On a Lotka-Volterra Competition Diffusion Model with Advection

Qi WANG

Abstract In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.

Keywords Competition-diffusion-advection, Stability, Dynamics

1 Introduction and Statement of Main Results

The Lotka-Volterra competition-diffusion system

models two competing species. Hereu(x,t) andv(x,t) denote respectively the population densities of two competing species at locationx∈Ω and timet>0, andd1,d2>0 are random diffusion rates of speciesuandvrespectively. The habitat Ω is a bounded region in RN, with smooth boundary?Ω,ndenotes the unit outer normal vector on?Ω, and the no flux boundary condition means that no individuals cross the boundary. The functionm(x) represents their common intrinsic growth rate or local carrying capacity, which is non-constant. 1 ≥b>0 and 1 ≥c >0 are interspecific competition coefficients. Then the maximum principle yields thatu(x,t)>0,v(x,t)>0 for everyx∈and everyt>0 (see [27]). By both mathematicians and ecologists, particular interests in two-species Lotka-Volterra competition models with spatially homogeneous or heterogeneous interactions are the dynamics of(1.1)(see[2—3,8—10,12,14—16,18—22,26]and the references therein). We say that a steady state(U,V)of(1.1)is a coexistence state if both components are positive, and it is a semi-trivial state if one component is positive and the other is identically zero.

Ifm(x) ∈Cα(Ω) (α∈(0,1)) withandm/≡0, then we denote by Θdthe unique positive solution of

One can refer to[2,25]for the proof of existence and uniqueness results of (1.2). (1.2)indicates that (1.1) has two semi-trivial steady states, denoted by (Θd1,0)and (0,Θd2), for everyd1>0 andd2>0.

Under above conditions, He and Ni [14] provided a complete classification on the global dynamics of system (1.1), which says that either one of the two semi-trivial steady states is globally asymptotically stable, or there is a unique coexistence steady state which is globally asymptotically stable, or the system is degenerate in the sense that there is a compact global attractor consisting of a continuum of steady states which connect the two semi-trivial steady states (see [14, Theorems 1.3 and 3.4]). We refer the interested readers to [14—16] for more investigations on system (1.1).

Besides random dispersal, it seems reasonable to argue that it is also plausible that species could move upward along the resource gradient (see e.g. [2, 7] and the references therein). A more general problem as follows was considered in [31],

where the non-constant functionP(x) ∈C2(Ω) is used to specify the advective direction, and the advection rates of two species are denoted byα1,α2>0, respectively. Here the movement strategies, growth rates and competition abilities of two species are taken into account and allowed to be different. Throughout this paper, we make the following basic hypotheses.

Assumption 1.1

Assumption 1.2

This paper is devoted to some dynamics of the following problem for all(d1,d2,η)in the special caseb=c=1,

where

Here all intra-and inter-specific competition coefficients are normalized to 1,which means that the two species have identical competition abilities.

1.1 Motivation and related work

System (1.3) has important applications in biological scenarios. For example, by lettingm1(x)=m2(x)=m(x)=P(x),b=c=1, one obtains the following model

Recently(1.8)has been frequently used as a standard model to study the evolution of conditional dispersal (see, e.g., [1, 6, 13] forα1,α2>0, [4—5, 23—24] forα1>0 =α2, and the textbook[11]). Basically speaking, system (1.8) models the competition between two species with the same population dynamics but different movement strategies as reflected by their diffusion and/or advection rates.

For system (1.3), it is known that Xiao and Zhou gave a complete classification on the global dynamics (see [31, Theorems 1.1—1.3]). We also know that whenα1=α2= 0, system(1.3) withm2=was considered in [15], where the effect of homogeneous versus heterogeneous distribution of resource was compared. Motivated by the above works,we hope to extend some arguments above to system (1.3). That is, we also look forward to comparing the effect of homogeneous versus heterogeneous distribution of resource. Moreover,compared with [15] and [31], we will show the influence of advection and do some further studies. For technical reasons, in this paper, we assume thatm2=Sincefor sufficiently smallη, it would be interesting to extend some of the results in this paper to the case wherem2=further in another paper.

The purpose of this paper is to consider some more dynamics of (1.6) by regarding the movement ratesd1,d2,α1,α2as variable parameters with others fixed.

The rest of this paper is organized as follows. In Subsection 1.2,we present some preliminary results, which may be helpful to verify our results. In Subsection 1.3, we establish our main results (Theorems 1.1—1.4). The proofs will be given in Section 2.

