An Elliptic Nonlinear System of Two Functions with Application

KANG Joon Hyuk and ROBERTSON Timothy

1 Department of Mathematics,Andrews University,Berrien Springs MI.49104-0350,U.S.A.

2 Department of Mathematics,University of Tennessee,Knoxville TN.37996-1320,U.S.A.

Abstract. The purpose of this paper is to give a sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain Ω in Rn.Also considered are the effects of perturbations on the coexistence state and uniqueness.The techniques used in this paper are super-sub solutions method,eigenvalues of operators,maximum principles,spectrum estimates,inverse function theory,and general elliptic theory.The arguments also rely on some detailed properties for the solution of logistic equations.These results yield an algebraically computable criterion for the positive coexistence of competing species of animals in many biological models.

Key Words: Competition system;coexistence state.

1 Introduction

One of the prominent subjects of study and analysis in mathematical biology concerns the competition of two or more species of animals in the same environment.Especially pertinent areas of investigation include the conditions under which the species can coexist,as well as the conditions under which any one of the species becomes extinct,that is,one of the species is excluded by the others.In this paper,we focus on the general competition model to better understand the competitive interactions between two species.Specifically,we investigate the conditions needed for the coexistence of two species when the factors affecting them are fixed or perturbed.In earlier literature,the models were concerned with studying those with homogeneous Neumann boundary conditions.Later on,the more important Dirichlet problems,which allow flux across the boundary,became the subject of study.

2 Literature review

Within the academia of mathematical biology,extensive academic work has been devoted to investigation of the simple competition model,commonly known as the Lotka-Volterra competition model.This system describes the competitive interaction of two species residing in the same environment in the following manner:

Suppose two species of animals,rabbits and squirrels for instance,are competing in a bounded domain Ω.Letu(x,t) andv(x,t) be densities of the two habitats in the placexof Ω at timet.Then we have the dynamic competition model

wherea,d＞0 are growth rates,b,f＞0 are self-limitation rates,andc,e＞0 are competition rates.Here we are interested in the time independent,positive solutions,i.e.the positive solutionsu(x),v(x) of

which are called the coexistence state or the steady state.The coexistence state is the positive density solution depending only on the spatial variablex,not on the time variablet,and so its existence means that the two species of animals can live peacefully and forever.

The mathematical community has already established several results for the existence,uniqueness and stability of the positive steady state solution to (2.1)(see[1-7]).

One of the initial important results for the time-independent Lotka-Volterra model was obtained by Cosner and Lazer.In 1984,they published the following sufficient conditions for the existence and uniqueness of a positive steady state solution to (2.1):

When they said that a visitor was asking for her, and then proceeded each one to tell breathlessly a different tale of wonder, in which she could only distinguish the words, oxen, gold, club, giant, lion, she thought they were all out of their minds

Theorem 2.1.(in[2])

(B)Furthermore,if

where θM(x)for M＞0is the unique positive solution to the logistic equation as mentioned in the Lemma3.4,then the positive solution is unique.

Biologically,the conditions in Theorem 2.1 implies that if the self-reproduction and self-limitation rates are relatively large,and the competition rates are relatively small,in other words,if members of each species interact strongly among themselves and weakly with members of the other species,then there is a unique positive steady state solution to (2.1),that is,the two species within the same domain will coexist indefinitely at unique population densities.

In 1989,Cantrell and Cosner extended these results by proving that the reproduction and self limitation rates may vary within bounds without losing the uniqueness result,given certain conditions.Biologically,Cantrell and Cosner’s theorem suggests that two species can relax ecologically and maintain a coexistence state.Their primary result is given below:

In our analysis we focus on the conditions required for the maintenance of the coexistence state of the model when bounded functionsgandhare slightly perturbed.Biologically,our conclusion implies that two species may slightly relax ecologically and yet continue to coexist at unique densities.

In Theorem 2.2,the condition 0＜c,e ＜1 biologically implies that the self-limitation rates of both species are relatively larger than competition rates.This condition plays an important role in the proof of Cantrell and Cosner’s theorem by implying the invertibility of the Frechet derivative (linearization) of (2.1) at a fixed reproduction rate (a,a).

