Traveling Wave Solutions for Lotka-Volterra System Re-Visited

Using a new method of monotone iteration of a pair of smooth lower- and upper-solutions, the traveling wave solutions of the classical Lotka-Volterra system are shown to exist for a family of wave speeds. Such constructed upper and lower solution pair enables us to derive the explicit value of the minimal (critical) wave speed as well as the asymptotic rates of the wave solutions at infinities. Furthermore, the traveling wave corresponding to each wave speed is unique modulo a translation of the origin. The stability of the traveling wave solutions with non-critical wave speed is also studied by spectral analysis of the linearized operator in exponentially weighted Banach spaces.


INTRODUCTION
We re-visit the classical Lotka-Volterra competition system: where u(x, t), v(x, t) are the population densities of two competing species, the constants d i , a i , b i , c i , i = 1, 2 are assumed to be positive. In this paper, we are trying to acomplish the following goals: providing a new and easy construction of upper-and lower-solutions to derive the traveling wave solutions of (1.1); obtaining an accurate description of the minimal wave speed and asymptotic behaviors (up to the first order) of the wave solutions; investigating the stability of the traveling wave solutions in various Banach spaces. System (1.1) has been extensively studied. In  and , there are many applications and treatments of solutions of (1.1) in bounded spatial domain under various initial and boundary conditions. As is well known, system (1.1) and its cooperative counter-parts admit traveling wave solutions. In , Tang and Fife showed the existence of the traveling wave solutions connecting the extinction state with the co-existent state, In , Kanel and Zhou studied the existence of the traveling wave solutions connecting the coexistent state to a semi-exitinction state. In , the traveling wave solution connectiong two semi-extinction states were studied, they also estimated the minimal wave speed. For the other treatment of the traveling wave solutions of system (1.1) and its generalizations, please see , , , .
Throughout this paper, we make the following assumptions: . a 2 c 2 < a 1 c 1 , Under these conditions system (1.1) has three equilibria (0, 0), ( a1 b1 , 0) and (0, a2 b2 ). We will use a new monotone iteration method to investigate the traveling wave solutions of (1.1). The traveling wave solution connects two of the above equilibria. The monotone iteration method has been widely used in the study of the traveling wave solutions of reaction diffusion system such as (1.1), but most constructed upper-and lower-solutions in literature are non-smooth. The relatively larger 'gap' between the non-smooth upper and lower-solutions creates certain difficulties in deriving the accurate asymptotic estimates of the traveling wave solutions at infinities. Such estimates are valuable in applications, and enable one fully exploit the cooperative or competitive structure of the Lotka Volterra system.
The smooth upper-solution in the monotone iteration as in section 3 is derived from the traveling wave solutions of the KPP equation. Observing that if we take fuction v to be a constant, then the first equation of (1.1) is a generalized KPP equation, the same consideration is also true for the second equation. The existence, uniqueness, asymptotics as well as the stability of the traveling wave solutions of the KPP system are well known, so are the properties of the upper-solution. The most difficult part in section 3 is to construct the lower solution for system (1.1). Since such constructed upper solution is 'nearly' a solution, a compromise is made to derive the smooth lower-solution. In fact, the lower-solution does not satisfy the boundary condition at ∞. Thanks to the realxed condition in , we can still apply the monotone iteration scheme as specified in , . The trade off of such 'shorter' lower solution is that we can have some freedom to choose the lower solution with desired asymptotic rate at negative infinity. This leads to an accurate (up to first order) asymptotic estimates of the traveling wave solution at −∞, and an exact value of the minimal wave speed.
The asymptotics of the traveling wave solutions at infinities are obtained by comparison principle. Once the asymptotic behaviors of the traveling wave solutions are known, we can use the Maximum principle and Sliding domain method to derive the uniqueness, strict monotonicity as well as the local stability of the traveling wave solutions.
The local stability of the traveling wave solutions is studied by means of spectral analysis in some weighted Banach spaces. We proceed to show that the linearized operator about the traveling wave solution has essential spectrum lying completely in the left complex plane, and that 0 is not an eigenvalue of the linearized operator in the weighted Banach spaces, whereas all the other eigenvalues of the linearized operator have negative real parts. This means the traveling wave solution is linearly exponentially stable. Since such linear operator is sectorial, the linear stability implies the local nonlinear stability of the traveling wave solutions , , . Though general theories on the stability of the traveling wave solutions are known , , , the verification of the conditions there is in fact a case by case study. The methods used in the stability study of the traveling wave solutions of (1.1) are similar in the spirit to those in , , , , , , , but in comparing to the above methods, we need to further overcome the 'unstable' component of the system. This is done by studying an equivalent form of the linearized operator in a smaller weighted Banach spaces.
