# Stability of Traveling Waves to the Lotka-Volterra Competition Model

Authors: Ahmad Alhasanat; Chunhua Ou
Journal: Complexity
 (2019)

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## Abstract

In this paper, the stability of traveling wave solutions to the Lotka-Volterra diffusive model is investigated. First, we convert the model into a cooperative system by a special transformation. The local and the global stability of the traveling wavefronts are studied in a weighted functional space. For the global stability, comparison principle together with the squeezing technique is applied to derive the main results.


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## Body

## 1. Introduction

We are concerned here with the diffusive Lotka-Volterra competition model(1)ϕt=d1ϕxx+r1ϕ1-b1ϕ-a1ψ,ψt=d2ψxx+r2ψ1-a2ϕ-b2ψ,with the initial data(2)ϕx,0=ϕ0x≥0,ψx,0=ψ0x≥0,∀x∈R.Here ϕ(x,t) and ψ(x,t) are the population densities at time t and location x; d1 and d2 are the diffusive coefficients; r1 and r2 are the net birth rates; a1 and a2 are the competition coefficients; 1/b1 and 1/b2 are the carrying capacities for each species. For derivation and biological interpretation of this model, we refer readers to [1, 2].Using the transformations(3)r1d1x→x,r1t→t,b1ϕx,t=ϕ~x,t,b2ψx,t=ψ~x,t,d=d2d1,r=r2r1,a1b2→a1,a2b1→a2,the nondimensional form of the system becomes(4)ϕ~t=ϕ~xx+ϕ~1-ϕ~-a1ψ~,ψ~t=dψ~xx+rψ~1-a2ϕ~-ψ~. By letting u=ϕ~,v=1-ψ~, this model can be further written as a cooperative system(5)ut=uxx+u1-a1-u+a1v,vt=dvxx+r1-va2u-v,with(6)ux,0=u0x=ϕ~x,0,vx,0=v0x=1-ψ~x,0,∀x∈R.For our study, we will assume that u0(x) and v0(x) are nonnegative. The existence and uniqueness of the solution of the above problem can be easily verified by a classical argument of Picard’s iteration. Throughout this paper, we assume that the conditionC10<a1<1<a2is satisfied. Under this condition, equilibria to system (5) in the region u,v∣0≤u≤1,0≤v≤1 are only (0,0),(0,1), and (1,1). In the absence of diffusion in the system (5), it can be shown that (0,0) is unstable and (1,1) is stable. For the system, we are particularly interested in the traveling wave solution, connecting (1,1) and (0,0), in the form(7)u,vx,t=U-,V-z, where z=x-ct is the wave variable, c≥0 is the wave speed, and (U-,V-) is called the wavefront and satisfies(8)0=U-zz+cU-z+U-1-a1-U-+a1V-,0=dV-zz+cV-z+r1-V-a2U--V-,subject to(9)U-,V--∞=1,1,U-,V-∞=0,0.This is equivalent to studying traveling waves for the original competition system (4) that connect the boundary equilibria (0,1) and (1,0).The existence of traveling waves to the above problem is well-studied in literature. It is known that there existsc∗≥0 so that problem (8)-(9) has a monotone solution (U-,V-)(z) for c≥c∗ and no wavefront exists for c<c∗; see [3–6]. c∗ is called the minimal wave speed for this system and satisfies c∗≥21-a1. When c∗=21-a1, we say that the minimal wave speed islinearly determined; see the details in [4].We know that(U-,V-)(x-ct) is a special pattern that only satisfies the first two equations in (5). For the stability of this pattern, we want to know if the solution of (5) tends to (U-,V-)(x-ct) for given initial data u0(x) and v0(x). To this end, we use the (z,t)-coordinate and(10)u,vx,t=U,Vz,t, to transform the uv-model (5) into the partial differential model(11)Ut=Uzz+cUz+U1-a1-U+a1V,Vt=dVzz+cVz+r1-Va2U-V,subject to(12)Uz,0=u0z,Vz,0=v0z,∀z∈R. It is easy to see that (U-,V-)(z) is the steady-state to the above new system.We should mention that dynamics for (4) is very rich. There are always three nonnegative equilibria (0,0),(1,0), and (0,1). In