A COMMENSAL SYMBIOSIS MODEL WITH HOLLING TYPE FUNCTIONAL RESPONSE AND NON-SELECTIVE HARVESTING IN A PARTIAL CLOSURE∗†

Yu Liu,Xinyu Guan

(College of Math.and Computer Science,Fuzhou University,Fuzhou 350116,Fujian,PR China)

1 Introduction

During the last decade,many scholars have focused on the study of the commensal symbiosis model[1-12],which describes a relationship only favorable to the one side and has no influence to the other side.

Sun and Wei[4]first proposed a intraspecific commensal model:

They investigated the local stability of all equilibrium points.However,they did not give any information about the global dynamic behaviors of the system.Zhu,Xu and Li[11]proposed a commensalism model,in which one of the species could not be survived independently.By giving the phase trajectories analysis of the system,they are able to analyse the stability property ofallthe equilibria of the system.Han and Chen[5]incorporated the feedback control variables into the commensal symbiosis modelsystem(1.1).By constructing some suitable Lyapunov function,they showed that the system admits a unique globally stable positive equilibrium;Xie et al.[6],Xue et al.[13,15]studied the dynamic behaviors of the discrete commensalsymbiosis model;Miao et al.[16]further studied the dynamic behaviors of the periodic Lotka-Volterra commensal symbiosis model with impulse.

One could see that all of the above systems are based on the traditional Lotka-Volterra model,which made the assumption that the influence of the second species to the first one is linearize.This may not be suitable.Recently,Wu,Liand Zhou[1]further assumed that the relationship between two species is of Holling type functional response,and they established the following two species commensalsymbiosis model

where ai,bi,i=1,2,p and c1are all positive constants with p≥1.By applying the Dulac criterion,they showed that the unique positive equilibrium of the system is globally asymptotically stable.

It brings to our attention that all of the above study did not consider the influence of harvesting.Indeed,to obtain the resource for the development of the human being,harvesting of the species is necessary.Stimulated by the works of Chakraborty,Das and Kar[13],we further incorporated the non-selective harvesting term into system(1.2),and this leads to the following system

where ai,bi,qi,i=1,2,p,E and c1are all positive constants with p≥1 and E being the combined fishing effort used to harvest,and m(0

The aim of this paper is to investigate the local and global stability property of the possible equilibria of system(1.3).We arrange the paper as follows:In the next section,we will investigate the existence and local stability property of the equilibria of system(1.1).In Section 3,we will investigate the global stability property of the equilibria;in Section 4,an example together with its numeric simulation is presented to show the feasibility of our main results.

2 The Existence and Local Stability of the Equilibria

The equilibria of system(1.3)is determined by equations

The system always admits the boundary equilibriums

where

Obviously,if a1>E mq1,then A2is nonnegative,and if a2>E mq2,then A3is nonnegative.To ensure the positivity of A4,one needs to assume that a1+

Concerned with the local stability property of the above four equilibria,we have the following theorem.

Theorem 2.1

(4)assume that

and

hold,thenA4(x∗,y∗)is locally asymptotically stable.

ProofThe Jacobian matrix of system(1.3)is calculated as

Then the Jacobian matrix of system(1.3)about the equilibriums

respectively.One could easily see that under the assumption of Theorem 2.1,all of the above three matrix have two negative eigenvalues,which means that

are all locally asymptotically stable.

Note that A4(x∗,y∗)satisfies the equations

The Jacobian matrix about the equilibrium A4is given by

The eigenvalues of the above matrix areλ1= −b1x∗<0,λ2= −b2y∗<0.Hence,A4(x∗,y∗)is locally stable.

This ends the proof of Theorem 2.1.

3 Global Stability of the Positive Equilibrium

In this section,we try to obtain some sufficient conditions which could ensure the globalasymptotical stability of the equilibria.

As a direct corollary of Lemma 2.2 of F.D.Chen[14],we have the following lemma.