1.2 Preliminaries

Before describing our results, we first introduce some notations and do some preparations.Under (1.5), (1.6) has two semi-trivial steady states for alld1,d2>0 andα1,α2≥0 (see [2]),denoted by (θd1,0), (0,θd2) respectively, whereθd1∈C2(Ω) is the unique positive solution of

andθd2∈C2(Ω) is the unique positive solution of

which immediately implies that

Following the approach in [14], we now define

where

Thus to study the dynamics of system(1.6),we should study the stability of semi-trivial steady states (θd1,0), (0,θd2). Mathematically, the stability of (θd1,0) is determined by the following linear eigenvalue problem

Similarly, the stability of (0,θd2) is determined by the linear eigenvalue problem as follows:

The Krein-Rutman theorem (see [17, p20, Theorems 7.1—7.2]) reads that problem (1.15) and(1.16)admit a principal eigenvalue, denoted byσ1(d2,η,?θd1),σ1(d1,η,m?θd2), and their corresponding eigenfunction can be chosen to be strictly positive in Ω. The following lemma characterizes the linear stability of the two semitrivial steady states of (1.6).

Lemma 1.1(see [2]) (θd1,0)is linearly stable if σ1(d2,η,?θd1)<0and is linearly unstable if σ1(d2,η,?θd1)>0. Similarly,(0,θd2)is linearly stable if σ1(d1,η,m?θd2)<0and is linearly unstable if σ1(d1,η,m?θd2)>0.

Hence, we obtain the following equivalent descriptions:

To understand the dynamics of system(1.6), we also need to consider the neutrally stable case,which leads us to further define

By definition, it is easy to see

Letλ1(η,h) denote the unique nonzero principal eigenvalue of

In fact,λ1(η,h) is also the nonzero principal eigenvalue of

We now collect some properties aboutλ1(η,h), which can be derived in [2].

Lemma 1.2The problem(1.19)has a nonzero principal eigenvalue λ1=λ1(η,h)if and only if h changes sign andh(x)eηPdx/=0. More precisely, if h changes sign, then

(iv)λ1(η,h1)>λ1(η,h2)if h1≤h2, h1/≡h2, and both h1,h2change sign.

(v)λ1(η,h)is continuous in h.

In order to analyze the principal eigenvalue of problems (1.15)and (1.16), it is more convenient to consider the following more general form of eigenvalue problem:

which is equivalent to

The principal eigenvalue of problem(1.21),denoted byσ1(d1,η,h),is expressed by the following variational formula (see, e.g. [2])

The following lemma collects a useful property ofσ1(d1,η,h) (see e.g. [2]).

Lemma 1.3The first eigenvalue σ1(d1,η,h)of(1.21)has the following property:If λ1(η,h)>0, then σ1(d1,η,h)<0 ?d1>

1.3 Main results

Based on the above preparations, we are now ready to state our main results concerning the steady states of (1.6). Before giving the first theorem, motivated by [14], we simply need to define

Since by (1.11)—(1.12), it is easy to verify that

and

Indeed, (1.11) is equivalent to

After dividing by eηP~θd1on both side in the first equation above and integrating over Ω, we obtain the first inequality in (1.27). The second inequality holds similarly. Meanwhile (1.28)can be deduced in [30]. Then we get that

Theorem 1.1Assume that(1.4)-(1.5)hold. Let Lu,Su,Lvand Svbe defined as in(1.24)and(1.25). Then the following statements hold for(1.6):

(i)ForΣu, we have thatΣu={(d1,d2,η):d1∈I,d2>d?2(d1,η)}, where I is defined as in(1.26)and d?2(d1,η)is defined as in(2.4).

(ii)ForΣv, we have thatΣv=?.

Combined with [31, Theorems 1.1—1.3]) and Theorem 1.1, we can characterize the sets Σ?and Σ0,0directly, i.e., Theorem 1.2. Thus the proof is omitted here.

Theorem 1.2Assume that(1.4)-(1.5)hold. Let Lu,Su,Lvand Svbe defined as in(1.24)and(1.25). Then the following statements hold for(1.6):

(i)ForΣ?, we have that

(ii)ForΣ0,0, we have the following characterization:

Hence,Σ0,0/=?if and only if there exists(d1,d2,η)∈Γsuch that θd1=θd2.

Based on Theorem 1.2, we will consider whether the set Σ?is empty for larged1. Furthermore, if Σ?is nonempty, we study what the asymptotic behavior of the unique coexistence steady state of (1.6)is asd1→+∞when (d1,d2)∈Σ?. To deal with these problems, we shall analyze the asymptotic behavior ofd?2(d1,η) asd1→+∞more carefully.