the case when a1<1,a2<1, or the case when a1>1,a2>1, there exists a unique positive coexistence equilibrium (13)ϕ~∗,ψ~∗=1-a11-a1a2,1-a21-a1a2.Based on the phase plane analysis to the ordinary differential system of (4) without diffusion terms, the nonlinearity of the model (4) when a1<1 and a2<1 is called the persistence case (or coexistence). Likewise, the nonlinearity is called the monostable case when a1<1 and a2>1 are satisfied, or the bistable case when a1>1 and a2>1. Traveling waves to (4) have been investigated considerably. For the bistable case, please see [7, 8] for the existence of traveling waves connecting (1,0) and (0,1), and [9] for the uniqueness and parameter dependence of wave speeds. For the monostable case, we refer to [3, 10] for the existence of traveling waves, and [11, 12] for the selection of the minimal speed. For the persistence (coexistence), the existence of traveling wave connecting (0,0) and (ϕ~∗,ψ~∗) has been studied in [13, 14]. When time delays are incorporated into (4) in the persistence case, Li et al. [15] and Gourley and Ruan [16] have proved the existence of traveling waves.The stability of traveling waves to a scalar partial differential equation has been well-studied, e.g., [17–27], the monograph [6, 28] and the survey paper [29]. Indeed, the extension of this study to a general system is not trivial. As we know, when time delays are directly incorporated in the competition terms in (4), the system becomes nonmonotone and the comparison principle cannot work. Alternatively, in [30, 31], the authors studied the stability of traveling waves for the so-called cooperative delayed reaction diffusion system by changing the signs of a1 and a2. To be exact, with putting delay=0, they studied the cooperative system (14)ϕt=d1ϕxx+r1ϕ1-b^1ϕ+a^1ψ,ψt=d2ψxx+r2ψ1+a^2ϕ-b^2ψ, where di,ri,a^i, and b^i are all positive. This corresponds to the persistence case in our model (4). Under the condition b^1b^2-a^1a^2>0, a positive equilibrium(15)ϕ+,ψ+=a^1+b^2b^1b^2-a^1a^2,b^1+a^2b^1b^2-a^1a^2 exists. They proved that the traveling wave fronts, connecting (0,0) and (ϕ+,ψ+), are exponentially stable in some weighted L∞ spaces, and obtained the decay rates by the weighted energy estimate.Despite the success in the study of the stability of traveling waves to the classical model (4) in the bistable and persistence cases, the stability of traveling wave in the monostable remains still unsolved. The purpose of this paper is to systematically study the local and the global stability of the steady-state (U-,V-)(z). Using the method of spectrum analysis in [32], we give the local stability. For the global stability, we construct an upper and a lower solutions to the system (11), and prove their convergence to the traveling wave (U-,V-)(z). In view of comparison together with the squeezing technique, we arrive at new results on the global stability of the traveling waves. We remark that our method is different from that in [30, 31] where weighted energy method was applied.The rest of the paper is organized as follows. Local analysis of the wave profile near the unstable point is studied in Section2. In Section 3, we study the local stability of the steady-state by applying the standard linearization. The resulting spectrum problem is studied by the method in [32]. A suitable weighted functional space is chosen to proceed the analysis. In Section 4, besides the weighted functional space, the upper-lower solution method together with the squeezing technique is applied to derive the global stability results. Conclusions are presented in Section 5.