Lemma 3.1Ifa>0,b>0and≥x(b−ax),whent≥0andx(0)>0,we have

Ifa>0,b>0and≤x(b−ax),whent≥0andx(0)>0,we have

Lemma 3.2[15]System

has a unique globally attractive positive equilibriumy∗=

Theorem 3.1

Proof(1)From a10 such that

From the second equation of(1.3)we have

Hence

For the aboveε>0,there exists a T1>0,such that

For t>T1,from the first equation of system(1.3),we have

Hence

(2)Similarly to the analysis of(1),for arbitrary enough smallε>0,there exists a T2>0,such that

For t>T2,from the first equation of system(1.3),we have

It follows from Lemma 3.1 that

On the other hand,from the first equation of system(1.3),we also have

It follows from Lemma 3.1 that

It follows from(3.7)and(3.8)that

Sinceεis any arbitrary small positive constant,setting ε→0 in the above inequality leads to

(3)Consider the function

noting that

Hence,f(x)is a monotonic function of x.Forε>0 enough small,without loss of generality,we may assume thatε<.From(3.3)and the monotonic increasing of the function

From the second equation of(1.3)we have

It follows from Lemma 3.2 that

Forε>0 enough small,there exists an enough large T3>0,such that

For t>T3,from the first equation of system(1.3),(3.9)and(3.10),we have

Hence

(4)Similarly to the analysis of(3),from the second equation of system(1.3)and Lemma 3.1,we have

Hence,for arbitrary small positive constantε>0(without loss of generality,we may assume thatε0 such that

For t>T4,from the first equation of system(1.3)and(3.11),we have

It follows from Lemma 3.1 and the monotonic of the function f(x)that

For t>T4,from the first equation of system(1.3)and(3.11),we also have

It follows from Lemma 3.1 and the monotonic of the function f(x)that

(3.12)together with(3.13)shows that

Settingε→0 in(3.14)leads to

Consequently,A4(x∗,y∗)is globally attractive.

This completes the proof of Theorem 3.1.

4 Numeric Simulations

Now let us consider the following example.

Example 4.1Consider the following system

In this system,corresponding to system(1.3),we take a1=a2=c1=1,b1=b2=2,E=4,q1=0.5,q2=1.For the system without harvesting,that is,m=0,from Wu,Li and Zhou[1],the system admits a unique positive equilibriumwhich is globally asymptotically stable.

(1)Take m=0.6,then

and so,from Theorem 3.1(1),A1(0,0)is globally attractive,see Figure 1;

(2)take m=0.3,then

holds.From Theorem 3.1(2),A2(0.2,0)is globally attractive,see Figure 2;

(3)take m=0.1,then

and

hold.From Theorem 3.1(4),A4(0.5153846154,0.3)is globally attractive,see Figure 3.

Figure 1:Numeric simulations of system(4.1)with m=0.6 and the initial conditions(x(0),y(0))=(0.4,2),(1,0.3),(0.02,0.02),(1,2)and(0.1,2),respectively.

Figure 2:Numeric simulations of system(4.1)with m=0.5 and the initial conditions(x(0),y(0))=(0.4,2),(1,0.3),(0.2,0.2),(1,2)and(0.1,2),respectively.

Figure 3:Numeric simulations of system(4.1)with m=0.1 and the initial conditions(x(0),y(0))=(0.4,2),(1,0.3),(0.2,0.2),(1,2)and(0.1,2),respectively.

5 Conclusion

A commensal symbiosis model with Holling type functional response and nonselective harvesting in a partial closure is proposed and studied in this paper.Already,Wu,Li and Zhou[1]showed that for the system without harvesting(that is,m=0),the system could admits a unique positive equilibrium,which means that the two species could be coexist in a stable state.However,our results shows that depending on the choice of the parameter m,which represents the fraction of the stock could be permit for harvesting,the system may have four kinds of dynamic behaviors:Extinction,partial survival of the one species,or the coexist of the two species in a stable state.That is,by introducing the harvesting,the dynamic behaviors of the system becomes complicated,and it is weakly in the sense that part or all of the species will be driven to extinction.

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