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 [

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 [

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

In this paper, for system (

Throughout this paper, we will introduce the following assumptions:

there exists constant

there exists constant

there exists constant

In addition, for a function

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:

Suppose that assumptions (

there exist positive constants

Further, we consider the following nonautonomous linear equation:

Suppose that assumption

there exists a positive constant

The proof of Lemma

Suppose that assumption

The proof of Lemma

Let

Motivated by the biological background of system (

Suppose that assumptions

Let

Consider the following auxiliary equation:

From the second equation of the system (

Suppose that assumptions

According to assumption

Let

We first prove that

Now, we prove the conclusion of Theorem

From Theorem

Assume that

So, for any

In Theorem

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:

Therefore, assumptions

This work is supported by the Sciences Foundation of Shanxi (2009011005-3) and the Major Subject Foundation of Shanxi.