## 2. The Local Analysis of the Wave Profile Near the Equilibrium0,0

In this section, we study the behavior of the traveling wave(U-,V-)(z) locally near the equilibrium (0,0). Assume that the solution has exponential decay as z→∞. Indeed this claim can be easily verified by the maximum principal coupled with a comparison near the neighborhood of infinity. Therefore, we set (16)U-,V-z~ζ1e-μz,ζ2e-μzasz→∞, for positive constants ζ1,ζ2, and μ. By substituting this into (8) and linearizing the equations we have(17)Aμζ1ζ2=00,where A(μ) is given by(18)Aμ=μ2-cμ+1-a10ra2dμ2-cμ-r.The system of algebraic equations (17) has a nontrivial solution if and only if det(A)=0. This implies μ=μ1,2,3>0, where(19)μ1c=c-c2-41-a12,μ2c=c+c2-41-a12,and(20)μ3c=c+c2+4dr2d.Indeed, a condition so that μ1 and μ2 are reals is(21)c≥21-a1≔c0.Forc>c0, obviously μ1<μ2. When 0≤d<1, we have also μ2<μ3 for all c>c0, i.e., e-μ1z dominates both of e-μ2z and e-μ3z. In this case, the eigenvector of A(μ) corresponding to μi, for i=1,2, is the strongly positive vector (ζ1(μi)ζ2(μi))T, where(22)ζ1μi=-dμi2-cμi-randζ2μi=ra2.It follows that(23)U-zV-z=C1ζ1μ1ζ2μ1e-μ1z+C2ζ1μ2ζ2μ2e-μ2z,asz→∞,for C1>0 or C1=0, C2>0. For the case when(24)1<d<2+r1-a1≔d^,the same behavior in (23) is still true if c∗<c≤c^, where(25)c^=r+1-a1d-1+1-a1d-1r+1-a1.If c>c^, then μ1<μ3<μ2 and we have(26)U-zV-z=C1ζ1μ1ζ2μ1e-μ1z+C2-ζ1μ2-ζ2μ2e-μ2z+C301e-μ3z,asz→∞,for C1>0 or C1=0, C2,3>0. Here, (01)T is the eigenvector of A(μ) corresponding to μ3, and note that ζ1(μ2)<0 in this case. On the other hand, when(27)d>d^,(U-,V-)(z) behaves like (26) if c>c^. For the case when c∗<c<c^, we have μ3<μ1<μ2. Hence,(28)U-zV-z=C1-ζ1μ1-ζ2μ1e-μ1z+C2-ζ1μ2-ζ2μ2e-μ2z+C301e-μ3z,asz→∞,for C1,3>0, or C1=0, C2,3>0. We summarize the above behaviors in Table 1.Table 1
The asymptotic behavior of the wave profile(U-,V-) near infinity.
Condition ond Condition onc The asymptotic behavior 0≤d<1 c > c ∗ (23) 1<d<d^ c ∗ < c < c ^ (23) 1<d c > c ^ (26) d>d^ c ∗ < c < c ^ (28)Kan-on in [3] derived the asymptotic behaviors of (U-,V-)(z) near infinity when c≥c∗. After deriving the behavior of U-(z), he used it into the V-equation to find the behavior of V-(z) when μ1≤μ2≤μ3 and when μ3≤μ1≤μ2. Our result here agrees with that in [3] when c>c∗. We further study the case when μ1<μ3<μ2.Finally, we have the asymptotic behavior for the solutionU-(z) when the wave speed is greater than the minimal speed c∗.Theorem 1.
Forc>c∗, the wavefront U- has the following behavior: (29)U-z~C1e-μ1z,asz→∞for some C1>0.Proof.
On the contrary, assume that for somec1>c∗, the wavefront U- has the following behavior:(30)U-z~C2e-μ2z,asz→∞for some C2>0. By this assumption, it follows that (U-,V-)(x-c1t) is a solution to the following partial differential equation:(31)ut=uxx+u1-a1-u+a1v,vt=dvxx+r1-va2u-v,with the initial conditions(32)ux,0=U-xandvx,0=V-x. We know that there exists a monotonic traveling wavefront to the system (31) for any c≥c∗. In particular, assume (U,V)(x-ct) is a solution for some c∈(c∗,c1) with the initial condition(33)ux,0=Uxandvx,0=Vx.By a simple computation of the asymptotic behavior of this solution to (8)-(9) near ±∞, we can always obtain (by shifting if necessary) U-(x)≤U(x) for all x∈(-∞,∞). From the second equation of (8), we have V-(x)≤V(x) for all x∈(-∞,∞). From (31), by comparison, we get(34)U-x-c1t≤Ux-ct,V-x-c1t≤Vx-ct,for all (x,t)∈(R,R+). On the other hand, fix ξ=x-c1t. Then U¯(ξ)>0 is fixed, and we have(35)Ux-ct=Uξ+c1-ct~U+∞=0ast→∞.By (34), this implies that U-(ξ)≤0, which is a contradiction. The proof is complete.