The work of Lazer,Cosner,and Cantrell provides insight into the competitive interactions of two species operating under the conditions described in the Lotka-Volterra model.However,their results are somewhat limited by a few key assumptions.In the Lotka-Volterra model that they studied,the rate of change of densities largely depends on constant rates of reproduction,self-limitation,and competition.The model also assumes a linear relationship of the terms affecting the rate of change for both population densities.

However,in reality,the rates of change of population densities may vary in a more complicated and irregular manner than can be described by the simple competition model.Therefore,in the last decade,significant research has been focused on the existence and uniqueness of the positive steady state solution of the general competition model for two species,

or,equivalently,the positive solution to

whereg,h ∈C2designate reproduction,self-limitation and competition rates that satisfy certain growth conditions (see[8-11]).

Because of its broader applicability,the general competition model has become a more popular subject of research within the mathematical community over the past few years.

The functionsgandhdescribe how species 1(u) and 2(v) interact among themselves and with each other.

The followings are questions raised in the general model with nonlinear growth rates.

Problem 1:What are the suffciient conditions for existence of positive solutions?

Problem 2:What are the suffciient conditions for uniqueness of positive solutions?

Problem 3:What is the effect of perturbation for existence and uniqueness?

Theorem 2.2.(in [4])If a=d＞λ1,b=f=1,and0＜c,e＜1,then there is a neighborhood V of(a,a)such that if(a0,d0)∈V,then(2.1)with(a,d)=(a0,d0)has a unique positive solution.

In Section 4,we establish sufficient conditions for the existence and non-existence of positive solution of the system that generalizes the Theorem 2.1.We also achieve solution estimates in the Section 5 to prove the uniqueness and the invertibility of linearization in Sections 6,7 and 8,where we investigate the effect of perturbation for existence and uniqueness that generalizes the Theorem 2.2.

An especially significant aspect of the global uniqueness result is the stability of the positive steady state solution,which has become an important subject of mathematical study.Indeed,researchers have obtained several stability results for the Lotka-Volterra model with constant rates (see[2,4,7,8]).However,the stability of the steady state solution for the general model remains open to investigation.

The research presented in this paper therefore begins the mathematical community’s discussion on the stability of the steady state solution for the general competition model.

3 Preliminaries

Before entering into our primary arguments and results,we must first present a few preliminary items that we later employ throughout the proofs detailed in this paper.The following definition and lemmas are established and accepted throughout the literature on our topic.

Definition 3.1.(Super and Sub solutions)The vector functionsforman super/sub solution pair for the system

These months of pregnancy5 have been bittersweet. I deeply love this coming child and delight in every little nudge and kick. But it has occurred to me that this is the last time in Nicky s childhood that it will be just the two of us. Soon enough he will have to learn to share . . . Mommy s lap, Mommy s hugs, Mommy s attention.

Lemma 3.1.If gi in the Definition3.1are in C1and the system admits an super/sub solution pair,then there is a solution of the system in3.1withIf

inΩfor i=1,...,N,then.

Lemma 3.2.(The first eigenvalue)Consider

where q(x)is a smooth function fromΩto R andΩis a bounded domain in Rn.

(A)The first eigenvalue λ1(q)of(3.1),denoted by simply λ1when q≡0,is simple with a positive eigenfunction ϕq.

23. Swans: Swans have a diverse history in folklore, including Greek mythology where Zeus assumes the form of a swan in Leda and the Swan. The bird is also an erotic symbol due to its association with Aphrodite/Venus. Swan maiden tales are found around the world, including the story of Swan Lake, the famous ballet. Since the swan moves in the three elements earth, water, and air, it has traditionally been associated with shape-shifting, especially with the form of a beautiful young woman. Like storks, swans were sometimes thought to assume human form when they migrated to other lands (Jones 1995, 408).

(B)If q1(x)＜q2(x)for all x∈Ω,then λ1(q1)＜λ1(q2).

(C)(Variational Characterization of the first eigenvalue)

Lemma 3.3.(Maximum principles)Consider

Oho! exclaimed the fox to himself, you think you will escape me that way, do you? We shall soon see about that, my friend, and very quietly and stealthily he prowled round the house looking for some way to climb on to the roof.

whereΩis a bounded domain in Rn with

(M1)∂Ω∈C2,α(0＜α＜1),

(A)If a(x)≡0,then.

Belshazzar appears in the Old Testament as a Babylonian general and son of Nebuchadnezzar II; according to the Old Testament he was warned of his doom by divine handwriting on the wall that was interpreted by Daniel (6th century BC) (WordNet).Return to place in story.