We remark that the stability is only for the traveling waves with non-critical wave speed. The stability of the traveling waves with critical wave speed is currently under investigation.
The paper is arranged as follows: in section 2, we study the steady states of the system (1.1) and obtain the attraction region of the stable steady state; in section 3, we show there are traveling wave solutions connecting the one of the unstable steady states with a stable one, corresponding to each wave speed, the traveling wave solution is unique and strictly monotone. The analysis is done by utilizing an accurate description of asymptotic behavior of the traveling wave solutions. Furthermore, we have obtained the estimate of the critical wave speed. In the last section of the paper, we show the traveling wave solutions are locally, nonlinearly exponentially stable.

THE EQUILIBRIA AND THEIR STABILITY
In this section, we analyze the constant steady states of system (1.1) under conditions H1-H3.
Consider system (1.1) with the initial conditions where u 0 and v 0 are non-negative bounded smooth functions on R.
We first start withũ = a1 b1 + Aρ(t). From (2.2), it needs to satisfy the differential inequality Also forṽ = Bρ(t), we see from (2.2) that it needs to satisfy the differential inequality and it suffices to show that . Finally we look atû = a1 b1 −ρ(t). Again from (2.2), it needs to satisfy the differential inequality For ρ(t) ≥ 0 it suffices to show that . From hypotheses (H1) and (H3) we observe the fact that From the hypotheses (H3), one can obtain that we can now conclude that the three differential inequalities (2.6)-(2.8) will all be satisfied if the function ρ(t) is a positive solution of the differential equation This leads to the function ρ(t) given in (2.5) with ρ(0) < α/β = a 1 /b 1 and lim t→∞ ρ(t) = 0 .
From the arbitrariness of constant A in Theorem (2.2), we then have the the attraction region for the equilibrium (a 1 /b 1 , 0). When the hypotheses (H1) and (H3) hold, for all the initial density functions (u 0 , v 0 ) in the rectangular area the solution (u, v) of the system (1.1)-(2.1) converges to the equilibrium (a 1 /b 1 , 0) uniformly on R as t → ∞ with the rate e −αt . In the meantime, by adding hypothesis (H2), we can quickly find that the equilibriums (0, a 2 /c 2 ) and (0, 0) are both unstable. For this purpose we construct a pair of upper-lower solutions (ũ,ṽ) = (M, a 2 /c 2 − ρ(t)) and (û,v) = (Aρ(t), 0), where M ≥ a 1 /b 1 is a constant. Constant A and function ρ(t) will be determined later. The differential inequalities in (2.2) are automatically satisfied byũ andv. Forû andṽ, the following relations need to hold: The above inequalities are equivalent to from hypothesis (H2) we have From the hypotheses (H3), one can obtain that Both the inequalities in (2.11) can be satisfied by choosing ρ(t) as the solution of the differential equation This results in the function (2.14) ρ(t) = γ δ + Ce −γt with an arbitrary constant C > 0. For arbitrarily small ǫ > 0, one can always find a constsnt C large enough such that ρ(0) = γ/(δ + C) < ǫ. The fact that lim t→∞ ρ(t) = γ/δ leads to the following theorem indicating that (0, a 2 /c 2 ) and (0, 0) are both unstable.

THE TRAVELING WAVES
In section 2, we showed system (1.1) has two unstable constant steady states: (0, 0), (0, a2 c2 ) and one asymptotically stable constant steady state ( a1 b1 , 0). We will show that there are traveling wave solutions of (1.1) having the form and connecting the unstable state (0, a2 c2 ) with ( a1 b1 , 0) as the variable a1 d x + ca 1 t runs from −∞ to +∞. The constant c in (3.2) is the wave speed and the minimal speed is also called the critical wave speed. Throughout the rest of the paper, we assume d 1 = d 2 = d.
To simplify notions, we introduce the following transformations to (1.1): , and q is a constant satisfying a 2 c −1 2 < q < a 1 c −1 1 . Under transformations (3.1) and (3.2), system (1.1) is changed into with the corresponding boundary conditions to change system (3.3) into the following monotone (cooperative) system: Remark 3.1. Note that from hypotheses [H1]-[H3] and relations (3.2), we have the following inequalities: Before showing the existence of the traveling wave solutions for (3.6) with boundary conditions (3.7), we first recall the following well known fact: (please see ,  for the proof) Let a function f be a C 2 function on the interval [0, β], β > 0, with f > 0 on (0, β), and has a unique monotonically increasing traveling wave solution ω c (ξ) , ξ ∈ R , where the lower index denotes the dependence of the wave solution ω on c.
We next show the existence of the traveling wave solution for system (3.6)-(3.7).