## 3. The Local Stability

To study the local stability, as usual, we add a small perturbation to the traveling wave and study the behavior of this perturbation for large time period. If this perturbation decays, then we say that the traveling wave is locally stable. Forδ≪1 and a parameter λ, let (36)Uz,t=U-z+δϕ1zeλt,Vz,t=V-z+δϕ2zeλt, where ϕ1 and ϕ2 are two real functions. Substitute these formulas into (11) and linearize the system about (U-,V-) to get the following spectrum problem:(37)λΦ=LΦ≔DΦ′′+cΦ′+JzΦ,where Φ=(ϕ1ϕ2)T, D and J(z) are 2×2 matrices given by(38)D=100dandJz=1-a1-2U-+a1V-a1U-ra21-V-r-1-a2U-+2V-.ForΦ in a suitable space, we shall find sign of the maximal real part to the spectrum (λ) of the operator L to determine the local stability of the traveling wave solution. To proceed, we introduce a weighted functional space Lwp,(39)Lwp=fz:wzfz∈LpR,p≥1with the norm(40)fzLwp=∫-∞∞wzfzpdz1/p, where(41)wz=1w1z,1w2zis the weight function with(42)w1z=e-αz-z0,z>z01,z≤z0,w2z=e-βz-z0,z>z01,z≤z0,for some positive constants α, β, and z0 to be chosen. Here, Lp(R), for p≥1, is the well-known Lebesgue space of integrable functions defined on R. Then we consider the operator L on this new space and find its spectrum. To do this, we write Φ(z) in the form(43)Φ=ϕ1ϕ2=w1ψ1w2ψ2,for Lp-functions ψ1 and ψ2. Substituting (43) into (37) gives a new spectrum problem in the weighted space Lwp,(44)λΨ=LwΨ≔DΨ′′+MzΨ′+NzΨ, where Ψ=(ψ1ψ2)T, M(z) and N(z) are 2×2 matrices defined by(45)Mz=c+2w1′w100c+2dw2′w2and (46)Nz=w1′′w1+cw1′w100dw2′′w2+cw2′w2+Yz, with the ik-element of the matrix Y(z), yik, being given in terms of the ik-element of the matrix J(z) as yik=wk/wijik; that is,(47)Nz=w1′′w1+cw1′w1+1-a1-2U-+a1V-a1U-w2w1ra21-V-w1w2dw2′′w2+cw2′w2+r-1-a2U-+2V-.The details to find the essential spectrum of the operatorLw can be finalized by using Theorem A.2 in [32] and are given below. After we choose the weight function so that the essential spectrum is on the left-half complex plane, we can determine the sign of the maximal real part of the point spectrum in the weighted space as well.First of all, to apply the method in [32], we need to choose α and β so that the matrix functions M(z) and N(z) are bounded; i.e., the limits(48)limz→∞⁡U-zw2zw1z=A1andlimz→∞⁡1-V-zw1zw2z=A2,for some constants A1 and A2, are satisfied. We choose(49)α-μ1<β≤α,where μ1 is defined in (19). This makes, by using Theorem 1, A1=0 and(50)A2=0whenβ<α,1whenβ=α.Now, we define(51)S±≔λ∣det-τ2D+iτM±+N±-λI=0,-∞<τ<∞,where M± and N± are the limits of M(z) and N(z) as z→±∞, respectively. Then the essential spectrum of the operator Lw is contained in the union of regions inside or on the curves S+ and S-; see [32, pp. 140]. By letting z→+∞, M+, and N+ are given as (taking condition (49) into account)(52)M+=c-2α00c-2dβandN+=α2-cα+1-a10A2dβ2-cβ-r.The equation det(-τ2D+iτM++N+-λI)=0 has two solutions λ=λ1,2, where (53)λ1=-τ2+iτc-2α+α2-cα+1-a1,λ2=-τ2d+iτc-2dβ+dβ2-cβ-r.This means that S+ is the union of two parabolas in the complex plane which are symmetric about the real axis; namely,(54)S+,1=λ1∣-∞<τ<∞andS+,2=λ2∣-∞<τ<∞.The most right points of these curves are α2-cα+1-a1 and dβ2-cβ-r, respectively, which are negative