## Qualitative Theory of Functional Differential and Integral Equations

View this Special IssueResearch Article | Open Access

# Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System

**Academic Editor:**Samir Saker

#### Abstract

This paper is concerned with the existence of multiple periodic solutions for discrete Nicholson’s blowflies type system. By using the Leggett-Williams fixed point theorem, we obtain the existence of three nonnegative periodic solutions for discrete Nicholson’s blowflies type system. In order to show that, we first establish the existence of three nonnegative periodic solutions for the -dimensional functional difference system , where is not assumed to be diagonal as in some earlier results. In addition, a concrete example is also given to illustrate our results.

#### 1. Introduction and Preliminaries

In 1954 Nicholson [1] and later in 1980 Gurney et al. [2] proposed the following delay differential equation model: where is the size of the population at time , is the maximum per capita daily egg production, is the size at which the population reproduces at its maximum rate, is the per capita daily adult death rate, and is the generation time.

Now, Nicholson’s blowflies model and its various analogous equations have attracted more and more attention. There is large literature on this topic. Recently, the study on Nicholson’s blowflies type systems has attracted much attention (cf. [3–8] and references therein). In particular, several authors have made contribution on the existence of periodic solutions for Nicholson’s blowflies type systems (see, e.g., [6, 7]). In addition, discrete Nicholson’s blowflies type models have been studied by several authors (see, e.g., [9–12] and references therein).

Stimulated by the above works, in this paper, we consider the following discrete Nicholson’s blowflies type system: where is a constant, is a nonnegative integer, and , , , and are all -periodic functions from to .

In fact, there are seldom results concerning the existence of* multiple* periodic solutions for Nicholson’s blowflies type equations. It seems that the only results on this topic are due to Padhi et al. [13–15], where they established several existence theorems about multiple periodic solutions of Nicholson’s blowflies type equations. In addition, recently, several authors have investigated the existence of almost periodic solutions for Nicholson’s blowflies type equations (see, e.g., [11, 16, 17] and references therein). However, to the best our knowledge, there are few results concerning the existence of multiple periodic solutions for Nicholson’s blowflies type systems. That is the main motivation of this paper.

Next, let us recall the Leggett-Williams fixed point theorem, which will be used in the proof of our main results.

Let be a Banach space. A closed convex set in is called a cone if the following conditions are satisfied: (i) if , then for any ; (ii) if and , then .

A nonnegative continuous functional is said to be concave on if is continuous and

Letting , , and be three positive constants and letting be a nonnegative continuous functional on , we denote In addition, we call that is increasing on if for all with .

Lemma 1 (see [18]). *Let be a cone in a Banach space , let be a positive constant, let be a completely continuous mapping, and let be a concave nonnegative continuous functional on with for all . Suppose that there exist three constants , , and with such that *(i)*, and for all ;*(ii)* for all ;*(iii)* for all with .**Then has at least three fixed points , , and in . Furthermore, , and .*

Throughout the rest of this paper, we denote by the set of all integers, by the set of all real numbers, and by the space of all -periodic functions , where is a fixed positive integer. It is easy to see that is a Banach space under the norm where . In addition, we denote

#### 2. Main Results

To study the existence of multiple periodic solutions for system (2), we first consider the following more general -dimensional functional difference system: where, for every , is -periodic and nonsingular matrix, and is -periodic in the first argument and continuous in the second argument.

To note that the existence of periodic solutions for system (7) and its variants had been of great interest for many authors (see, e.g., [19–25] and references therein) is needed. However, in some earlier works (see, e.g., [21]) on the existence of periodic solutions for system (7), the matrix is assumed to be diagonal. In this paper, we will remove this restrictive condition by utilizing an idea in [22], where the authors studied the existence of periodic solutions for a class of nonlinear neutral systems of differential equations.

Let ,

We first present some basic results about and .

