicholson’s blowflies model involving a linear harvesting term. Adv Differ Equ 2012, 158 (2012). https://doi.org/10.1186/1687-1847-2012-158
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DOI: https://doi.org/10.1186/1687-1847-2012-158