if(55)α∈μ1,μ2andβ∈0,μ3,where μ1,μ2, and μ3 are defined in (19)-(20). Hence, when the above condition satisfies, S+=S+,1∪S+,2 is on the left-half complex plane.Similarly, we findS- by solving the equation det(-τ2D+iτM-+N--λI)=0, with(56)M-=c00candN-=-1a10r1-a2.This gives two solutions λ=λ3,4, where (57)λ3=-τ2+iτc-1,λ4=-τ2d+iτc+r1-a2.From C1, S-=λ3∣-∞<τ<∞∪λ4∣-∞<τ<∞ is on the left-half complex plane.The above analysis shows that the essential spectrum ofLw is on the left-half complex plane as long as conditions (49) and (55) are satisfied. In fact, there are many choices of α and β satisfying these conditions depending on μ1,μ2, and μ3. We choose them by the following algorithm.Algorithm 2.
Two mechanisms are valid to chooseα and β so that all conditions in (49) and (55) hold: (1)
Ifμ1<μ3, then we choose β=α for any α∈(μ1,min{μ2,μ3}).(2)
Ifμ1≥μ3, then we choose ϵ<β<μ3 and α=μ1+ϵ for small ϵ>0. In particular, we can choose β=2ϵ and α=μ1+ϵ, for ϵ<min{μ2-μ1,μ3/2}.
Finally, in order to get a local stability result, we need to check the sign of the principal eigenvalue in the point spectrum for (37)-(38). Consider the associated linear partial differential system(58)ut=Duzz+cuz+Jzu,where u(z,t)=(u1(z,t),u2(z,t)). The eigenpair (λ,Φ) of (37) implies a solution eλtΦ to the above system. Let Qt=u(t,z,ϕ) denote the solution semiflow of (58) for any given initial data ϕ in Lp. It is easy to see Qt is compact and strongly positive. By the well-known Krein-Rutman theorem (see, e.g., [33]), Qt has a simple principal eigenvalue λmax with a strongly positive eigenvector, and all other eigenvalues eλt must satisfy(59)eλt<eλmaxt. For any c>c∗, we have from Theorem 1 that U-(z)~C1e-μ1z, C1>0, as z→∞. λ=0 is an eigenvalue to the operator L defined in (37) with the one-sign (strongly positive) eigenvector (-U-′,-V-′)(z). By the choice of the weighted functional space Lwp, the one-sign eigenvector (U-′,V-′)(z) is not inside. Hence, the real parts of point spectrum of the operator Lw in Lwp are all negative. We can also explain this in a simple analysis. Assume to the contrary that (λ,Φ) is an eigenpair of the eigenvalue problem (37)-(38) with λ>0 and Φ∈Lwp. Obviously, the one-sign function Φ¯=(-U-′,-V-′)(z) satisfies (58). For Φ in the Lwp-space, we have essentially (or except for a set of zero measure) Φ¯(z)>Φ(z) as z→∞. On the other hand, when z→-∞, we can apply the method of asymptotic analysis and assume that the eigenfunction of (37) behaves like keμz for some positive values k and μ. By substituting it into the eigenvalue problem and using the behavior of J(z), we obtain that μ is increasing with respect to λ. This implies that Φ¯(z)>Φ(z) as z→-∞. Hence, by choosing k¯ sufficient large, we can have k¯Φ¯≥Φ. By comparison, from the partial differential system (58), we obtain k¯Φ¯(z)≥Φeλt, which contradicts λ>0. This implies that for Φ∈Lwp, the real parts of all eigenvalues λ of (37) should be nonpositive.
Now we are in a position to state the local stability result.Theorem 3.
For anyc>c∗, the wavefront (U-,V-)(z) is locally stable in the weighted functional space Lwp with the weight function w(z) defined in (41)-(42), where α and β in the formula of w(z) are chosen by Algorithm 2.