Lemma 2. *For all with , the following assertions hold: *(i)*,*(ii)*,*(iii)*,*(iv)*.*

*Proof. *One can show (i) and (ii) by some direct calculations and noting that . So we omit the details. In addition, the assertion (iii) follows from the assertion (i) and the assertion (iv) follows from the assertion (ii).

By using Lemma 2, we can get the following result.

Lemma 3. *A function is a -periodic solution of system (7) if and only if is a -periodic function satisfying
*

*Proof. **Sufficiency*. Assume that is a -periodic function satisfying (9); that is,
Then, we have
where
Thus, we conclude that is a -periodic solution of system (7).*Necessity*. Let be a -periodic solution of system (7). Then, we have
which yields
For all , we have
which yields
Letting and noting that is -periodic, we get
Noting that
we conclude
That is, (9) holds. This completes the proof.

Let

Now, we introduce a set It is not difficult to verify that is a cone in . Finally, we define an operator on by

Theorem 4. *Assume that and the following assumptions hold. **(H0)**, for all and , and for all , , and .**(H1)**There exist two constants such that
**(H2)**There exists a constant such that , and
**Then system (7) has at least three nonnegative -periodic solutions.*

*Proof. *Firstly, by (H0) and noting that , is an operator from to . Secondly, noting that is continuous for the second argument, by similar proof to [21, Lemma 2.5], one can show that is completely continuous.

Let
It is easy to see that is a concave nonnegative continuous functional on and .

Now, we show that maps into . For every , we have and for all . Then, by (H1), we have
Similarly, for every , it follows from (H1) that
That is, condition (ii) of Lemma 1 holds.

Let . Next, let us verify condition (i) of Lemma 1. It is easy to see that the set

In addition, for every , we have , , and for all . Then, by (H2), we get
which means that condition (i) of Lemma 1 holds.

It remains to verify that condition (iii) of Lemma 1 holds. Let with ; we have and
which yields
Then, we have

Then, by Lemma 1, we know that has at least three fixed points in . Then, it follows from Lemma 3 that system (7) has at least three nonnegative -periodic solutions.

Now, we apply Theorem 4 to Nicholson’s blowflies system (2). Let , and let , , , , and be as in Theorem 4.

Corollary 5. *Assume that , and (), , and are all nonnegative for and . Then the system (2) has at least three nonnegative -periodic solutions provided that , and
*

*Proof. *We only need to verify that all the assumptions of Theorem 4 are satisfied. Firstly, it is easy to see that (H0) holds. Let
Secondly, let us check (H1). In fact, one can choose sufficiently small such that, for all with , there holds
In addition, for all , we have
So, letting
we conclude that (H1) holds.

It remains to verify (H2). For all with and , by using (34), we have
This completes the proof.

Next, we give a concrete example for Nicholson’s blowflies type system (2).

*Example 6. *Let , , , , and
By a direct calculation, we can get
Then, we have and . In addition, we have and
So, by Corollary 5, we know that system (2) has at least three nonnegative -periodic solutions.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are grateful to the referees for valuable suggestions and comments. Hui-Sheng Ding acknowledges support from the NSF of China (11101192), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province.