Theorem 3.2. Let the parameters ǫ 1 , ǫ 2 , b and r satisfy conditions in Remark 3.1, then corresponding to every c ≥ 2 1 − r 1+ǫ2 system (3.6) has a monotone traveling wave solution satisfying the boundary condition (3.

7). (Recall hypotheses [H1]-[H3]
imply all the conditions in Remark 3.1 are valid.) Proof. The proof will be done by monotone iterating a pair of smooth upper-and lower-solutions. We first construct a twice differentiable smooth upper-solutions.
According to lemma 3.1, for every c ≥ 2 1 − r 1+ǫ2 , there is correspondingly a For 0 ≤ u 2 (ξ) ≤ū 2 (ξ), we readily verify that The last inequality is true provided b < 1− r 1+ǫ2 +ǫ 1 , which is valid due to hypothesis H. It is also straightforward to verify that (ū 1 ,ū 2 ) satisfies the boundary conditions (3.7). We next construct a twice continuously differentiable lower solution for the system (3.6)-(3.7). Let the function Z(ξ), ξ ∈ R be the solution of (3.13) Here l is some number in the interval (0, 1) to be determined. One can readily verify that the solutions of (3.9) and (3.13) are related by the following (3.14) Since 0 < l < 1, we have We define a lower solution of (3.6), (3.7) by setting where l ∈ (0, 1) is to be determined. We readily verify that they satisfy The last inequality is valid provided that Also by the limiting boundary conditions of (3.13) we see ). Inequality (3.17) along with (3.18) show that (ũ 1 ,ũ 2 ) consists of a pair of lower-solutions for system (3.6), (3.7).
Noting that such constructed upper-and lower-solution pairs are ordered. We can apply the monotone iteration methods provided in  or  to derive the conclusion of this Theorem. Here we only sketch the ideas.
To further study the asymptotics of the traveling wave solutions as obtained in Theorem 3.2, we shall need the following Lemma concerning the scalar problem (3.8).
Lemma 3.3. The solution w c (ξ) to (3.8), described in Lemma 3.1, has the following asymptotic behaviors: 1. Corresponding to the wave speed c > 2 √ α 1 , where a ω and b ω are positive constants; 2. Corresponding to minimal wave speed c = 2 √ α 1 , Proof. The conclusion follows ,  with slight changes.
Based on Lemma 3.3, we study the asymptotic behaviors of the traveling wave solutions of system (3.6), (3.7) at infinities.
The limit system of (3.31) at +∞ is a constant coefficient system, and is given by The exponential growth rates of the traveling wave solutions of (3.6)-(3.7) are determined by those of the solutions of (3.32). The justification is as follows: first we note that (3.32) admits exponential dichotomy. By the roughness of exponential dichotomy, solutions of (3.31) grow/decay exponentially (possibly with a different exponential rate) , . Since the derivative (w 1 , w 2 ) of the traveling wave solution solves (3.31), the traveling wave solutions of (3.6) approach exponentially to the steady state (1, 1 1+ǫ2 ). To find out the exact asymptotic rates of the solutions of (3.31), we first change (3.31), (3.32) into the first order systems. letting The exponential growth of the traveling wave solution (u 1 (ξ), u 2 (ξ)) at infinity implies that ∞ t0 |R + − R ∞ |dξ < +∞ for some large t 0 > 0.
It is easy to check that the matrix R ∞ has 4 distinct eigenvalues, then by and , the asymptotic exponential rates of the solutions of (3.31) are the same as those of the solutions of (3.32).
Theorem 3.6. Assume hypotheses [H1] to [H3]. The traveling wave solution to system (3.6)-(3.7), obtained for each wave speed c ≥ 2 1 − r 1+ǫ2 , with properties described in Corollary 3.4 and 3.5, is unique up to a translation of the origin.
2. Forθ ≥ 0, there exists ξ 1 ∈ R, such that one of the components of Uθ and U 2 are equal at the point ξ 1 ; and for all ξ ∈ R, we have Uθ 1 (ξ) ≥ U 2 (ξ). We then consider the system (3.47) on (−N, N ) and θ =θ in the definition for W . To fix ideas, we suppose that the first component of Uθ 1 and U 2 is equal at the point ξ 1 . The maximum principle for this component implies that the first component of Uθ 1 (ξ) is identically equal to that of U 2 (ξ). Also, we readily obtain that for large +ξ, the limiting equation for (3.47) is the same as (3.32) . Since the first component of W is identically zero and the off diagonal limit coefficient r in the first equation in (3.32) is not equal to zero, we conclude from the first equation in (3.47) that the second component of W must vanish for all large ξ. By the maximum principle for the second equation, we conclude that the second component of W is also identically zero for all ξ ∈ R. Similarly, we next consider the case that the second component of Uθ 1 and U 2 are equal at the point ξ 1 . We first obtain the limiting equation of (3.47) for large −ξ. The off diagonal limit coefficient of the second equation will be b 1+ǫ2 = 0. We first deduce that the second componet of W is indentically zero, and then the first component must also be identically zero for ξ ∈ R.