For eachD >0, we set ΓD:= {(d1,d2,η) ∈Γ :d1> D}. Denote byρm,η,Pthe unique solution satisfying:

and

Theorem 1.3Assume that(1.4)-(1.5)hold. Then there exists a Dm,η,P>0depending only on m,η,P such that the followings hold for(1.6):

(i)If+C(m,η,P)>0, then for all d1> Dm,η,P,(θd1,0)is linearly stable, i.e.,

(ii)If+C(m,η,P)=0, thenfor all d1>Dm,η,P.

(iii)If+C(m,η,P)<0, then for all d1>Dm,η,P, there exist two numbers

andΠm,η,P∈Rdepending only on m,η,P(x)such that

which implies thatis nonempty.

Finally, we state a result which characterizes the asymptotical behavior of the coexistence steady state in details for the case infΩρm,η,P+C(m,η,P)≤0 asd1→+∞andd2→0.

Theorem 1.4Assume that(1.4)-(1.5)hold and+C(m,η,P)≤0. Let(d1,d2,η)∈Σ?and(U,V)be the corresponding unique coexistence steady state of(1.6). Then there exists a constant Dm,η,P>0depending only on m,η,P such that the following holds:

If we assume further that+C(m,η,P)<0, then

where p∈[0,Λm,η,P), and C1and C2are two positive constants depending only on η,P and m.

In fact, in [31], the existence and the globally asymptotic stability of co-existence steady state of (1.3) has been considered. Hence, combined with the results in this paper, some properties of steady states of (1.6) are clear.

2 Proofs of the Main Results

Proof of Theorem 1.1The proof is divided into three steps.

Step 1(d1,d2,η)∈Σuindicates that (d1,η)∈I.

Suppose that (d1,η)/∈I, whereIis defined as in (1.26). Then

which implies thatσ1(d2,η,?θd1) ≥0, i.e., (d1,d2,η) /∈Σu. Hence (d1,d2,η) ∈Σuimplies that (d1,η)∈I.

We next characterize the setIin detail.

Step 2I1/=?if and only ifLu<1

Indeed,it suffices to show thatLu<10,y0∈, such that

That is, (d′1,η)∈I1/=?, which finishes the proof of Step 2.

Step 3SinceLu<1< Su, it immediately follows thatI=I0∪I1?R+× R+. If(d1,η)∈I0, thenσ1(d2,η,?θd1)<0 by (1.23). If (d1,η)∈I1, then

by Lemmas 1.2—1.3. Hence after defining

we obtain that (d1,d2,η)∈Σuif and only if (d1,η)∈Iandd2>d?2. This finishes the proof of Theorem 1.1(i). The proof of Theorem 1.1(ii) is in fact the same as (i) and is thus omitted.

Next in order to establish Theorem 1.3, motivated by [15], we need to verify the following asymptotic expansion ofθdasd→+∞,which will be used later.

Proposition 2.1Assume that(1.4)-(1.5)hold. Let θdbe the unique solution of

Then there exists a constant Dm,η,P>0depending only on m,η,P such that

for all d > Dm,η,P, where ρm,η,P, C(m,η,P)are defined as in(1.32)-(1.33), and γm,η,P,K(m,η,P)are defined below:

ProofMultiplying the equation ofρm,η,Pbyγm,η,PeηPand the equation ofγm,η,Pbyρm,η,PeηP, and then we see by (1.33), from integrating by parts that

This, together with (2.8), implies that

Hence there exists a uniqueθ3satisfying

Let nowθ4be the unique solution to

By some straightforward computations, we have

Thusθ±is a pair of upper and lower solutions to (2.5) for alldsufficiently large. Note that for alldsufficiently large, 0<θ?<θ+, by the upper/lower solution method (see [29]) and the uniqueness ofθd, we have thatθ?≤θd≤θ+. This thus finishes the proof of (2.6).

Now we are going to prove Theorem 1.3.

Proof of Theorem 1.3We divide this proof into several cases.

Case 1+C(m,η,P)>0.

It follows from (2.6) that there exists a constantDm,η,P>0 such that<θd1onfor alld1> Dm,η,P. Hence,σ1(d2,?θd1)< σ1(d2,0) = 0 by (1.23), which implies that (θd1,0) is linearly stable.

Case 2+C(m,η,P)=0. Thusρm,η,P+C(m,η,P)≥0.

