The Permanence in a Single Species Nonautonomous System with Delays and Feedback Control
We consider a single species nonautonomous system with delays and feedback control. A general criterion on the permanence for all positive solutions is established. The results show that the feedback control does not influence the permanence of species.
As we well know, a single species without feedback control is very important on mathematical ecology and has been studied in many articles. Many important results on the permanence, extinction, global asymptotical stability, and their special cases of periodic and almost periodic system can be found in [1–6].
However, we note that ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecosystem is the question of whether or not an ecosystem can withstand those unpredictable forces which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. In 1993, Gopalsamy and Weng  introduced a feedback control variable into the delay logistic model and discussed the asymptotic behavior of solution in logistic models with feedback controls, in which the control variables satisfy certain differential equation. In the recent years, the population dynamical systems with feedback controls have been studied in many articles, for example, see [8–13] and references cited therein.
Motivated by the previous works, we focus our attention on the permanence of species for the following single specie non-autonomous systems with delays and feedback control
The main purpose of this paper is to establish a new general criterion for the permanence of system (1.1), which is described by integral form and independent of feedback control. The paper is organized as follows. In the next section, we will give some assumptions and useful lemmas. In Section 3, some new sufficient conditions which guarantee the permanence of all positive solutions for system (1.1) are obtained. In the last section, we will give an example to illustrate the conclusions obtained in this paper.
In this paper, for system (1.1) we denote that is the density of the species at time , is the control variable, and , , represent the intrinsic growth rate and density-dependent coefficient of the species at time . The function is bounded continuous defined on ; functions , , , , , and are continuous, bounded, and nonnegative defined on ; functions and are nondecrease defined on and satisfy , and . Furthermore, there exist positive constants , and such that and for all .
Throughout this paper, we will introduce the following assumptions:()there exists constant such that ()there exists constant such that
()there exists constant such that
In addition, for a function defined on set , we denote
Now, we state several lemmas which will be useful in the proving of main results in this paper.
First, we consider the following nonautonomous logistic equation: where functions , are bounded and continuous on . Furthermore, for all . We have the following result which is given in  by Teng and Li.
Further, we consider the following nonautonomous linear equation:
where functions and are bounded continuous defined on , and for all . We have the following result.
Lemma 2.3. Suppose that assumption holds. Then for any constants and there exist constants and such that for any and with when for all one has where is the solution of (2.7) with initial condition .
3. Main Results
Let . We define the Banach space of bounded continuous functions with the supremum norm defined by . By the fundamental theory of functional differential equation, we know that for any system (1.1) has a unique solution satisfying the initial condition
Motivated by the biological background of system (1.1), in this paper we are only concerned with positive solutions of system (1.1). It is not difficult to see that the solution of system (1.1) is positive, if the initial functions satisfy and and for all
Theorem 3.1. Suppose that assumptions hold. Then there exists constant such that for any positive solutions of system (1.1).
Proof. Let be any positive solution of system (1.1). Since
for all where is the initial time.
Consider the following auxiliary equation:
from assumptions and and according to Lemma 2.1, there exists constant such that for the solution of (3.3) with initial condition By the comparison theorem, we have From this, we further obtain Then, we obtain that for any constant there exists constant such that
From the second equation of the system (1.1), we have
Hence, we further have for all Consider the following auxiliary equation: from assumption and the conclusions of Lemma 2.2, we can get constant such that for any there exists constant such that for the solution of (3.10) with initial condition By the comparison theorem, we have Hence, we further obtain Choose the constant , then we finally obtain This completes the proof.
Theorem 3.2. Suppose that assumptions hold. Then there exists constant such that for any positive solutions of system (1.1).
Proof. According to assumption we can choose constants and such that for all we have
We consider the following equation:
where is parameter. By Lemma 2.3, for given and positive constant there exist constants and such that for any and when for all we have
where is the solution of (3.17) with initial condition
Let such that for all
We first prove that
In fact, if (3.20) is not true, then there exists a positive solution of system (1.1), a constant such that for all Then for all we have Let be the solution of (3.17) with initial condition by the comparison theorem, we have In (3.18), we choose and since for all we obtain Hence, from (3.22) we further obtain Considering the first equation (1.1), for any we have Integrating (3.25) form to we have Obviously, from (3.19) and (3.26), we obtain as which leads to a contradiction.
Now, we prove the conclusion of Theorem 3.2. In fact, if it is not true, then there exists a sequence of initial functions such that, for the solution of the system (1.1)
where and satisfies the initial condition From (3.20) and (3.27), for every m there are two time sequences and , satisfying and such that
From Theorem 3.1, we can choose a positive constant such that and for all Further, there is an integer such that for all Let for any we have
where Integrating the above inequality from to we further have Consequently, We can choose a large enough such that For any , and from (3.30) we can obtain
Assume that is the solution of (3.17) with the initial condition , then we have
In (3.18), we choose and , since for all we have for all Using the comparison theorem it follows that for all , and .
So, for any , and from (3.30) and (3.38), it follows
Integrating the above inequality from to then from (3.19), (3.29), and (3.30), we obtain which leads to a contradiction. Therefore, this contradiction shows that there exists constant such that for any positive solutions of system (1.1). This completes the proof.
Remark 3.3. In Theorem 3.2, we note that decided by (1.1), which is independent of the feedback controls. So, the feedback controls have no influence on the permanence of system (1.1). That is, there is the permanence of the species as long as feedback controls (human activities) should be kept in a certain range. In the range, the permanence of the species will not be influenced by the controls.
4. An Example
In this section we will give an example to illustrate the conclusion obtained in the above section. We will consider the following single species system with delays and feedback control:
where we have
Obviously, , , , , are continuous, bounded, and nonnegative defined on ; functions and are nondecrease defined on and satisfy , and . Choose the constants then we easily obtain
Therefore, assumptions hold, we obtain that the species is persistent.
This work is supported by the Sciences Foundation of Shanxi (2009011005-3) and the Major Subject Foundation of Shanxi.