## Theory and Applications of Periodic Solutions and Almost Periodic Solutions

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# Permanence, Extinction, and Almost Periodic Solution of a Nicholson's Blowflies Model with Feedback Control and Time Delay

**Academic Editor:**Yonghui Xia

#### Abstract

A Nicholson's blowflies model with feedback control and time delay is studied. By applying the comparison theorem of the differential equation and fluctuation lemma and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence, extinction, and existence of a unique globally attractive positive almost periodic solution of the system are obtained. It is proved that the feedback control variable and time delay have no influence on the permanence and extinction of the system.

#### 1. Introduction

Let be any continuous bounded function defined on ; we set

In order to describe the dynamics of Nicholson’s blowflies, Gurney et al. [1] proposed the following mathematical model in 1980: where is the size of the population at time , is the maximum per capita daily egg production rate, 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. Kulenović and Ladas [2], Győri and Ladas [3], and Győri and Trofimchuk [4] investigated the oscillatory behaviors of the solutions of (2). For the attractivity, Kulenović et al. [5] and So and Yu [6] have shown that, when , every positive solution of (2) tends to a positive equilibrium as if Reference [5] further showed that, for , every nonnegative solution of (2) tends to zero as , and for , (2) is uniformly persistent. Furthermore, Li and Fan [7] considered the following nonautonomous equation: where , and are all positive periodic functions. The authors show that (4) has a unique globally attractive periodic positive solution if Their results improved the results of Saker and Agarwal [8] who considered system (4) with ( is a constant).

Recently, Wang and Fan [9] proposed the following discrete Nicholson’s blowflies model with feedback control: Sufficient conditions are established for the permanence and the extinction of the system (6). They show that the bounded feedback terms do not have any influence on the permanence or extinction of (6). The authors in [9] also proposed the following continuous model: however, they did not discuss the dynamic behaviors of the system (7). Considering that continuous models can excellently show the dynamic behaviors of those populations who have a long life cycle, overlapping generations, and large quantity, sufficient conditions for the permanence, global attractivity, and the existence of a unique, globally attractive, strictly positive almost periodic solution of the system (7) with are obtained by Yu [10]. As pointed out by Nindjin et al. [11], time delay plays an important role in many biological dynamical systems, being particularly relevant in ecology and a model with time delay is a more realistic approach to the understanding of dynamics. Hence, it is necessary to study the model (7) which contains time delay.

In the following discussion, we always assume that are all continuous, positive almost periodic functions. Also, from the viewpoint of mathematical biology, we consider (7) together with the following initial conditions: where and are continuous on . It is not difficult to see that solutions of (7) and (8) are well defined for all and satisfy

The aim of this paper is, by constructing a suitable Lyapunov functional and applying the analysis technique of Feng et al. [12], to obtain sufficient conditions for the existence of a unique globally attractive positive almost periodic solution of the system (7) with initial condition (8).

This paper is organized as follows. In Section 2, by applying the analysis technique of [13, 14] and Fluctuation lemma [15, 16], we present the permanence and the extinction of model (7) and (8). In Section 3, by constructing a suitable Lyapunov functional, a sufficient conditions for the existence of a unique globally attractive positive almost periodic solution of the system (7) and (8). Examples together with their numeric simulations are stated in Section 4. For more works on almost periodic solutions of the ecosystem with feedback control, one could refer to [17–23] and the references cited therein.

#### 2. Permanence and Extinction

Now let us state several lemmas which will be useful in proving the main result of this section.

Lemma 1 (see [13]). *Assume that , is a boundedness continuous function and . Further suppose that *(i)*
then for all ,
Particularly, if is bounded above with respect to , then
*(ii)*Also suppose that
then for all ,
Particularly, if is bounded above with respect to , then
*

Lemma 2 (see [20]). *If , and , when and , one has
**
If , , and , when and , one has
*

Lemma 3. *Let be any solution of system (7) with initial condition (8); there exists positive numbers and , which are independent of the solution of the system, such that
*

*Proof. *Let be any solution of system (7) satisfying initial condition (8). Since for , according to the positivity of solution and the first equation of system (7), for ,
where is the mathematical constant.

