THE STABILITY OF A REACTION-DIFFUSION LOTKA-VOLTERRA COMPETITIVE SYSTEM WITH NONLOCAL DELAYS AND FEEDBACK CONTROLS??

Zhong Li,Zhenliang Zhu

(College of Math.and Computer Science,Fuzhou University,Fuzhou 350108,Fujian,PR China)

Abstract In this paper,we consider a Lotka-Volterra competitive system with nonlocal delays and feedback controls.Using the Lyapunov functional and iterative technique method,we investigate the global stability and extinction of the system.Also,we show the in fluence of feedback controls on dynamic behaviors of the system.Some examples are presented to verify our main results.

Keywords extinction;feedback control;reaction-diffusion;Lotka-Volterra;nonlocal delays

1 Introduction and Main Results

In this paper,we consider the following reaction-diffusion Lotka-Volterra competitive system with nonlocal delays and feedback controls

for t>0,x ∈ (0,π),under the homogeneous Neumann boundary conditions

and initial conditions

In system(1.1),uidenotes the population density of the i-th species;videnotes the feedback control variable;b1and b2are the intrinsic growth rates;a11and a22are the rates of the intra-specific competition of the first and second species respectively;a12and a21are the rates of the inter-specific competition of the first and second species respectively;ci,ei,diare coefficients of the feedback control variable;Diis the diffusion rate.All the parameters in system(1.1)are positive constants.The boundary conditions(1.2)imply that the populations and feedback control variable do not move across the boundary x=0,π.We assume that the kernel Gi(x,y,t)fi(t)depends on both the spatial and the temporal variables.The delay in this type of model formulation is called a spatio-temporal delay or nonlocal delay(as we shall show below how Giare chosen).

The following two-species autonomous competitive system

where bi,aij,i,j=1,2 are positive constants,has been discussed in many books on mathematical ecology(for example[1]).If the coefficients of system(1.4)satisfy,then system(1.4)has a unique positive equilibriumwhich is globally attractive,that is,all positive solutions of system(1.4)satisfy.If the coefficients of system(1.4)satisfy,then system(1.4)is extinct,that is,all positive solutions of system(1.4)satisfy

In[2],the authors argued that in some situation,the equilibrium is not the desirable one(or a ff ordable)and a smaller value is required,which can be explained logically especially in a food limited environment since the circumstance can only withstand a certain amount of populations.Thus we must alter the system structurally by introducing a feedback control variable(Aizerman and Gantmacher[3]or Lefschetz[4]).On the other hand,ecosystem in the real world are continuously disturbed by unpredictable forces which can result in some changes of the biological parameters such as survival rates.We call the disturbance functions to be control variables.Gopalsamy and Weng[5]introduced a feedback control variable into a two species competitive system and discussed the existence of the globally attractive positive equilibrium of the system with feedback controls.For more details in this direction,please see[6-9].

In[10],Li,Han and Chen studied the following two-species autonomous Lotka-Volterra competitive system with in finite delays and feedback controls:

where bi,aij,ci,ei,di,i,j=1,2,are positive constants;xi(t)denotes the density of the population xi;ui(t)denotes the feedback control variable.By constructing suitable Lyapunov functional,the authors investigated the extinction and global stability of the equilibriums,and showed that the suitable feedback controls can retain or change the stability of system(1.5).

However,as argued in[11],in many ecological systems,the species under consideration may disperse spatially as well as evolve in time.This spatial dispersal or diffusion arises from the natural tendency of each species to diffuse to areas of lower population density.The role of diffusion in the ecological system has been extensively studied in[12-18].

In more realistic ecological models,any delays should be spatially inhomogeneous,that is,the delay a ff ects both the temporal and spatial variables,due to the fact that any given individual may not necessarily have been at the same spatial location at the previous times.Such delays are called a spatio-temporal delay or nonlocal delay.In[19],Gourley and So considered the following food-limited reaction-diffusion population model with nonlocal delay

with homogeneous Neumann boundary conditions=0,x=0,π,where the convolution f?u is defined by

here

is solution of

subject to

the function f(t)in(1.6)is called the delay kernel and satisfies f(t)≥0 for all t≥ 0 together with the normalization condition(t)dt=1.The authors[19]studied the the linear stability,boundedness,global convergence of solutions and bifurcations of system(1.6).

Gourley and Ruan[20]considered a two-species competition model described by a reaction-diffusion system with nonlocal delays.Using the energy function method,they studied the extinction and stability of the equilibria of the system.By employing linear chain techniques and geometric singular perturbation theory,they investigated the existence of traveling front solutions of the system.

