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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 101764, 6 pages
http://dx.doi.org/10.1155/2013/101764
Research Article

Global Attractivity of a Diffusive Nicholson's Blowflies Equation with Multiple Delays

Department of Mathematics and System Science, College of Science, National University of Defense Technology, Changsha 410073, China

Received 29 January 2013; Accepted 3 April 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Xiongwei Liu and Xiao 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.

Abstract

The present paper considers a diffusive Nicholson's blowflies model with multiple delays under a Neumann boundary condition. Delay independent conditions are derived for the global attractivity of the trivial equilibrium and the positive equilibrium, respectively. Two open problems concerning the stability of positive equilibrium and the occurrence of Hopf bifurcation are proposed.

1. Introduction

Since blowflies are important parasites of the sheep industry in some countries such as Australia, based on the experimental data of Nicholson [1, 2], Gurney et al. [3] first proposed Nicholson’s blowflies equation where is the size of the adult blowflies population at time ; is the maximum per capita daily egg production rate; is the size at which the blowflies population reproduces at its maximum rate; is the per capita daily adult death rate; is the generation time. For this equation, global attractivity and oscillation of solutions have been investigated by several authors (see [49]).

It is impossible that the size of the adult blowflies population is independent of a spatial variable; therefore, Yang and So [10] investigated both temporal and spatial variations of the diffusive Nicholson’s blowflies equation under Neumann boundary condition and gave the similar sufficient conditions for oscillation of all positive solutions about the positive steady state. Whereafter, many authors studied the various dynamical behaviors for this equation; we refer to Lin and Mei [11], Saker [12], Wang and Li [13], and Yi and Zou [14].

Meanwhile, one can consider a nonlinear equation with several delays because of variability of the generation time; for this purpose, Györi and Ladas [15] and Kulenović and Ladas [6] proposed the following generalized Nicholson’s blowflies model:

Luo and Liu [16] studied the global attractivity of the nonnegative equilibria of (3).

It is of interest to investigate both several temporal and spatial variations of the blowflies population using mathematical models. Hereby, in this paper, we consider the following system: with Neumann boundary condition and initial condition where , and , , are all positive constants, is a bounded domain with a smooth boundary , , denotes the exterior normal derivative on , and is Hölder continuous in with .

Though the global attractivity of the nonnegative equilibria of (2) has been studied by Yang and So [10] and Wang and Li [13, 17], they just gave some sufficient conditions. Furthermore, as far as we know, the stability for partial functional differential equations with several delays was investigated by few papers. Motivated by the above excellent works, in this paper, we consider the global attractivity of the nonnegative equilibria of the systems (4)–(6) and present some conditions which depend on coefficients of the systems (4)–(6). When , our results complement those in Yang and So [10] and Wang and Li [13].

It is not difficult to see that if , then (4) has a unique nonnegative equilibrium and if , then (4) has a unique positive equilibrium .

The rest of the paper is organized as follows. We give some lemmas and definitions in Section 2 and state and prove our main results in Section 3. In Section 4, several simulations are obtained to testify our results, and some unsolved problems are discussed.

2. Preliminaries

In this section, we will give some lemmas which can be proved by using the similar methods as those in Yang and So [10].

Lemma 1. (i) The solution of (4)–(6) satisfies for .
(ii) If on , then the solution of (4)–(6) satisfies for .

Next, we will introduce the concept of lower-upper solution due to Redlinger [18] as adapted to (4)–(6).

Definition 2. A lower-upper solution pair for (4)–(6) is a pair of suitably smooth function and such that(i) in ,(ii) and satisfy for all with , and(iii).

The following lemma is a special case of Redlinger [19].

Lemma 3. Let be a lower-upper solution pair for the initial boundary value problem (4)–(6). Then, there exists a unique regular solution of (4)–(6) such that on .

The following lemma gives us boundedness of the solution .

Lemma 4. (i) The solution of (4)-(6) satisfies
(ii) There exists a constant such that on .

Proof. Let be the solution of the following Cauchy problem:
Solving the equation, we have
Taking then is a lower-upper solution pair for (4)–(6). In fact, for any with , , one can get
By Lemma 3, there is a unique regular solution such that
Note that
Therefore, the formula (8) is correct, and there exists one such that for any and
So we complete Lemma 4.

3. Main Results and Proofs

Theorem 5. Assume that , then every solution of (4)–(6) tends to (uniformly in ) as .