By applying Lemma 1(i) to (19), we have
Hence, there exists such that
Equation (21) together with the second equation of (7) leads to
Using Lemma 1(i) again, one has
Obviously, are independent of the solution of system (7). Equations (20) and (23) show that the conclusion of Lemma 3 holds. The proof is completed.

Lemma 4. *Assume that **
**holds. Then there exists positive constants and , which are independent of the solution of system (7), such that
*

*Proof. *Let be any solution of system (7) satisfying initial condition (8). From Lemma 3, there exists a such that for all ,, , where . According to the first equation of system (7) and the positivity of solution, for ,
where .

Integrating both sides of (26) from to leads to
or
Particularly, taking , one can get
Substituting (29) into the second equation of system (7) leads to
Applying Lemma 1(i) to the above differential inequality, for , one has
where we used the fact .

Note that there exists a , such that , as , where . In fact, we can choose . And so, fixing , combined with (31), we can obtain
for all , where .

Considering that , for , from the first equation of system (7) and the positivity of the solution, for , we can get
By Lemma 2, we have
Thus, there exists such that for all ,
Equation (35) together with the second equation of (7) leads to
By applying Lemma 1(ii) to the above differential inequality, we have
Obviously, are independent of the solution of system (7). Equations (34) and (37) show that the conclusion of Lemma 4 holds. The proof is completed.

From Lemmas 3–4 and the definition of permanence, we can obtain the following conclusion.

Theorem 5. *Assume that holds; then system (7) with initial condition (8) is permanent.*

As a direct corollary of Theorem 2 in [24], from Theorem 5, we have the following.

Corollary 6. *Suppose that holds; then system (7) admits at least one positive -periodic solution if are all continuous positive -periodic functions.*

Theorem 7. *Let be any positive solution of the system (7) with initial condition (8). Assume that
**
holds; then
*

*Proof. *Firstly, from the the first equation of (7),
If the former case of (38) holds, then
which shows that .

If the latter case of (38) holds, from (40) we have or is decreasing; therefore, . Hence . We only need to show that . Otherwise, if , then there exists a , such that for . According to the Fluctuation lemma, there exists a sequence as such that , , as . We can choose a large enough number such that for ; hence, for all .

For , together with the first equation of (7) leads to
Let ; we obtain that or which is impossible. Hence, or

Now, we come to prove that
For any , according to (43), there exists a , such that
Then, for ,
Thus, by applying Lemma 1(i) to the above differential inequality, we have
which implies that
The proof is complete.

#### 3. Existence of a Unique Almost Periodic Solution

Now, we give the definition of the almost periodic function.

*Definition 8 (see [25, 26]). *A function , where is an -vector, is a real scalar, and is an -vector, is said to be almost periodic in uniformly with respect to , if is continuous in and , and if for any , there is a constant , such that in any interval of length , there exists such that the inequality
is satisfied for all . The number is called an -translation number of .

*Definition 9 (see [25, 26]). *A function is said to be an asymptotically almost periodic function if there exists an almost periodic function and a continuous function such that

We denote by the set of all solutions of system (7) satisfying for all .

Lemma 10. * One has *

* Proof. *Since are almost periodic functions, there exists a sequence , as such that
as uniformly on . Suppose is a solution of (7) satisfying , for . Obviously, the sequence is uniformly bounded and equicontinuous on each bounded subset of . Therefore, by the Ascoli-Arzela theorem, there exists a subsequence of , which we still denote by , such that uniformly on each bounded subset of . For any , we may assume that for all . For , we have
Applying Lebesgue’s dominated convergence theorem and letting in to previous equations, we obtain
for all . Since is arbitrarily given, is a solution of system (7) on . It is clear that , for . That is to say, . This completes the proof.

