Research Article | Open Access
Positive Periodic Solutions of Nicholson-Type Delay Systems with Nonlinear Density-Dependent Mortality Terms
This paper is concerned with the periodic solutions for a class of Nicholson-type delay systems with nonlinear density-dependent mortality terms. By using coincidence degree theory, some criteria are obtained to guarantee the existence of positive periodic solutions of the model. Moreover, an example and a numerical simulation are given to illustrate our main results.
In the last twenty years, the delay differential equations have been widely studied both in a theoretical context and in that of related applications [1–4]. As a famous and common delay dynamic system, Nicholson’s blowflies model and its modifications have made remarkable progress that has been collected in  and the references cited there in. Recently, to describe the dynamics for the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics which belong to the Nicholson-type delay differential systems, Berezansky et al. , Wang et al. , and Liu  studied the problems on the permanence, stability, and periodic solution of the following Nicholson-type delay systems: where , and , .
In , Berezansky et al. also pointed out that a new study indicates that a linear model of density-dependent mortality will be most accurate for populations at low densities and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates. Consequently, Berezansky et al.  presented an open problem: to reveal the dynamics of the following Nicholson’s blowflies model with a nonlinear density-dependent mortality term: where is a positive constant and function might have one of the following forms: or with positive constants .
Most recently, based upon the ideas in [5–8], Liu and Gong  established the results on the permanence for the Nicholson-type delay system with nonlinear density-dependent mortality terms. Consequently, the problem on periodic solutions of Nicholson-type system with has been studied extensively in [10–13]. However, to the best of our knowledge, there exist few results on the existence of the positive periodic solutions of Nicholson-type delay system with . Motivated by this, the main purpose of this paper is to give the conditions to guarantee the existence of positive periodic solutions of the following Nicholson-type delay system with nonlinear density-dependent mortality terms: under the admissible initial conditions where , and are all bounded continuous functions, and , , .
For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function defined on , let and be defined as We also assume that and are all -periodic functions, , and , .
Let be the set of all (nonnegative) real vectors; we will use to denote a column vector, in which the symbol denotes the transpose of a vector. We let denote the absolute-value vector given by and define . For matrix , denotes the transpose of . A matrix or vector means that all entries of are greater than or equal to zero. can be defined similarly. For matrices or vectors and , (resp. ) means that (resp. ). We also define the derivative and integral of vector function as and .
The organization of this paper is as follows. In the next section, some sufficient conditions for the existence of the positive periodic solutions of model (1.3) are given by using the method of coincidence degree. In Section 3, an example and numerical simulation are given to illustrate our results obtained in the previous section.
2. Existence of Positive Periodic Solutions
In order to study the existence of positive periodic solutions, we first introduce the continuation theorem as follows.
Lemma 2.1 (continuation theorem ). Let and be two Banach spaces. Suppose that is a Fredholm operator with index zero and is L -compact on , where is an open subset of . Moreover, assume that all the following conditions are satisfied:(1), for all , ;(2), for all ;(3)the Brouwer degree Then equation has at least one solution in .
Our main result is given in the following theorem.
Theorem 2.2. Suppose Then (1.3) has a positive -periodic solution.
Proof. Set and . Then (1.3) can be rewritten as
As usual, let for all be Banach spaces equipped with the supremum norm . For any , because of periodicity, it is easy to see that is -periodic. Let
It is easy to see that
Thus, the operator is a Fredholm operator with index zero. Furthermore, denoting by the inverse of , we have It follows that Obviously, and are continuous. It is not difficult to show that is compact for any open bounded set by using the Arzela-Ascoli theorem. Moreover, is clearly bounded. Thus is -compact on with any open bounded set .
Considering the operator equation , we have Suppose that is a solution of (2.10) for some .
Firstly, we claim that there exists a positive number such that . Integrating the first equation of (2.10) and in view of , it results that which together with (2.4) implies that Similarly, we have It follows from (2.12) and (2.13) that
Since , there exist such that It follows from (2.12) and (2.14) that which implies that Using (2.14) yields In particular, It follows that Again from (2.14), we have Similarly, we can obtain Since , from (2.10), we have Hence, from (2.24) and the fact that , we have Noting that is strictly monotone increasing on and it is clear that there exists a constant such that In view of (2.25) and (2.27), we get In the same way, there exists a constant such that Again from (2.14), (2.15), (2.28), and (2.29), we get Then, we can choose two sufficiently large positive constants and such that
Let be a fix constant such that Then (2.22), (2.23), and (2.31) imply that , if is solution of (2.10). So we can define an open bounded set as such that there is no and such that . That is to say for all .
Secondly, we prove that for all . That is for all .
If , then is a constant vector in , and there exists some such that . Assume , so that . Then, we claim If for , it follows from (2.2) and (2.8) that Hence, which implies This is a contradiction and implies that for .
If for , it follows from (2.2) and (2.8) that Consequently, a contradiction to the choice of . Thus, for .
Similarly, if , we obtain Consequently, (2.33) and (2.39) imply that for all .
Furthermore, let and define continuous function by setting
For all , then there exists some such that . There are two cases: or . When or , from (2.33) and (2.39), it is obvious that or . Similarly, if or , it results that or . Hence for all .
Finally, using the homotopy invariance theorem, we obtain It then follows from the continuation theorem that has a solution which is an -periodic solution to (2.4). Therefore is a positive -periodic solution of (1.3) and the proof is complete.
3. An Example
In this section, we give an example to demonstrate the results obtained in the previous section.
Example 3.1. Consider the following Nicholson-type delay system with nonlinear density-dependent mortality terms: Obviously, , , , , , , , , then which means the conditions in Theorem 2.2 hold. Hence, the model (3.1) has a positive -periodic solution in , where . The fact is verified by the numerical simulation in Figure 1.
Remark 3.2. Equation (3.1) is a form of Nicholson’s blowflies delayed systems with nonlinear density-dependent mortality terms, but as far as we know there are no that results can be applicable to (3.1) to obtain the existence of positive -periodic solutions. This implies the results of this paper are essentially new.
The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. In particular, the authors expresses the sincere gratitude to Prof. Bingwen Liu for the helpful discussion when this work is carried out. This work was supported by National Natural Science Foundation of China (Grant nos. 11201184 and 11101283), Innovation Program of Shanghai Municipal Education Commission (Grant no. 13YZ127), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant nos. Y6110436, LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).
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Copyright © 2012 Wei Chen and Lijuan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.