#### References

- A. J. Nicholson, “An outline of the dynamics of animal populations,”
*Australian Journal of Zoology*, vol. 2, pp. 9–65, 1954. View at: Google Scholar - W. S. Gurney, S. P. Blythe, and R. M. Nisbet, “Nicholson's blowflies revisited,”
*Nature*, vol. 287, pp. 17–21, 1980. View at: Google Scholar - B. Liu, “The existence and uniqueness of positive periodic solutions of Nicholson-type delay systems,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 6, pp. 3145–3151, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Liu and S. Gong, “Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 4, pp. 1931–1937, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Berezansky, L. Idels, and L. Troib, “Global dynamics of Nicholson-type delay systems with applications,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 1, pp. 436–445, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Chen and L. Wang, “Positive periodic solutions of Nicholson-type delay systems with nonlinear density-dependent mortality terms,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 843178, 13 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Zhou, “The positive periodic solution for Nicholson-type delay system with linear harvesting terms,”
*Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems*, vol. 37, no. 8, pp. 5581–5590, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. H. Saker and S. Agarwal, “Oscillation and global attractivity in a periodic Nicholson's blowflies model,”
*Mathematical and Computer Modelling*, vol. 35, no. 7-8, pp. 719–731, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. H. Saker, “Oscillation of continuous and discrete diffusive delay Nicholson's blowflies models,”
*Applied Mathematics and Computation*, vol. 167, no. 1, pp. 179–197, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. W.-H. So and J. S. Yu, “On the stability and uniform persistence of a discrete model of Nicholson's blowflies,”
*Journal of Mathematical Analysis and Applications*, vol. 193, no. 1, pp. 233–244, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Alzabut, Y. Bolat, and T. Abdeljawad, “Almost periodic dynamics of a discrete Nicholson's blowflies model involving a linear harvesting term,”
*Advances in Difference Equations*, vol. 2012, article 158, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. O. Alzabut, “Existence and exponential convergence of almost periodic solutions for a discrete Nicholson's blowflies model with a nonlinear harvesting term,”
*Mathematical Sciences Letters*, vol. 2, pp. 201–207, 2013. View at: Google Scholar - S. Padhi, C. Qian, and S. Srivastava, “Multiple periodic solutions for a first order nonlinear functional differential equation with applications to population dynamics,”
*Communications in Applied Analysis*, vol. 12, no. 3, pp. 341–351, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Padhi, S. Srivastava, and J. G. Dix, “Existence of three nonnegative periodic solutions for functional differential equations and applications to hematopoiesis,”
*Panamerican Mathematical Journal*, vol. 19, no. 1, pp. 27–36, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Padhi and S. Pati, “Positive periodic solutions for a nonlinear functional differential equation,”
*Georgian Academy of Sciences A. Razmadze Mathematical Institute: Memoirs on Differential Equations and Mathematical Physics*, vol. 51, pp. 109–118, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. O. Alzabut, “Almost periodic solutions for an impulsive delay Nicholson's blowflies model,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 1, pp. 233–239, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-S. Ding and J. J. Nieto, “A new approach for positive almost periodic solutions to a class of Nicholson's blowflies model,”
*Journal of Computational and Applied Mathematics*, vol. 253, pp. 249–254, 2013. View at: Publisher Site | Google Scholar | MathSciNet - R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,”
*Indiana University Mathematics Journal*, vol. 28, no. 4, pp. 673–688, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. G. Dix, S. Padhi, and S. Pati, “Multiple positive periodic solutions for a nonlinear first order functional difference equation,”
*Journal of Difference Equations and Applications*, vol. 16, no. 9, pp. 1037–1046, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. N. Raffoul, “Positive periodic solutions of nonlinear functional difference equations,”
*Electronic Journal of Differential Equations*, no. 55, pp. 1–8, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. N. Raffoul and C. C. Tisdell, “Positive periodic solutions of functional discrete systems and population models,”
*Advances in Difference Equations*, no. 3, pp. 369–380, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. N. Islam and Y. N. Raffoul, “Periodic solutions of neutral nonlinear system of differential equations with functional delay,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 2, pp. 1175–1186, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Jiang, J. Wei, and B. Zhang, “Positive periodic solutions of functional differential equations and population models,”
*Electronic Journal of Differential Equations*, no. 71, pp. 1–13, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zeng, L. Bi, and M. Fan, “Existence of multiple positive periodic solutions for functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 2, pp. 1378–1389, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Zhang, D. Zhu, and P. Bi, “Existence of periodic solutions of a scalar functional differential equation via a fixed point theorem,”
*Mathematical and Computer Modelling*, vol. 46, no. 5-6, pp. 718–729, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Hui-Sheng Ding and Julio G. Dix. 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.