Now without loss of generality, we may assume that?θd1changes sign in Ω for alld1large enough. By (1.27) and Lemma 1.2,λ1(η,?θd1)>0. Hence in order to prove (ii), it suffices to show that there exist two constantsCm,η,P>0 andDm,η,P>0 depending only onm,η,Psuch that

i.e.,λ1> Cm,η,Pfor alld1> Dm,η,P. By (2.6), there exists a constantDm,η,P>0 such that

For eachL>0,

Hencechanges sign and furthermoreis defined and positive for alld1andLsufficiently large. Moreover, by Lemma 1.2,Since+C(m,η,P)=0, (2.16) and (2.17) imply that there exist two constantsC2L

Moreover,choosingLandDm,η,Peven larger if necessary,one can see that=Lfor alld1> Dm,η,P. Applying a similar approach in [28], one can see that there exists a constantCm,η,P>0 depending only onm,η,Psuch thatλ1(η,ΘLd1)>Cm,η,Pfor alld1>Dm,η,P.

Case 3+C(m,η,P)<0.

by (1.27) and Lemma 1.2, we then observe that bothλ1(η,?θd1) andλ1(η,(?ρm,η,P?C(m,η,P))eηP) are defined and positive for alld1> Dm,η,P.Now letφ?>0 be the principal eigenfunction corresponding toλ1(η,(?ρm,η,P?C(m,η,P))eηP) normalized such that=1 and define

Clearly,λ1(η,?θd1)=d1λ1(η,d1?d1θd1).

We next verify that in fact

for alld1> Dm,η,P, where Λm,η,P:=Since by (2.6), we have

the continuity ofλ1(·) in Lemma 1.2 then leads to

Denoteφ >0 the principal eigenfunction corresponding toThen one can easily check that

asd1→+∞. Rewrite thatwhereψis the unique solution of

Denote

We then obtain that

and it follows from some calculations thatωsatisfies the following equations:

Using (2.6), one sees

Multiplying the equation forωbyφeηPand the equation forφbyωeηP, integrating over Ω and subtracting, we deduce that

Combining (2.23), (2.27) and (2.29) together, we see

asd1→+∞. By (2.24), we have

asd1→+∞. Hence dividing both sides of (2.30) bydxand lettingd1→+∞, using (2.23)—(2.24), (2.27), (2.29) and the above estimate again, we derive that

which implies thatλ1(η,d1?d1θd1)?This together with (2.27), (2.29)and(2.31) implies that

Therefore dividing both sides of (2.30) by?C(m,η,P))φφ?e2ηPdxand lettingd1→+∞,we obtain(2.21). This in turn indicates thatλ1(η,m?θd1)=and

The proof of Theorem 1.3 is thus finished.

Proof of Theorem 1.4Let (U,V) be the coexistence steady state of (1.6). Then(,)=(Ue?ηP,Ve?ηP) satisfies

By the maximum principle, we have that

Integrating the equation forover Ω, we obtain from H¨older inequality that

where we have used the identity(m?U?V)dx=dxobtained by multiplying the equation ofby eηP, and dividing by, integrating over Ω. Since there exists a constantC >0 such that

which can be derived by a similar method of the proof of Poincar′e’s inequality in [11], then(2.38) gives rise to

The inequality above implies

Since ‖V‖∞0 depending only onm,η,Psuch that both ‖ρm?V,η,P‖∞andC(m?V,η,P) are uniformly bounded ind2for alld1>Dm,η,P. Therefore, similar to (2.6), together with (2.41), we obtain that

and

On the other hand, from the equation forand the maximum principle, we have

Thus we have verified (1.36). It only remains to prove (1.37). We claim that

Letrbe the unique solution to

Multiplying the equation forrbyreηPand integrating over Ω, we obtain that

where we have used Young’s inequality. By means of a similar inequality to (2.39), we have that

Choosingε>0 small enough, we then derive from (1.36) and the above estimate that

Since

by (2.47) and (1.36), it is easy to see from (1.32) and (2.47) that

This together with (2.51)and the estimate ofVin (1.36)implies the second equality of (2.46).By (1.33) and the above identity, we have

We now assume that infΩρm,η,P+C(m,η,P)<0 and prove (1.37). By (1.36), it suffices to show that

The equation forVreads thati.e.,

By (2.42) and (2.46),

uniformly ind2, for alld1> Dm,η,P. Assuming for contradiction that (2.54) does not hold.By (1.36), passing to a subsequence ofd1andd2if necessary, we get thatd1‖V‖∞→0 asd1→+∞andd1d2→p∈[0,Λm,η,P), which further implies that

However, taking limits on both sides of (2.55) asd1→+∞andd1d2→p∈[0,Λm,η,P),we obtain from the above estimate, (2.56) and Lemma 1.2 that=λ1(η,?eηP(ρm,η,P+C(m,η,P)))=Λm,η,P, which is a contradiction. This finishes the proof of (1.37).

AcknowledgementThe author sincerely thanks the reviewers for their kind comments.