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A set of easily verifiable sufficient conditions are derived to guarantee the existence and the global stability of positive periodic solutions for two-species competitive systems with multiple delays and impulses, by applying some new analysis techniques. This improves and extends a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.

Throughout this paper, we make the following notation and assumptions:

let

then those spaces are all Banach spaces. We also denote that

In this paper, we investigate the existence, uniqueness, and global stability of the positive periodic solution for two corresponding periodic Lotka-Volterra competitive systems involving multiple delays and impulses:

In [

They had assumed that the net birth

In [

They had assumed that the net birth

Alvarez and Lazer [

They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (

However, the ecological system is often deeply perturbed by human exploitation activities such as planting, harvesting, and so on, which makes it unsuitable to be considered continually. For having a more accurate description of such a system, we need to consider the impulsive differential equations. The theory of impulsive differential equations not only is richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modeling of many real world phenomena [

For the sake of generality and convenience, we always make the following fundamental assumptions.

A function

for each

Under the above hypotheses (

The following lemmas will be used in the proofs of our results. The proof of Lemma

Suppose that (

if

if

From Lemma

The organization of this paper is as follows. In Section

In this section, we will introduce some concepts and some important lemmas which are useful for the next section.

Let

The set

Let

The region

By the definition of

Suppose that

Since

As

Let

The set

Assume that

Suppose that

Let

In the following section, we only discuss the existence and global asymptotic stability of positive periodic solutions of systems (

Since

In addition to (

Then systems (

Since the solutions of systems (

Clearly,

We now proceed to the discussion on the uniqueness and global stability of the

In addition to (

Then systems (

Letting