## 4. The Global Stability

We study here the global stability of the steady-state(U-,V-)(z) in a special choice of the weighted functional space Lwp(R). Let p=∞ and define the norm fLw∞=esssupz∈Rwzfz, for some weight function w(z). Assume μ1<μ3. By Algorithm 2, we choose α=β∈(μ1,min{μ2,μ3}). Specifically, let α=β=μ1+ϵ, for small positive number ϵ. Also, we assume that the functions U-(z) and V-(z) satisfy the conditionC2V-zU-z≤mina2,1a1,∀z∈-∞,+∞.Theorem 4.
Supposec>c∗, μ1<μ3, and conditions C1-C2 hold true. Assume that the initial data U(z,0)=U0(z) and V(z,0)=V0(z) satisfy (60)0,0≤U0,V0z≤1,1,∀z∈R,limz→-∞inf⁡U0,V0z>0,0,and(61)U0z-U-z∈Lw∞R,V0z-V-z∈Lw∞R. Then the solution (U,V)(z,t) to (11) exists globally with(62)0,0≤U,Vz,t≤1,1,∀z,t∈R×R+, and converges to the steady-state (U-,V-)(z) exponentially in the sense of (63)supz∈R⁡Uz,t-U-z≤ke-ηt,t>0,supz∈R⁡Vz,t-V-z≤ke-ηt,t>0, for positive constants k and η.To prove Theorem4, we will find an upper and a lower solution to the partial differential equations system (11). For z∈R, define (64)U0+z=maxU0z,U-z,V0+z=maxV0z,V-z,U0-z=minU0z,U-z,V0-z=minV0z,V-z. It is easy to see that the following inequalities are true:(65)0,0≤U0-,V0-z≤U0,V0z≤U0+,V0+z≤1,1,0,0≤U0-,V0-z≤U-,V-z≤U0+,V0+z≤1,1.Denote (U+,V+)(z,t) and (U-,V-)(z,t) as the solutions to the system (11) with the initial data (U0+,V0+)(z) and (U0-,V0-)(z), respectively; that is,(66)Ut±=Uzz±+cUz±+U±1-a1-U±+a1V±,Vt±=dVzz±+cVz±+r1-V±a2U±-V±,U±,V±z,0=U0±,V0±z.By the comparison principle, one gets(67)0,0≤U-,V-z,t≤U,Vz,t≤U+,V+z,t≤1,1,∀z,t∈R×R+,0,0≤U-,V-z,t≤U-,V-z≤U+,V+z,t≤1,1,∀z,t∈R×R+.In the following lemmas we shall prove the convergence of(U+,V+)(z,t) and (U-,V-)(z,t) to the wavefront (U-,V-)(z). Then we apply the squeezing theorem to obtain the result in Theorem 4.Lemma 5.
Under the conditions in Theorem4, (U+,V+)(z,t) converges to (U-,V-)(z).Proof.
For(z,t)∈R×R+, define(68)Pz,t=U+z,t-U-zandQz,t=V+z,t-V-z.These functions, P and Q, satisfy the initial value conditions(69)Pz,0=U0+z-U-zandQz,0=V0+z-V-z.By (65) and (67), for all z∈R and t≥0, we have(70)0,0≤P,Qz,t≤1,1.By (8) and (66) and using condition C2, we can verify that P and Q satisfy(71)Pt≤Pzz+cPz+1-a1P+P+U--P+a1Q,Qt≤Qzz+cQz+ra2P-Q+rQ+V--a2P+Q.To study the stability in the weighted functional space Lw∞, with w(z) defined in (41), we first let (72)PQz,t=e-αz-z0P-Q-z,t,for  allz,t∈R×R+,where P- and Q- are functions in L∞(R) and z0 is the same number used in the weight function w(z). This gives(73)P-Q-t≤DP-Q-zz+MP-Q-z+AαP-Q-+U-+e-αz-z0P--P-+a1Q-rV-+e-αz-z0Q--a2P-+Q-≔L1P-,Q-L2P-,Q-,where A(α) is the same matrix defined in (18) and M=diag⁡(c-2α,c-2dα).
DefineP-1(z,t) and Q-1(z,t) as(74)P-1z,t=k1ζ1e-η1tandQ-1z,t=k1ζ2e-η1t,∀z,t∈R×R+, for some constants k1,η1>0 to be chosen and (ζ1,ζ2)=(ζ1(α),ζ2(α)) is the eigenvector of the matrix A(α) associated with the eigenvalue α2-cα+1-a1. Simple computations give (75)ζ1α=α2-cα+1-a1-dα2-cα-r=μ12+ϵ1-d+1-a1+r,ζ2α=ra2, which are positive for small ϵ and μ1<μ3. Since the initial values P-(z,0) and Q-(z,0) are in the space Lw∞, we can choose k1≥maxz∈R{P-(z,0)/ζ1,Q-(z,0)/ζ2}. Direct computations and using condition C2 show that both of L1(P-1,Q-1) and L2(P-1,Q-1) are negative. This allows choosing a positive value to η1 so that the inequality(76)P-1Q-1t=-η1k1ζ1ζ2e-η1t≥L1P-1,Q-1L2P-1,Q-1holds. Hence, since (P-1,Q-1)(0,z)≥(P-,Q-)(0,z) and by comparison on unbounded domain, see, e.g., [34, Proposition 2.1],(77)P,Qz,t=P-,Q-e-αz-z0≤k1ζ1,ζ2e-αz-z0-η1t,∀z,t∈R×R+. In particular, this is true when z∈[z0,∞), for any fixed z0.