Motivated by the works of Gourley and So[19]and Gourley and Ruan[20],in this paper,we discuss the extinction and global stability of a reaction-diffusion Lotka-Volterra competitive system(1.1),and show the e ff ect of nonlocal delay and feedback control on system(1.1),that is,feedback control can retain or change the stability of system(1.1).

In system(1.1),we assume that

is the weight function describing the distribution at the past times of the individual of the species uiat position x and time t,and satisfiessubject toand,i=1,2;the delay kernel fi(t)satisfies

The organization of this paper is as follows.In Section 2,we introduce some definitions and lemmas.In Section 3,we study the extinction and global stability of system(1.1).To illustrate the feasibility of our main results,Section 4 is devoted to giving some numerical simulations.At last,we give a brief discussion of our result.

2 Preliminaries

In this section,we present some preliminary results required in the sequel.

Let R=(?∞,+∞),? =(0,π).For 1≤ p≤ ∞,let Lp(?)denote the Banach space of Lesbegue measurable functions u on ? satisfying

In particular,if p=2,L2(?)becomes a Hilbert space with the usual inner productandLetdenote the norm in L2((0,T);L2(?;R)),that is,

Further,for m∈N,1≤p≤+∞,the Sobolev space Wm,p(?)is defined by

where α=(α1,···,αn),|α|=α1+···+αn,and the derivativesare taken in a weak sense.When endowed with the norm

Wm,p(?)is a Banach space(see,for example[21]).

It can be easily seen that(0,0,0,0)and(M1,M2,N1,N2)are a pair of coupled upper and lower solutions of problem(1.1)-(1.3),where

with

Hence,the global existence of solutions(u1(t,x),u2(t,x),v1(t,x),v2(t,x))of(1.1)-(1.3)can be derived based on the theory of upper-lower solution pairs(see,for example,Redlinger[22]or Pao[23]).It follows that 0≤ui(t,x)≤Mi,0≤vi(t,x)≤Ni(i=1,2)for(t,x) ∈ R × [0,π].In addition,if ?i(0,x)0, ψi(0,x)0(i=1,2),then it follows from the strong maximum principle that ui(t,x)>0,vi(t,x)>0(i=1,2)for all t>0,x ∈ [0,π].

By simple computation,system(1.1)has a trivial steady state solution E0(0,0,0,0),two semi-trivial steady state solutions

If

then system(1.1)has a unique positive steady state,where

Lemma 2.1 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Then

Proof It follows from the first and second equations of system(1.1)that

Let zi(t)be a solution of the following ordinary differential equation

From the above inequalities,for any ε>0 sufficiently small,there exists a T1>0 such that for any x∈ [0,π]and t≥ T1,ui(t,x)≤+ε.Therefore,if follow from the third and fourth equations of system(1.1)that

Let wi(t)be a solution of the following ordinary differential equation

It is easy to see that

It follows from the comparison principle that vi(t,x)≤wi(t),then

Setting ε→ 0,we obtain

This ends the proof of Lemma 2.1.

Similar to the proof of Lemma 2.5 in[14],we obtain the following lemma.

Lemma 2.2 If u(t,x)is a bounded nonnegative function,where(t,x)∈ (0,+∞)×(0,π),and Gi(x,y,t),fi(t),i=1,2 are defined by(1.7)and(1.8)respectively,then for i=1,2,

uniformly for x ∈ [0,π].

Lemma 2.3 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Assume thatholds,then there exists an α>0 such that

It follows from Lemma 2.1 that

According to Lemma 2.2,we obtain

uniformly for x ∈ [0,π].

Hence,for ε>0 sufficiently small satisfying(2.1),there is a t1>0 such that

From the above inequalities and the first equation of system(1.1),for x∈[0,π],t≥ t1,we have

From(2.1),a standard comparison argument shows that

Setting ε→ 0,it follows that

This ends the proof of Lemma 2.3.

Lemma 2.4 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Assume that=0 uniformly for x ∈ [0,π],then

uniformly for x ∈ [0,π].