Proof. By Lemma 4, without loss of generality, let for . Under the condition , we can get
Define and to be the solutions of the following two delay equations, respectively:
By using the similar methods to prove Lemma 4, we can get that under the condition , and here and are the solutions of (17).
Because of , for any , , one can get
Therefore, from Definition 2, is a lower-upper pair of (4)-(5) with initial condition on . Consequently, by Lemma 3, we have
By Theorem 1 of Luo and Liu [16], it follows from that the solutions and of (17) both satisfy
Hence, we complete the proof of Theorem 5.

Theorem 6. If , then every nontrivial solution of (4)–(6) satisfies

Proof. Let , then the function is increasing on and decreasing on , , for . Let , then it is not difficult to verify that the function satisfies the following conditions: the function is increasing on and decreasing on , , for and for .
There are now two possible cases to consider.
Case  1 (). In view of Lemma 4, we may also assume without loss of generality that every solution of (4)–(6) satisfies
Let , , and . By (23), we have
From Lemma 1(ii), let
Let . Now, we define two sequences and to satisfy, respectively,
We prove that and are monotonic and bounded. First of all, we prove that is monotonically increasing, and is the least upper bounded. Note and , we have
By induction and direct computation, we have
Similarly, we have
Define and to be the solutions of the following differential equations, respectively:
It follows from (24) and (25) that for any . Consider (30), for any , we have
Therefore, from Definition 2, is a lower-upper pair of (4)-(5) with initial condition on . Consequently, by Lemma 3, we have
Note that is monotonically decreasing for and , while is monotonically increasing for and . Hence,
Define and to be the solutions of the following differential equations, respectively:
Repeating the above procedure, we have the following relation:
By (28) and (29), and taking limits on both sides of (35), we have which implies
Case  2 (). Similarly, let and be the same as in the proof of Case 1; we can also get (35). Hence, the proof of Theorem 6 is complete.

Remark 7. Our main results are also valid when does not depend on a spatial variable in (4).

4. Numerical Simulations and Discussion

In this section, we will give some numerical simulations to verify our main results in Section 3 and present several interesting phenomena by simulations that we cannot give a theoretical proof. We just consider the case in (4).

4.1. Numerical Simulations

Different parameters will be used for simulations, and some data come from [20]. Figure 1 corresponds to the case with , , , , , and , and under the above conditions, we have . We choose the initial condition , , and the solution is decreasing and almost zero at time 160.

101764.fig.001
Figure 1: Parameters: , , , , , and . Initial condition is , .

Figure 2 corresponds to the case with , , , , , and , and under the above conditions, we have and . Choose the initial condition , . From Figure 2, we can observe that the solution oscillates around 13 and 14 days; however, tends to as time tends to 100 days. Therefore, Figures 1 and 2 support our main results (Theorems 5 and 6).

101764.fig.002
Figure 2: Parameters: , , , , , , and . Initial condition is , .
4.2. Discussion

In Section 3, we obtain two main results under the conditions and , which are independent of the delays , . A natural problem is what will happen when and the delays , are changed.

It is similar to Theorem 3 in Luo and Liu [16]; we present the following open problems.

Open Problem 1. If and , then every nontrivial solution of (4)–(6) satisfies

Figure 3 corresponds to the case with , , , , , , and , and initial condition is , . Under the above conditions, we have and . Sufficient conditions are dependent on coefficients and delay for the global attractivity of equilibria , and Figure 3 shows that the Open Problem 1 is right, but we cannot prove that.

101764.fig.003
Figure 3: Parameters: , , , , , , and . Initial condition is , .

From Figure 4, we have and . The condition is not satisfied, but is still globally attractive.

101764.fig.004
Figure 4: Parameters: , , , , , , and . Initial condition is , .

From Figure 5, we have and . The condition is not satisfied, but the global attractivity is not true. Moreover, Figure 5 shows that there is a periodic solution, which is very interesting. We guess that the reason is that the system brings Hopf bifurcation as the parameters change. Therefore, we state the following open problem.

101764.fig.005
Figure 5: Parameters: , , , , , , and . Initial condition is , .

Open Problem 2. Under suitable conditions, the systems (4)–(6) will lead to Hopf bifurcation.

Remark 8. Now, we have not intensively studied these two problems. Because the nonmonotonicity of the nonlinear term in (4) makes it very difficult for us to solve Open Problem 1, and we cannot prove Open Problem 2 because of multiple delays.

Acknowledgment

Project was supported by Hunan Provincial Natural Science Foundation of China (12jj4012) and Research Project of National University of Defense Technology (JC12-02-01).

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