Now, we introduce the weight functionw(z) defined in (41)-(42) with α=β=μ1+ϵ. By the above analysis, we need to prove the convergence of (P,Q)(z,t) to (0,0) for z∈(-∞,z0]. Note that the full system of (P,Q)(z,t) can be expressed as(78)PQt=DPQzz+cPQz+JzPQ+-P+a1QPr-a2P+QQ.Here, J(z) is the same 2×2 matrix defined in (38). Let z0 be chosen so that(79)Jz≤-1+ϵ1a1+ϵ1ϵ1r1-a2+ϵ1≔Jϵ1,for some given small ϵ1>0, when z≤z0. This is equivalent to require that (U-,V-)(z) is close to (1,1) for all z≤z0. Define (P^,Q^)(t) as the solution of the autonomous system(80)P^Q^t=Jϵ1P^Q^+-P^+a1Q^P^r-a2P^+Q^Q^,with the initial data(81)P^0≥P-z,0,Q^0≥Q-z,t,∀z∈R.Then (P^,Q^) is an upper solution to system (78).
Now we need to prove the convergence of(P^,Q^)(t) to (0,0) as t→∞. The Jacobian matrix J(0,0)=Jϵ1 of system (80) at the fixed point (0,0) has two eigenvalues, λ^2<λ^1<0. By the phase plane analysis, there exists 0<δ≤1 so that the flow in the P^Q^-space converges to origin for any initial data (P^,Q^)(0) in the box [0,1]×[0,δ]. Hence, we conclude that(82)P^,Q^=k^1C^1,C^2eλ^1tast→∞, for positive constant k^1 and (C^1C^2)T is the eigenvector of Jϵ1 corresponding to λ^1. For the maximal possible choice of the constant δ so that we have the convergence result inside the box [0,1]×[0,δ]; see Remark 6.
We can choosek^1 large and λ¯1=min{η1,-λ^1} so that, at the boundary z=z0, we have(83)P,Qz0,t≤k1ζ1,ζ2e-η1t≤k^1ζ1,ζ2e-λ¯1t. Hence, by comparison on the domain (-∞,z0]×[0,∞), see, e.g., [35, Lemma 3.2],(84)P,Qz,t≤k^1ζ1,ζ2e-λ¯1t,∀z,t∈-∞,z0×R+.This completes the proof.Remark 6.
The maximal possible value of the constantδ, which could be 1, depends on the location of the fourth fixed point to the system (80) near or inside the box [0,1]×[0,1]. See Figure 1 for all possible different cases. In Figure 1(a), the positive fixed point is far away from the box [0,1]×[0,1] and does not affect the flow. This happens when a2>2. Hence we set δ=1. Figure 1(b) shows the effect of the positive fixed point on the flow, which still outside the box. The maximal choice of δ for this case exists in the interval (a2-1-ϵ1/r,1). The number a2-1-ϵ1/r is the positive Q^-intercept of the nullcline Q^t=0. A fixed point exists inside the box [0,1]×[0,1] in Figure 1(c), where δ becomes close to the value a2-1-ϵ1/r.Figure 1
The phase portrait of system (80) when ϵ1=0.003 and r=1.875.
(a)
a 1 = 0.5 and a2=2.4. We choose δ=1 (b)
a 1 = 0.5 and a2=1.4. The maximal possible value of δ is in (0.3984,1) (c)
a 1 = 0.3 and a2=1.4. The maximal possible value of δ becomes close to 0.3984Lemma 7.
Under the conditions in Theorem4, (U-,V-)(z,t) converges to (U-,V-)(z).Proof.