From Lemma 2.1,for any ε>0 sufficiently small,there exists a T1>0 such that for any x ∈ [0,π]and t≥ T1,

From the above inequalities,(2.2)and the first equation of system(1.1),for x∈[0,π],t≥ t1=max{T1,τ1},we have

Let w1(t)be a solution of the following ordinary differential equation

For any ε>0 sufficiently small,it follows fromthat

Hence,for any ε>0 sufficiently small,there exists asuch that for any x∈ [0,π]and t≥,

It follows from(2.5)and the third equation of system(1.1)that

Let p1(t)be a solution of the following ordinary differential equation

Then solutions of the above equality satisfy.By the comparison theorem,we have u1(t,x)≥p1(t).Then

Hence,for any ε>0 sufficiently small,there exists a T2≥such that for any x ∈ [0,π]and t≥ T2,

By(2.2),(2.6)and the first equation of system(1.1),we have

It follows from(2.3),(2.4)and(2.6)that

Therefore,by the similar arguments as above,we have

For any ε>0 sufficiently small,there exists a>T2such that for any x∈[0,π]and t≥,

If follows from(2.7)and the third equation of system(1.1)that

By the similar arguments as above,we have

Hence,for any ε>0 sufficiently small,there exists a T3>such that for any x ∈ [0,π]and t≥ T3,

From(2.2),(2.8)and the first equation of system(1.1),there exists a t2=max{T3,τ2}such that for any x ∈ [0,π]

It follows from(2.3),(2.4),(2.7)and(2.8)that

By the similar arguments as above,one has

Therefore,for any ε>0 sufficiently small,there exists a≥t2such that for any x ∈ [0,π]and t≥,

If follows from(2.9)and the third equation of system(1.1)that

By the similar arguments as above,we have

Hence,for any ε>0 sufficiently small,there exists a T4≥such that for any x ∈ [0,π]and t≥ T4,

Obviously,from(2.3),(2.4)and(2.5)-(2.10),for any x ∈ [0,π]and t≥ T4,we have

Repeating the above procedure,we get four sequencesand,n=1,2,···,such that for n ≥ 2

Clearly,we have

We claim that the sequencesare non-increasing,and the sequencesare non-decreasing.To prove this claim,we will carry out by induction.Firstly,we immediately get

Assume that our claim is true for n,that is,

After a tedious but straightforward computation,we obtain that

It follows from(2.12)that

Subtracting the first equality of(2.13)from the second equality,we obtain

Therefore,we have

Setting ε→ 0,we obtain

This completes the proof of Lemma 2.4.

Take c1=c2=0 in system(1.1),that is,consider system(1.1)without feedback controls.Then system(1.1)is reduced to the following system

Without lose of generality,it follows from Theorem 2.3 in[20]that we have the following theorem.

Theorem 2.1 Let(u1(t,x),u2(t,x))Tbe a solution of system(2.14)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?1(0,x)0 and ?2(0,x)0.

3 Main Results

The trivial steady state solution E0is of no interest here.In this paper,we discuss the stability of the equilibria E1,E2and E?,and shows the in fluence of feedback controls on the global stability of system(1.1).More precisely,we present the main results of this paper.

Theorem 3.1 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Assume further that(H1)and

Proof Define

It is easy to see that the equations of(1.1)can be rewritten as

Calculating the derivative of V1along the solution of system(3.1),it follows that

Using the property of Ki(x,y,t),i=1,2 as described in(1.7)and(1.8),we have

Substituting the above equalities into(3.3)leads to

Now,define a new Lyapunov functional

It is derived from(3.4)and(3.5)that

can lead to

From(H2),we have δi>0,i=1,2.It is easy to see that

For any T>0,integrating(3.7)over[0,T],we derive that

From(3.8)we can conclude that

and

for some constants Ci,Di,Ei,Fi,i=1,2 independent of T.

Noting that ui(t,x),vi(t,x),i=1,2 are bounded,it follows from(3.9)that

for some constants Qi,i=1,2 independent of T.We derive from(3.9)-(3.11)thati=1,2 thus

We obtain from the Sobolev compact embedding theorem(see,for example[21])that

This completes the proof of Theorem 3.1.

Now,we study the stability of semi-trivial steady state solution of system(1.1).

Theorem 3.2 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Assume further that

Proof From Lemma 2.1 we have

Hence,for any ε>0 sufficiently small,from Lemma 2.2,there exists a T1>0 such that for any x ∈ [0,π]and t≥ T1,

where

For any ε>0 sufficiently small,it follows from Lemma 2.2 that there exist positive constants τn(τn< τn+1),n=1,2,···such that

From(3.12),(3.13)and the first equation of system(1.1),for x ∈ [0,π],t≥ t1=max{T1,τ1},we have

For any ε>0 sufficiently small,it follows from(H3)that

then from the comparison principle,we obtain

Hence,for any ε>0 sufficiently small,from Lemma 2.2,there exists a T2≥ t1such that for any x ∈ [0,π]and t≥ T2,

where

It follows from(3.15)and the second equation of system(1.1)that

Hence,for any ε>0 sufficiently small,from Lemma 2.2,there exists a T3>T2such that for any x ∈ [0,π]and t≥ T3,

where

By(3.13),(3.16)and the first equation of system(1.1),for x ∈ [0,π],t≥ t2=max{T3,τ2},we have