(M3)L is uniformly elliptic in,with ellipticity constant γ,i.e.,for every x∈and every real vector ξ=(ξ1,...,ξn)

And yet standing10 beside the hospital bed watching the life ebb11 from my sleeping father was painful. I felt like a little girl at his bedside, unable to talk to him yet again. I became fixated with his fingers – fat and soft, lying gently curled beside him. Slowly they transformed from plump sausages to stone – white and immovable. It was his fingers that told me he had gone from this life, not the bleeping of monitors or the bustling12 of nursing staff.

Then there are neighborhoods U of u0and V of v0such that the equation A(u,v)=0has exactly one solution v∈V for every u∈U.The solution v depends continuously on u.

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(M2)|aij(x)|α,|ai(x)|α,|a(x)|α ≤M(i,j=1,...,n),

(B)If a(x)≡0,Ωis connected and u attains its maximum (minimum) at an interior point ofΩ,then u is identically a constant inΩ.

(C)If a(x)≤0,then-min (u,0).[Thus a nonnegative maximum (nonpositive minimum) must be attained on the boundary.] In particular,if Lu=0inΩ,then.

(D)If a(x)≤0,Ωis connected and u attains a nonnegative maximum (nonpositive minimum) at an interior point ofΩ,then u is identically a constant inΩ.

20. That is my house and yours, where we shall live together: A house can represent the family line (Biedermann 179). According to Jung, What happens inside it, happens within ourselves (Biedermann 179). Freud associates the house with the woman, the mother, in a sexual or childbearing sense (Biedermann 179).

In our proof,we also employ accepted conclusions concerning the solutions of the following logistic equations.

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Lemma 3.4.Consider

“It is gold! it is gold!” cried they, rushing forward, and seizing the horses. Then they struck the little jockeys, the coachman, and the footman dead, and pulled little Gerda out of the carriage.

where f is a decreasing C1function such that there exists c0＞0such that f(u)≤0for u≥c0andΩis a bounded domain in Rn.

(1)If f(0)＞λ1,then the above equation has a unique positive solution,where λ1is the first eigenvalue of -Δwith homogeneous boundary condition as in the Lemma3.2.We denote this unique positive solution as θf.

(2)If f(0)≤λ1,then u≡0is the only nonnegative solution to the above equation.

The main property about this positive solution is thatθfis increasing asfis increasing.Especially,fora＞λ1,we denoteθaas the unique positive solution of

Hence,θais increasing asa＞0 is increasing.

We need the following lemmas for the perturbation of uniqueness in the Sections 7 and 8.

Lemma 3.5.(Implicit Function Theorem)Let X,Y,Z be Banach spaces.For a given(u0,v0)∈X×Y and a,b＞0,let S={(u,v):‖u-u0‖≤a,‖v-v0‖≤b}.Suppose A:S →Z satisfies the following:

Lemma 3.7.(Fredholm Alternative)Let T:X→X be a compact linear operator on the normed space X and let S:X →X be a linear operator such that S-1∈B(X),where B(X)is the set of bounded linear operators on X.Then either either one of the followings is true:

(2)Av(·,·)exists and is continuous in S (in the operator norm).

(3)A(u0,v0)=0.

(4)[Av(u0,v0)]-1exists and belongs to B(Z,Y).

Let u∈C2(Ω)∩C()be a solution of Lu≥0(Lu≤0)inΩ.

Lemma 3.6.(Schauder’s Boundary Estimate)We consider

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During the first few days they noticed signs of much disturbance51 and distress52 in the streets, but about the fourth day, as they sat on the roof of the palace, they perceived a splendid procession passing below them along the street

for some c＞0that is independent of u.

(1)A is continuous.

(a)The homogeneous equation T(x)-S(x)=0has a nontrivial solution x∈X.

(b)For each y∈X the inhomogeneous equation T(x)-S(x)=y has a unique solution x∈X.Furthermore,in case (b),the operator(T-S)-1∈B(X).

with(M1),(M2),and(M3)in the Lemma3.3.If,then

Having established these preliminaries,we now commence our investigation of the general competition model.

4 Existence,nonexistence

The most general type of elliptic interacting system of two functions with homogeneous boundary condition is

where we assume that theC2functionsgandhare relative growth rates satisfying the following so-called growth rate conditions:

The following inequalities will be useful for later use which are true for anyC2functions satisfying only (EN3).