For(z,t)∈R×R+, define (85)Rz,t=U-z-U-z,tandSz,t=V-z-V-z,t. These functions, R and S, satisfy the initial value conditions(86)Rz,0=U-z-U0-zandSz,0=V-z-V0-z.From (65) and (67), for all z∈R and t≥0, we have(87)0,0≤R,Sz,t≤1,1. From (8) and (66), R and S satisfy the system(88)RSt=DRSzz+cRSz+JzRS--R+a1SRr-a2R+SS,with J(z) defined in (38). By condition C2, we have(89)Rt≤Rzz+cRz+1-a1R+R-U-R-a1S,St≤dSzz+cSz+ra2R-S+rS-V-a2R-S.Similar to the previous analysis in the proof of Lemma 5, and making a use of the facts R<U- and S<V-, we can prove that there exist η2>0 and(90)k2≥eαz-z0maxz∈R⁡Rz,0ζ1,Sz,0ζ2 so that(91)R,Sz,t≤k2ζ1,ζ2e-η2t,∀z,t∈R×R+.
For the choice ofz0 in proof of Lemma 5, we study the stability in the weighted space Lw∞. To this end, define (R^,S^)(t) as the solution of the system(92)R^S^t=Jϵ1R^S^-w1-R^+a1S^R^r-a2R^+S^S^,with the initial data(93)R^0≥Rz,0,S^0≥Sz,0,∀z∈R.It is easy to see that (R^,S^) is an upper solution to the system (88). The phase plane analysis shows that (R^,S^)(t) converges to origin for any initial data in the region [0,1]×[0,1] except the point (1,1). Similar to the previous lemma,(94)R,Sz,t≤k^2ζ1,ζ2e-λ¯2t,∀z,t∈-∞,z0×R+. for some positive constants k^2 and λ¯2. This completes the proof.Now, we are ready to give the proof of Theorem4.Proof of Theorem4.
From (67), for all (z,t)∈R×R+, we have(95)Rz,t≤Uz,t-U-z≤Pz,t,Sz,t≤Vz,t-V-z≤Qz,t.By Lemmas 5 and 7 and the squeezing theorem, it follows that there exist k>0 and η>0 so that (96)Uz,t-U-z≤ke-ηt,Vz,t-V-z≤ke-ηt, for all (z,t)∈R×R+. This proves the desired result.ConditionC2 is used in the previous analysis to construct the upper solutions in the proof of Lemmas 5 and 7. It implies that, at c=c0 and z→+∞,(97)ζ2μ1ζ1μ1≤mina2,1a1, and it can be guaranteed by(98)d≤2,a1a2-1r≤2-d1-a1. This condition arose in the linear speed selection studies; see [36]. To see that the condition C2 can be realized for all z∈R, we prove the following claim.Claim 8.
d = 0 and a1a2≤1 imply C2.Proof.
In the case whend=0, the V--equation can be written in the form(99)V-′=rc1-V-V--a2U-,V--∞=1,V-+∞=0.Since a1a2≤1, we need to prove V-(z)≤a2U-(z) for all z∈R. Assume, for contrary, this is not true for some z¯∈R. By (99), V- is increasing at the neighborhood of z¯. Since U-(z) is a decreasing function, we have V-(z¯+δ)>V-(z¯)>a2U-(z¯)>a2U-(z¯+δ), for some δ>0. Similarly, we can show that V-(z) is increasing for all z≥z¯, which contradicts the fact V-(+∞)=0. This implies that condition C2 holds true.


## 5. Conclusions

The local and the global stability of traveling waves to the two-species Lotka-Volterra competition model (5) under the condition C1 are investigated. Using the linearization and the essential spectrum analysis in [32], we find that the traveling wavefront is stable in some weighted functional space; see Theorem 3. Many choices of the exponential weight functions are valid; see Algorithm 2.Under some further conditionC2, we apply the upper-lower solution method to obtain a global stability result. Indeed, we prove that both the upper and the lower solutions tend to the wavefront. Our main results are presented in Theorem 4.



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*Source: 6569520-2019-01-01.xml*