For any ε>0 sufficiently small,it follows from(3.12),(3.14)and(3.16)that

By the comparison principle,we have

Hence,for any ε>0 sufficiently small,from Lemma 2.2,there exists a T4≥ t2such that for any x ∈ [0,π]and t≥ T4,

where

From(3.12)and(3.15)-(3.17),for any x ∈ [0,π]and t≥ T4,we have

Repeating the above procedure,we get two sequencesand,n=1,2,···,such that for n≥2

Without loss of generality,we assume,n=1,2···.Then

We claim that the sequencesare non-increasing,and the sequencesare non-decreasing.To prove this claim,we will carry out by induction.Firstly,we immediately get

Assume that our claim is true for n,that is,

By computation,we have

Note that(H3)holds,then if follows from Lemma 2.3 thatα,that is≥ α>0.Obviously≥0.To proveuniformly for x ∈ [0,π],it suffices to show that=0.Otherwise,we suppose that>0.Letting n→+∞in(3.18),we obtain

Multiplying the second equation of(3.19)byand adding it to the first equation of(3.19),we obtain

From the first inequality in condition(H3),>0 and(3.20),we have

It follows from the second inequality in condition(H3)and(3.21)that u2≤0,which is a contradiction,then we obtainuniformly for x ∈ [0,π].Hence,by Lemma 2.4,we haveuniformly for x ∈ [0,π].This ends the proof of Theorem 3.3.

Using Lyapunov functional method,another sufficient conditions which guarantee the stability of semi-trivial steady state solution of system(1.1)are obtained.

Theorem 3.3 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Assume further that

Proof Define

Ki(x,y,t)=Gi(x,y,t)fi(t), ?=(0,π),i=1,2,

System(1.1)can be rewritten as

If follows from the first inequality in condition(H4)that b2?a21u1<0.Using similar arguments to those in the proof of Theorem 3.1,we have

This ends the proof of Theorem 3.3.

Similar to the proofs of Theorems 3.2 and 3.3,we have the following theorem.

Theorem 3.4 Let(u1(t,x),u2(t,x),v1(t,x),v2(t,x))be a solution of system(1.1)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?i(0,x)0 and ψi(0,x)0,i=1,2.Assume further that

or

Note that when ci=0,i=1,2,system(1.1)is reduced to system(2.14).Similar to the analysis of Theorems 3.1,3.2 and 3.4,we have the following corollary.

Corollary 3.1 Let(u1(t,x),u2(t,x))Tbe a solution of system(2.14)with the boundary conditions(1.2)and the initial conditions(1.3)satisfying ?1(0,x)0 and ?2(0,x)0.

Remark 3.5 When ci=0,i=1,2,conditions(H3)and(H4)are changed intoand,respectively.Obviously,andare weaker than.Then,the conditions of Corollary 3.1 are weaker than those of Theorem 2.1.Hence,Theorems 3.1-3.4 and Corollary 3.1 generalize and improve the results of[20].

Remark 3.6 If system(1.1)is reduced to system(1.5),Theorems 3.1-3.4 generalize the main results of[10].Especially,it is hard to construct the extinction of Lyapunov functional to study the extinction of system(1.1)as in[10].Hence,in Theorem 3.2,we use the iterative technique method to investigate the extinction of system(1.1).

4 Example

In this section,we give some examples to show the feasibility of our results.

In the following,we always takeand

However,it is difficult for us to carry out numerical simulations directly because of nonlocal term.Define

Similar to[19],the equations of(1.1)are rewritten as:

Each component is considered with homogeneous Neumann boundary conditions;additionally,we need the following initial condition

Similar to(4.1)-(4.3),the equations of(2.14)are rewritten as:

Consider the following system

where b1=4;a11=2;a12=2;b2=1;a21=1;a22=2;τ1=1;τ2=2.Obviouslyholds,it follows from(i)of Theorem 2.1 that system(4.5)has a semi-trivial steady state(3,0),which attracts all positive solutions of system(4.5).

Now,we show the in fluence of feedback controls on dynamic behaviors of system(4.5),and consider the following feedback controls system(4.6)

Example 4.1 In system(4.6),set c1=7;e1=1;d1=0.5;c2=2;e2=2;d2=1.By computation,one has

then conditions(H1)and(H2)hold.It follows from Theorem 3.1 that system(4.6)has a unique positive steady state E?(0.6897,0.1034,0.3448,0.0517),which attracts all positive solutions of system(4.6).Note that species u2is extinct in system(4.5).However,species u2is globally stable in system(4.6),that is,feedback controls can make an extinct species in system(4.5)become globally stable.Figure 1 shows the dynamics behavior of system(4.6).