Lemma 4.1.For all u,v with u≥0,v≥0,

Proof.By (EN3) and the Mean Value Theorem,there issuch thatand

But no sooner had they finished their first bows and curtseys than a slight breeze sprung up, and began to sway the princess, whose equerries had retired8 out of respect

But,by the monotonicity ofgu,we have

Similarly,we can prove the second inequality forh.

The following Lemma says that if there is no competition between the species,then we can easily conclude their peaceful coexistence.

Lemma 4.2.If we consider

then the conditions gu(0,0)＞λ1,hv(0,0)＞λ1(i.e,reproductions are relatively large) are sufficient to guarantee the existence of a positive density solution to the above equations.

Proof.be a constant such thatM＞2c0.Then by (EN2),(EN4),the monotonicity ofguand the Lemmas 3.4,4.1,

and so,uandare subsolution and supersolution to

respectively.So,by the Lemma 3.1,there is a positive solutionuto

Similarly,there is a positive solutionvto

But,if there is some competition between the two species,then as we notice in the following Theorem,we should have much larger reproduction rates,i.e,we have stronger conditions to guarantee their coexistence.

We establish the following two existence results:

Theorem 4.1.(A)If gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,then(4.1)has a solution(u,v)with u＞0,v＞0.

(B)If,then(4.1)has a solution(u,v)with u＞0,v＞0.

Biologically,we can interpret the conditions in Theorem 4.1 as follows.The functionsg,hand their partial derivatives describe how species 1(u) and 2(v) interact among themselves and with each other.Hence,the conditions in both (A) and (B) imply that each species must have large enough reproduction capacity for their peaceful coexistence.

Proof.(A) Let.Then by the Mean Value Theorem,there iswithsuch that

and so,by the Lemma 4.1,

which is a contradiction to the Maximum Principles.Therefore,by the monotonicity ofgu,we conclude

Similarly,we have

Furthermore,by (EN2) and the monotonicity ofgu,

and similarly

(B) Let,whereϕ0＞0 is the eigenfuntion of-Δ with homogeneous boundary condition corresponding to the smallest eigenvalueλ1as in the Lemma 3.2.Then by (EN2) and the monotonicity ofgu,

Similarly,we have

Sincegu[0,,by the continuity ofguandhv,for sufficiently smallϵ＞0,

so,by the Lemma 4.1 and monotonicity ofgu,

and similarly,

We also have the following nonexistence result.

Theorem 4.2.If either gu(0,0)≤λ1or hv(0,0)≤λ1,then(4.1)does not have any positive solution.

It says that if one of the species has small reproduction,then it may be extinct,which means that the two species can not coexist.

Proof.Supposegu(0,0)≤λ1and (u,v) is a solution to (4.1) withu≥0,v≥0.Then by (EN2) and the monotonicity ofg,

and sou/2 is a subsolution to

Any large constantM＞c0is a supersolution to

Hence,by the Lemma 3.1 there is a solutionwith

But,sincegu(0,0)≤λ1,by the Lemma 3.4,,sou≡0.Similarly,we can provev≡0 ifhv(0,0)≤λ1.

5 Solution estimates

In this section,we build up upper and lower bounds of solutions under certain conditions to establish uniqueness results in the next sections.

We have the following solution estimates.

Theorem 5.1.If gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,then for any positive solution(u,v)to(4.1),

Proof.Suppose (u,v) is a positive solution to (4.1).Then by (EN2) and the monotonicity ofg,

sou/2 is a subsolution to

Any constantM＞c0is a supersolution to

Hence,by the Lemma 3.1 and uniqueness of positive solution in the Lemma 3.4,u/2≤θgu(·,0),in other words,

Similarly,we have

For sufficiently smallϵ＞0,by the monotonicity ofhv,

Sincehv(0,c0)≤0,c0is a supersolution to

Therefore,by the Lemma 3.1 and uniqueness of positive solution in the Lemma 3.4,

Hence,by (5.2),monotonicity ofgu,and Lemma 4.1,

Δ(u)+ugu(u,2c0)≤Δ(u)+ugu(u,)≤Δ(u)+ugu(u,v)≤Δ(u)+g(u,v)=0,souis a supersolution to

Sincegu(0,2c0)＞λ1,by the continuity ofgu,gu(ϵϕ0,2c0)＞λ1for sufficiently smallϵ＞0,and so

soϵϕ0is a subsolution to

Hence,by the Lemma 3.1 and uniqueness of positive solution in the Lemma 3.4,we have

Similarly,we have

By (5.1),(5.2),(5.3) and (5.4),we conclude the desired inequalities.

