Dynamics of a Two Species Competitive System with Pure Delays

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Received 6 May 2016; accepted 1 July 2016; published 4 July 2016

1. Introduction

Population competition systems of Lotka-Volterra type have been investigated extensively in recent years [1] - [5] . The basic and the simplest two species nonautonomous competitive system for Lotka-Volterra type is as following form

(1)

There is an extensive literature concerned with the properties of system (1) that has been discussed by many authors [1] - [4] .

However, in the real world, the growth rate of a natural species will not often respond immediately to changes in its own population or that of an interacting species, but will rather do so after a time lag [6] . Recently, many people are doing research on the dynamics of population with time delays, which is useful for the control of the population of mankind, animals and the environment. Therefore, it is essential for us to investigate population systems with time delays. In this paper, we investigate the following two species Lotka-Volterra type competitive systems with pure discrete time delays

(2)

By using the technique of comparison method and Liapunov function method, we will establish some sufficient conditions on the boundedness, permanence, existence of positive periodic solution and global attractivity of the system.

The organization of this paper is as follows. In the next Section, we will present some basic assumptions and main definition and lemmas. In Section 3, conditions for the positivity and boundedness are considered. In the final Section, we considered the conditions for the permanence, existence of positive periodic solution and global attractivity of the system.

2. Preliminaries

In system (2), we have that represent the density of two competitive species at time t, respectively; represent the intrinsic growth rate of species at time t, respectively; and represent the intra patch restriction density of species at time t, respectively; and represent the competitive coefficients between two species at time t, respectively. represent the time delay in the model. In this paper, we always assume that

are positive constants, , are continuous positive functions.

are positive constants, , are continuous positive ω-peri- odic functions.

From the viewpoint of mathematical biology, in this paper for system (2) we only consider the solution with the following initial conditions

(3)

where are nonnegative continuous functions defined on satisfying with.

In this paper, for any continuous function we denote

Now, we present some useful definitions.

Definition 1. (see [7] ) System (2) is said to be permanent if there exists a compact region such that every solution of system (2) with initial conditions (3) eventually enters and remains in the region D.

Definition 2. (see [8] ) System (2) is said to be global attractive, if for any two positive solutions

and of system (2), one has

The following two lemmas will be used in the proof of the main results of system (2).

Lemma 1. (see [9] ) Consider the following equation:

where, , we have

1) If, then.

2) If, then.

Lemma 2. (see [10] ) Let be a nonnegative function defined on, such that is integrable on and uniformly continuous on. Then,.

3. Positivity and Boundedness

In this section, we will obtain positivity and boundedness of system (2). The following Lemma is about the positivity of system (2).

Lemma 1. Set is positively invariant for system (2).

The proof of Lemma1 is simple, and here we omit it.

The following theorem is about the boundedness of system (2).

Theorem 1. Suppose that assumption (H_{1}) holds, then there exist positive constants such that

for any positive solution of system (2).

Proof: Let be a solution of system (2). Firstly, it follows from the first equation of system (2) that for, we have

We consider the following auxiliary equation

By Lemma 2, we derive

By comparison, there exists a such that for.

Next, by using an argument similar in the above, there exist a such that, where

This completes the proof.

The following theorem is about the global attractivity of system (2). Firstly, for convenience we denote the following functions

where,

(4)

where, and are constants.

4. Permanence, Existence of Positive Periodic Solution and Global Attractivity

In this section, we will obtain the permanence, existence of positive periodic solution and global attractivity of system (2). First we obtain the global attractivity of system (2).

Theorem 2. Suppose that (H_{1}) and there exists a constant such that

Then system (2) has a positive solution which is globally attractive.

Proof: Let and are any two positive solutions of system (2). From Theorem 1, choose positive constants such that

(5)

for all. Let

Calculating the upper right derivation of along system (2) for all, we have

(6)

Define

where

Calculating the upper right derivative of and from (6), we have

(7)

Define

where

Further, we define a Liapunov function as follows

Calculating the upper right derivation of, from (6) and (7) we finally can obtain for all

(8)

From assumption (H_{2}), there exists a constant and such that for all we have

(9)

Integrating from to t on both sides of (8) and by (9) produces

(10)

hence, bounded on and we have

(11)

From the boundedness of and (11), we can obtain that and their derivatives remain bounded on. Therefore is uniformly continuous on. By Barbalat’s theorem it follows that

Therefore,

This completes the proof of Theorem 2.

From the global attractivity of system (2), we have the following result.

Corollary 1. Suppose that the conditions of Theorem 2 hold, then system (2) is permanent.

As a direct corollary of [11] (Theorem 2), from Corollary 1, we have the following result.

Corollary 2. Suppose that the conditions of Theorem 2 and () hold, then system (2) has a positive ω-peri- odic solution which is globally attractive.

Acknowledgements

This work was supported by the Natural Science Foundation of Xinjiang University (Starting Fund for Doctors, Grant No. BS130102, BS150202) and the National Natural Science Foundation of China (Grant No. 11401509, 11261056).

NOTES

^{*}Corresponding author.

References

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