Figure 1:Dynamics behavior of system(4.6)with c1=7;e1=1;d1=0.5;c2=2;e2=2;d2=1.

Example 4.2 In system(4.6),set c1=3;e1=1;d1=0.5;c2=2;e2=2;d2=1.By computation,(H4)holds,but(H3)is not satisfied,then it follows from Theorem 3.2 that system(4.6)has a semi-trivial steady state E1(1.1429,0,0.5714,0),which attracts all positive solutions of system(4.6).Hence,by choosing suitable feedback controls variables,the extinct species u2in system(4.5)is still extinct.Figure 2 shows the dynamics behavior of system(4.6).

Figure 2:Dynamics behavior of system(4.6)with c1=3;e1=1;d1=0.5;c2=2;e2=2;d2=1.

Example 4.3 In system(2.14),set b1=3;a11=1;a12=2;b2=1;a21=1;a22=1; τ1=1; τ2=2.Obviously,which does not satisfy the condition of Theorem 2.1(ii),thus(ii)of Theorem 2.1 fails to study system(2.14).Buthold,then it follows from(ii)of Corollary 3.1 that system(2.14)has a semi-trivial steady state(3,0),which attracts all positive solutions of system(2.14).Figure 3 shows the dynamics behavior of system(2.14).

Figure 3:Dynamics behavior of system(2.14)with b1=3;a11=1;a12=2;b2=1;a21=1;a22=1;τ1=1;τ2=2.

Example 4.4 In system(1.1),bi,aij,τi,i,j=1,2 are the same as those in Example 4.3.Let c1=0.5;e1=1;d1=0.5;c2=2;e2=2;d2=1.Obviously(H4)holds,but(H3)is not satisfied,then it follows from Theorem 3.3 that system(1.1)has a semi-trivial steady state E1(2.4,0,1.2,0),which attracts all positive solutions of system(1.1).Then,the extinct species u2in system(1.1)retains the property of extinction under suitable feedback controls variables.Figure 4 shows the dynamics behavior of system(1.1).

Figure 4:Dynamics behavior of system(1.1)with b1=3;a11=1;a12=2;b2=1;a21=1;a22=1;c1=0.5;e1=1;d1=0.5;c2=2;e2=2;d2=1;τ1=1;τ2=2.

Consider the following system

where b1=2;a11=3;a12=1;b2=1;a21=1;a22=1;τ1=1;τ2=2.Thenholds.It follows from(i)of Theorem 2.1 that system(4.7)has a unique positive steady state(0.5,0.5),which attracts all positive solutions of system(4.7).

Example 4.5 In system(1.1),bi,aij,τi,i,j=1,2 are chosen the same as those in system(4.7).Let c1=1;e1=1;d1=0.5;c2=2;e2=2;d2=1.Note thatimplies(H1)and(H2)hold,then it follows from Theorem 3.1 that system(1.1)has a semi-trivial steady state E?(0.5,0.25,0.25,0.125),which attracts all positive solutions of system(1.1).Hence,in this case,feedback controls have no in fluence on the stability of system(1.1),that is,feedback controls only change the value of the positive steady state and keep the property of stability.Figure 5 shows the dynamics behavior of system(1.1).

Figure 5:Dynamics behavior of system(1.1)with b1=2;a11=3;a12=1;b2=1;a21=1;a22=1;c1=1;e1=1;d1=0.5;c2=2;e2=2;d2=1;τ1=1;τ2=2.

Consider the following system

where b1=1;a11=3;a12=2;b2=1;a21=1;a22=1; τ1=1; τ2=2.Thenholds.It follows from(iii)of Theorem 2.1 that system(4.8)has a semi-trivial steady state(0,1),which attracts all positive solutions of system(4.8).

Example 4.6 In system(1.1),bi,aij,τi,i,j=1,2 are chosen the same as those in system(4.8).Let c1=1;e1=1;d1=0.5;c2=3;e2=2;d2=5.Obviously,(H1)and(H2)hold,then it follows from Theorem 3.1 that system(1.1)has a semi-trivial steady state E?(0.23,0.09,0.11,0.23),which attracts all positive solutions of system(1.1).Hence,suitable feedback controls can make an extinct species u1in system(4.8)become globally stable.Figure 6 shows the dynamics behavior of system(1.1).

Figure 6:Dynamics behavior of system(1.1)with b1=1;a11=3;a12=2;b2=1;a21=1;a22=1;c1=1;e1=1;d1=0.5;c2=3;e2=2;d2=5;τ1=1;τ2=2.