Applied and Computational Mathematics
Volume 5, Issue 5, October 2016, Pages: 193-201

The Stability Analysis of Two-Species Competition Model with Stage Structure and Diffusion Terms

Wang Hailing

Institute of Information Science and Technology, Xiamen University Tan Kah Kee Colledge, Xiamen, China

Wang Hailing. The Stability Analysis of Two-Species Competition Model with Stage Structure and Diffusion Terms. Applied and Computational Mathematics. Vol. 5, No. 5, 2016, pp. 193-201. doi: 10.11648/j.acm.20160505.12

Received: September 6, 2016; Accepted: October 7, 2016; Published: October 13, 2016

Abstract: In this paper, the author proposed and considered a reaction-diffusion equation with diffusion terms and stage structure. We discussed the stability of the positive equilibrium. By using the upper-lower solutions and monotone iteration technique, we obtained the zero steady state and the boundary equilibrium were linear unstable and the unique positive steady state was globally asymptotic stability. The traditional results are improved and this result applies to broader frameworks.

Keywords: Stage Structure, Reaction-Diffusion Equations, Equilibrium, Stability

Contents

1. Introduction

The growth of the species, often needs a process of development. Meanwhile, different species with different growth stage showed the different characteristics. So, studying the population model with stage structure has practical significance. But, a single population is not too much, each species must be affected by the other populations and the environment, so, in recent years, the literature on the two phase structure of single population was more, see [1, 2 ,3, 4, 5] etc. Among them, Chen Lansun in literature [1] listed some scholars’ research results. But, the stability of the equilibrium point on two population model was studied through ordinary differential equations in [2, 3, 4, 5]. Later, many scholars began to research the structure of three phase single population model. Gao Shujing [6] set up for a class of population model according to the young adults aged three stages. In 2005, Liang etc [7] established a class of population model which was divided into a pupa Larvae, adult three phase structure, Wu [8]studied the global asymptotic stability of a weakly-coupled reaction diffusion system in the three-species model. But in these papers, the authors did not notice the time delay. In fact, the development of population reproduction has some lag process, literature [9, 10, 11] considered the time delay on the basis of [6, 7], but, in these papers the author did not notice the free diffusion phenomena of the population. Literature [12] studied two species predator-prey with stage structure and diffusion terms which considered the effect of diffusion and phase structure. But the authors considered the species only spreading in the local scope. The local operator did not accurately describe the object of study behavior of space and time, therefore, we must introduce the convolution operator to describe the spatial diffusion process. On the basis of [12], we considered the following competition model with diffusion terms and stage structure:

(1)

Where,  respectively represent the population densities of the juvenile and adult of A at time t and location x,  represent the population densities of B at time t and location x,  represent the diffusion rates,  represent the death rate,  represent the birth rate,  represent the density coefficient,  represent the competition coefficient of adult A, B, The kernel  are inter-tribal and non-negative function satisfying

,

All the parameters are positive.

The paper is organized as follows: in section 2, we discuss the locally asymptotic stability between the zero balance and the boundary of equilibrium; in section 3, we use the upper-lower solutions and monotone iterative methods to discuss the global stability of the positive equilibrium point.

2. Equilibrium and Local Asymptotic Stability

The variables  of the system (1) have nothing with , so we need to consider the subsystems of the system (1):

In (1), let , , then the system (1) can rewrite following:

(2)

Lemma 2.1 Assume that , , or, , then the system(2)has four non-negative equilibrium:

, , ,

Where,

To study the asymptotic stability of the equilibrium by using of constant linear methods similar to [13], We introduce the transformation ,

, Then the system (2) Can translate into PFDE of :

(3)

Where,, , Then the characteristic equation for the linear part of system(3)can become

=0

Let

Then the characteristic equation becomes ,

Theorem 2.1 When , the equilibrium  is unstable.

Proof: for the equilibrium , the characteristic equation for the system (2) can become

If , which yields.

So, we get,

if ,then  holds.

In the same way, if , we can have .

Thus,  is unstable.

Theorem 2.2 When , the equilibrium  is local asymptotic stability and  is unstable.

Proof: for the equilibrium , the characteristic equation for the system (2) can become .

(i)    If , which yields, .

if , then

is contradiction with the suppose, so ;

(ii)  if , then  holds.

if , then

is contradiction with the suppose, so ;

Therefore, the equilibrium  is local asymptotic stability

By similar way, we can prove the balance  of that the system (2) is not stable.

Theorem 2.3 If , the equilibrium  is local asymptotic stability.

Proof: for the equilibrium , the characteristic equation for the system (2)can become

.

Since ,

we have

.

let

So we can get .

which yields , Further, we get .

Therefore, . if, then , so

Thereby,  which is contradiction with the suppose.

Therefore, the equilibrium  is local asymptotic stability.

The methods are also appropriate for a class of food chain system with stage structure. Such as

Where, are positive,  represents the population densities of the juvenile and adult at time t and location x, respectively represent the middle and top predators.

3. Global Stability

Using the upper-lower solution method and the monotone iterative method to consider the stability the following equations with the initial-boundary value problem:

(4)

Definiton 3.1 A pair of smooth function are said to be the coupled upper and lower solutions of problem (4),if in ,and the following differential inequalities hold

Lemma 3.1 If there exists a pair of upper and lower  of problem (4) and  is continue in , then the system (4) has the unique solution  satisfying .

Lemma 3.2 If  is continue in , and , then the system (4) has the unique solution.

Lemma 3.3 With the assuming of Lemma 3.2, if is the solution of the following problem

where,   then, .

Theorem 3.1 If , and let be the solution of the system (4), then , uniformly for.

Proof: let

Let  be the solutions of

(5)

Clearly  and  are the upper and lower solutions of problem (4), and by Lemma 3.1, we get

On the other hand, by Lemma 3.3, we have

(6)

And thus for any sufficiently small , there exists , such that ,

(7)

Let  be the solutions of

(8)

Then  and are a pair of upper and lower solutions of problem (4), and by Lemma 3.1, we get

By (7) and (8), we have

(9)

By the comparison principle, we get , where  are the upper and lower solutions of problem

(10)

by Lemma 3.3, we have

,

Therefore, we can conclude that

(11)

Where,

Furthermore, for any sufficiently small , there exists , such that

(12)

Let  be the solutions of

(13)

By definition 3.1,  and  are a pair of upper and lower solutions of problem (4), and by Lemma 3.1, we get

By (12) and (13), we have

(14)

By the comparison principle, we get , where  are the upper and lower solutions of problem

(15)

by Lemma 3.3, we have

,

Therefore we can conclude that

(16)

where,

It is obvious that .

Continue this process, we can get the following sequences

(17)

And satisfying

We need to testify the following

Let  in (17), we derive that

(18)

Which yields

since , system (4) has only zero solution with respect to .Therefore, from(18), we can get .

This completes the proof.

The methods are also appropriate for a class of cooperation model and Epidemic model with stage structure, and so on. for example:

Where, the parameters have the same the meanings with consistent.

4. Conclusion

Using of constant linear methods, we considered the local asymptotic stability; Employing the upper-lower solutions and monotone iterative methods, we considered the global stability of the positive equilibrium point about the competition model with diffusion terms and stage structure. The conclusions are also appropriate for the corresponding parabolic-ordinary differential system ( for some or all ). Besides, The conclusions are also appropriate for the predator-prey model and epidemic model and so on. So, The traditional results are improved and this result adds to the previous results and applies to broader frameworks. But, with the increase of the invasive species, we can study the multi-group reaction diffusion model in the next few years.

Acknowledgments

The paper was funded by the field incubation project 2015L02.

References

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