6 Uniqueness

For our convenience we denoteA(a,b) for the average of real numbersaandb:

Definition 6.1.We say that a real valued function f has the Mean Value Property in the sense of average if A(x1,x2)≥A(w1,w2),then there arebetween x1and x2,andbetween w1and w2such that

In this section,we prove the uniqueness of positive solution to (4.1) with the following additional growth conditions:

(U1)guu,guv,huv,hvvare bounded.

(U2)gv(0,v)=hu(u,0)=0 for allu,v.

(U3)g,h∈C2are such thatg(·,y) andh(x,·) have the Mean Value Property in the sense of average for allx,y∈R.

We have the following uniqueness result.

Theorem 6.1.If

(A)gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,and

(B)

then(4.1)has a unique posotive solution.

The condition in (B) implies that species 1 interacts strongly among themselves and weakly with species 2.Similarly for species 2,they interact more strongly among themselves than they do with species 1.

Proof.The existence was already proved in the last section.We prove the uniqueness.

Let (u1,v1),(u2,v2) be positive solutions to (4.1),and letp=u1-u2,q=v1-v2.We want to show thatp≡q≡0.

Since (u1,v1),(u2,v2) are solutions to (4.1),

Hence,p≡q≡0 if the integrand is positive definite,in other words,

which is true if

which is true if

which is true if the condition is satisfied by the solution estimates in the Theorem 5.1.

7 Uniqueness with perturbation

We consider the model

where Ω is a smooth,bounded domain inRnand theC2functionsgandhare relative growth rates satisfying the following conditions:

(P1)guu,guv,gvv,huu,huv,hvvare bounded.

(P2)guu ＜0,hvv ＜0.

(P3) There exists a constantc0＞0 such thatgu(u,0)＜0,hv(0,v)＜0 foru＞c0andv＞c0.

(P4) For allu,v,there areMu,Mvsuch thatgu(u,Mu)≤0,hv(Mv,v)≤0.

(P5)g(0,v)=h(u,0)=0 for allu,v.

Define

Then by the functional analysis theory,(B,‖·‖B) is a Banach space containing (g,h).

LetC⊆Bbe the constraint set such that for all (α,β)∈C,

The following theorem is our main result about the perturbation of uniqueness.

Theorem 7.1.Suppose the following conditions are satisfied

(A)gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,

(B)(7.1)has a unique coexistence state(u,v),

(C)the Frechet derivative of (7.1) at(u,v)is invertible.

Then there is a neighborhood V of(g,h)in B such that ifthen(7.1)withhasa unique positive solution.

Biologically,the first condition in Theorem 7.1 indicates that the rates of reproduction are relatively large.Similarly,the third condition,which requires the invertibility of the Frechet derivative,signifies that the rates of self-limitation are relatively larger than the rates of competition,a relationship that will be established in Lemma 7.1.When these conditions are fulfilled,the conclusion of our theorem asserts that small perturbations of the rates do not affect the existence and uniqueness of the positive steady state.That is,the two species implied can continue to coexist even if the factors determining the population densities vary slightly.

Now,at first glance,Theorem 7.1 may appear to be a consequence of the Implicit Function Theorem.However,the Implicit Function Theorem only guarantees local uniqueness.In contrast,our results in Theorem 7.1 guarantee global uniqueness.The techniques we will use in the proof of Theorem 7.1 include the Implicit Function Theorem and a priori estimates on solutions of (7.1).

Proof.Since the Frechet derivative of (7.1) at (u,v) is invertible,by the Implicit Function Theorem there is a neighborhoodVof (g,h) inBand a neighborhoodWof (u,v) insuch that for allthere is a unique positive solution∈Wof (7.1) with.Thus,the local uniqueness of the solution is guaranteed.