Lemma 11 (see [27]). *Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then, .*

Theorem 12. *In addition to , further suppose that * * there exists a , such that
**where is defined in (23); then system (7) with initial conditions (8) is globally attractive. That is to say, for any two positive solutions, one has
*

*Proof. *Let and be any two positive solutions of system (7)-(8). Theorem 5 implies there exist positive constants ,, and such that for
where and are defined in Lemma 3 and Lemma 4. Set , where
Calculating the upper right derivatives of along the solution of (7) leads to
where we used the elementary mean value theorem of differential calculus and lies between and . Then, for , we have
Hence, by (58), we can have
Calculating the upper right derivatives of along the solution of (7), one has
According to (58), (66), and condition , we can obtain
Integrating both sides of (62) from to leads to
Then,
Hence, . By system (7) and Theorem 5, we get , and their derivatives are bounded on , which implies that is uniformly continuous on . By Lemma 11, we obtain
The proof of Theorem 12 is complete.

Theorem 13. *Suppose all conditions of Theorem 12 hold; then there exists a unique almost periodic solution of systems (7) and (8).*

*Proof. *According to Lemma 10, there exists a bounded positive solution of (7) with initial condition (8). Then there exists a sequence , as , such that is a solution of the following system:
According to Theorem 5 and the fact that are all continuous, positive almost periodic functions, we know that both and its derivative function are uniformly bounded; thus, are uniformly bounded and equi-continuous. By Ascoli’s theorem, there exists a uniformly convergent subsequence such that for any , there exists a with the property that if , then
That is to say, are asymptotically almost periodic functions. Hence there exists two almost periodic functions and two continuous functions such that
where
are also almost periodic functions.

Therefore,
On the other hand,
So exist. Moreover,
These show that satisfied system (7). Hence, is a positive almost periodic solution of (7). Then, it follows from Theorem 12 that system (7) has a unique positive almost periodic solution. The proof is completed.

Without the feedback terms, that is , and (7) becomes the following equation: Equation (73) with periodic coefficients has been studied by Li and Fan [7] and Saker and Agarwal [8] with . Since the periodic case is a special case of almost periodic, hence, as a direct corollary of Theorem 13, we have the following.

Corollary 14. *Suppose are all continuous positive -periodic functions and for ; then (73) has a unique globally attractive periodic positive solution.*

*Remark 15. *Li and Fan in [7] show that (73) has a unique globally attractive periodic positive solution if , which is the same as Corollary 14. Thus, Theorem 13 supplements and generalizes results in [7, 8].

#### 4. Examples and Numeric Simulations

Now we give several examples together with their numeric simulations to show the feasibility of our main results.

*Example 16. *Consider the following example:
In this case, corresponding to system (7), we have . According to the proof of Lemmas 3 and 4, one has
Hence,
Thus, all the conditions of Theorem 13 are satisfied, and so, there exists a unique almost periodic solution of systems (74). Figure 1 shows this property.

*Example 17. *Consider the following example:

In this case, we have
Hence, By Theorem 7, we know that any positive solution of system (77) satisfies . Numerical simulation also confirms our result (see Figure 2).

#### 5. Conclusion

In this paper, we consider a Nicholson’s blowflies model with feedback control and time delay. It is shown that feedback control variable and time delay have no influence on the permanence and extinction of the system. Also, by constructing a suitable Lyapunov functional, a set of sufficient conditions which ensure the existence of a unique globally attractive positive almost periodic solution of the system is established. Moreover, compared with the main result of the relative discrete model (see [9]), we can see that the continuous and discrete models have similar results on permanence and the extinction of the Nicholson’s blowflies model with feedback control and time delay. At the end of this paper, two examples together with their numerical simulations show the verification of our main results. Our results supplement and generalize the results in [7, 8].

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Copyright © 2013 Haihui Wu and Shengbin Yu. 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.