To prove global uniqueness,suppose that the conclusion of Theorem 7.1 is false.Then,there are sequences (gn,hn,un,vn),such that (un,vn) andare positive solutions of (7.1) with (gn,hn),and (gn,hn)→(g,h).By Schauder’s estimate in elliptic theory,the convergence of (gn,hn),and the solution estimate in the Theorem 5.1,there are constantsk1＞0,k2＞0,k3＞0,k4＞0,k5＞0,k6＞0 such that

for alln=1,2,...,and so we conclude that|un|2,αand|vn|2,αare uniformly bounded.Therefore,there are uniformly convergent subsequences ofunandvn,which again will be denoted byunandvn.

Thus,let

and so by the monotonicity

which is a contradiction.By the same procedure with the sequence,we also have

In biological terms,the proof of our theorem indicates that if one of two species living in the same domain becomes extinct,that is,if one species is excluded by the other,then the reproduction rates of both must be small.In other words,the region condition of reproduction rates (A) is reasonable.

Now,the condition (C) in Theorem 7.1 requiring the invertibility of the Frechet derivative is too artificial to have any direct biological implications.We therefore turn our attention to more applicable conditions that will guarantee the invertibility of the Frechet derivative.We then obtain the following relationship:

We consider the model

where Ω is a smooth,bounded domain inRn,C2functionsgandhare such thatg(·,y) andh(x,·) have the Mean Value Property in the sense of average for allx,y∈R,and

(I1)guu,guv,hvv,huvare bounded,

(I2)gv ＜0,guu ＜0,guv ＜0,hu ＜0,huv ＜0,hvv ＜0,

(I3)g(0,v)=h(u,0)=0 for allu,v∈R,

(I5) there isc0＞0 such thatgu(u,0)＜0,hv(0,v)＜0 for allu,v＞c0.

Lemma 7.1.Suppose(u,v)is a positive solution to(7.2).If

(A)gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,

(B)

then the Frechet derivative of(7.2)at(u,v)is invertible.

Proof.The Frechet derivative of (7.2) at (u,v) is

We need to show thatN(A)={0}by the Lemma 3.7,whereN(A) is the null space ofA.By the assumption,there aresuch thatand

Hence,by the Lemma 3.2,we have

Therefore,

by (I3).But,by the Mean Value Theorem,there aresuch that

Hence,ϕ≡ψ≡0 if the integrand is positive definite,in other words,

which is true if

which is true if the condition (B) is satisfied by the solution estimate in Theorem 5.1.

Combining Theorem 6.1,Theorem 7.1,and Lemma 7.1,we obtain the following theorem,which is the main result in this section.The importance of Theorem 7.2 is that it improves the results as described in Theorem 2.2.

Theorem 7.2.If(P1)-(P5)and(I1)-(I5)are satisfied

(A)gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,and

(B)

then there is a neighborhood V of(g,h)in B such that ifthen(7.1)withhas a unique positive solution.

In biological terms,the results obtained in Corollary 7.2 confirm that under certain conditions,two species who relax ecologically can continue to coexist at fixed rates.The requirements given in (A) and (B) simply state that each species must interact strongly with itself and weakly with the other species.

8 Uniqueness with perturbation of region

We consider the model

where Ω is a smooth,bounded domain inRnandg,h∈C2.

Let Γ be a closed,bounded,convex region inBsuch that for all (g,h)∈Γ,

(PR1)guu,guv,gvv,huu,huv,hvvare bounded,

(PR2)guu ＜0,hvv ＜0,

(PR3) there exists a constantc0＞0 such thatgu(u,0)＜0,hv(0,v)＜0 foru＞c0andv＞c0,

(PR4) for allu,v,there areMu,Mvsuch thatgu(u,Mu)≤0,hv(Mv,v)≤0,

(PR5)g(0,v)=h(u,0)=0 for allu,v.

The following Theorem is the main result.

Theorem 8.1.Suppose

(A)For all(g,h)∈Γ,gu(0,2c0)＞λ1,hv(2c0,0)＞λ1,

(B)(8.1)has a unique positive solution for every(g,h)∈∂LΓ,where

(C)for all(g,h)∈Γ,the Fr`echet derivative of(8.1)at every positive solution to(8.1)is invertible,

(D)Γ-∂LΓ⊆C.

Then for all(g,h)∈Γ,(8.1)has a unique positive solution.Furthermore,there is an open set W in B such thatΓ⊆W and for every(8.1)has a unique positive solution.

Theorem 8.1 goes even further than Theorem 7.1 which states uniqueness in the whole region of (g,h) whenever we have uniqueness on the left boundary and invertibility of the linearized operator at any particular solution inside the domain.

Proof.For each fixedh,consider (g,h)∈∂LΓ and.We need to show that for all 0≤t ≤1,(8.1) withhas a unique positive solution.Since (8.1) with (g,h) has a unique positive solution (u,v) and the Frechet derivative of (8.1) at (u,v) is invertible,Theorem 7.1 implies that there is an open neighborhoodVof (g,h) inBsuch that ifthen (8.1) with (g0,h0) has a unique positive solution.Letλs=sup{0≤λ≤1|(8.1) withhas a unique coexistence state for 0≤t≤λ.}.We need to show thatλs=1.Supposeλs ＜1.From the definition ofλs,there is a sequence{λn}such thatand there is a sequence (un,vn) of the unique positive solutions of (8.1) withThen by elliptic theory,there is (u0,v0) such that (un,vn) converges to (u0,v0) uniformly and (u0,v0) is a solution of (8.1) with.

But,by the same proof as in the Section 7,u0＞0,v0＞0.

We claim that (8.1) has a unique coexistence state with.In fact,if not,assume thatis another coexistence state.By the Implicit Function Theorem,there existsand very close toλssuch that (8.1) withhas a coexistence state very close to,which means that (8.1) withhas more than one coexistence state.This is a contradiction to the definition ofλs.But,since (8.1) withhas a unique coexistence state and the Frechet derivative is invertible,Theorem 7.1 implies thatλscan not be as defined.Therefore,for each (g,h)∈Γ,(8.1) with (g,h) has a unique coexistence state (u,v).Furthermore,by the assumption,for each (g,h)∈Γ,the Frechet derivative of (8.1) with (g,h) at the unique solution (u,v) is invertible.Hence,Theorem 7.1 concluded that there is an open neighborhoodV(g,h)of (g,h) inBsuch that ifthen (8.1) withhas a unique coexistence state.LetThenWis an open set inBsuch that Γ⊆Wand for each(8.1) withhas a unique coexistence state.

Apparently,Theorem 8.1 generalizes Theorem 7.1.

9 Conclusions

In this paper,our investigation of the effects of perturbations on the general competition model resulted in the development and proof of Theorem 7.1,Lemma 7.1,and Theorem 7.2,as detailed above.The three together assert that given the existence of a unique solution (u,v) to the system (7.1),perturbations of the functions (g,h),within a specified neighborhood,will maintain the existence and uniqueness of the positive steady state.Indeed,our results specifically outline conditions sufficient to maintain the positive,steady state solution when the general competition model is perturbed within some region.

Applying this mathematical result to real world situations,our results establish that the species residing in the same environment can vary their interactions,within certain bounds,and continue to survive together indefinitely at unique densities.The conditions necessary for coexistence,as described in the theorem,simply require that members of each species interact strongly with themselves and weakly with members of the other species.

The research presented in this paper has a number of strengths,which confirm both the validity and the applicability of the project.First,the mathematical conditions required in Theorem 7.2 are identical to those required in Theorem 6.1.However,in the Theorem 6.1,we used these conditions to prove the existence and uniqueness of the positive steady state solution for the general competition model.In contrast,the Theorem 7.2 employs the same conditions to establish that the existence and uniqueness of this solution is maintained when the model is perturbed within some neighborhood.Thus,our findings extend and improve established mathematical theory.

Secondly,perturbations of the general model render its implications more applicable both mathematically and biologically.Because our theorem extends the steady state to any functions within some neighborhood of (g,h),results for the general model pertain to a far wider variety of functions.Biologically,perturbations extend the model’s description to species affected by factors that vary slightly yet erratically.Thus,the description of competitive interactions given by the model becomes a closer approximation of real-world population dynamics.

While our research therefore represents a progression in the field,the results obtained have an important limitation.Theorem 7.1,Lemma 7.1,and Theorem 7.2 establish that a region of perturbation exists within which the coexistence state is maintained for the general competition model.However,the exact extent of that region remains unknown.

Therefore,the results presented in this paper may serve as a platform for research of the question given above.Mathematicians should now attempt to establish the exact extent of the perturbation region in which coexistence is maintained for the general model.Such information would prove very useful not only mathematically but also biologically.Specifically,knowledge of the extent of the region would imply exactly how far the species can relax and yet continue to coexist.Thus,the results achieved through our research will enable the field to continue the development of theory on competitive interaction of populations.