Consequently, in either situation, there exists aθ ≥ 0, such that for all ξ ∈ R.
Theorem 3.7. Assume hypotheses [H1] to [H3]. System (3.6)-(3.7) does not have strict monotonic traveling wave solution tending to (0, 0) T as ξ → −∞ for c < 2 √ α. Here, α = 1 − r 1+ǫ2 . Proof. Suppose there is a constant c with 0 < c < 2 √ α and a corresponding solution V (ξ) = (v 1 (ξ), v 2 (ξ)) T of (3.6) tending to (0, 0) T as ξ → ∞. Similar to the proof of Corollary 3.4, we can deduce by integrating the asymptotic approximation of its derivative that the asymptotic behaviors of V (ξ) at −∞ must be of the form: where (A s , B s ) T and (Ā s ,B s ) can not be both zero, and h.o.t. is the short notation for the higher order terms. The condition 0 < c < 2 √ α implies that V (ξ) is oscillatting. This says that such solution of (3.6) with c < 2 √ α is not monotone.

STABILITY OF THE TRAVELING WAVES WITH NON-CRITICAL SPEEDS
In this section, we always hypotheses [H1] to [H3] for system (1.1); thus all the conditions in Remark 3.1 are satisfied for (3.6) and subsequent systems. We first show that the traveling wave solutions with the non-critical speed obtained in Theorem 3.2 is unstable in the space of continuous function C(R)×C(R) (Please see definitions below). This motivates us to investigate the stability in the "smaller" exponentially weighted Banach spaces. We will concentrate on the stability of the traveling waves with non-critical wave speeds.
Let σ p (L) = {λ ∈ σ(L) | λ is an eigenvalue of L}, and σ e (L) be the essential spectrum of L, which are points in σ(L) outside σ p (L) ∩ {isolated eigenvalues of L with f inite multiplicity}. Note that σ e (L) includes the continuous spectrum of L. , , . Let C(R) be the space of all continuous functions on the real line and C 0 (R) be its subspace We also need the following weighted Banach spaces: for non-negative numbers σ 1 , σ 2 , the space C σ1,σ2 is defined as: on which we define the norm Similarly, we can define C (i) σ1,σ2 , i = 1, 2, ... as well, for example: It can be readily verified that these spaces are Banach spaces.
Proof. This theorem holds for the traveling wave solutions with critical and noncritical wave speeds. We need to prove the trivial solution of (4.2) is unstable. Thus, it suffices to show that in the space C 0 the operator L in (4.3) has essential spectrum with positive real part. As is well known ( , ) the location of the continuous spectrum of the operator L is bounded by the spectrum of L at ±∞, which we denote by L + and L − respectively. More precisely, we let (4.5) (4.6) Here, U * ± respectively denote the limit of U * (ξ) as ξ → ±∞. Now consider the equation Following  and , we replace V by e (λt+iζξ) I, where I is an identity matrix and λ is a complex number and ζ is real. We then have The spectrum of the operator L + consists of curves given by: Solving (4.8), we have Letting λ = x + yi for x, y ∈ R, then by (4.9) we have (4.11) x = − y 2 (c * ) 2 − 1, or by (4.10), Similarly, the spectrum of L − consists of curves: in the complex plane. Consequently, by theory described in , we have Hence, by  again, the traveling wave solution U * (ξ) of (4.1) is essentially unstable in C 0 (R).
In order to obtain stability for the traveling solution U * , we will restrict the initial conditions and the operator L to a "smaller" Banach space C σ1,σ2 with σ 1 ≥ 0, σ 2 ≥ 0 and σ 2 1 + σ 2 2 = 0. To relate the operator L in C 0 (R) to an equilvalent operator in C σ1,σ2 , we introduce the mapping T : C σ1σ2 → C 0 as follows: (4.15) T V := (e σ1ξ + e −σ2ξ )V.
Lemma 4.2. Suppose σ 1 and σ 2 satisfying then the essential spectrum of the operatorL in the space C 0 (R) is contained in some closed sector in the left half complex plane with vertex on the horizontal axis left of the origin. Outside this sector, there are only a finite number of eigenvalues ofL.
Proof. As in the proof of Theorem 4.1, we first study the opearatorL at infinity. We have where σ 2 1 + c * σ 1 + ∂F ∂U (U * ) and σ 2 2 − c * σ 2 + ∂F ∂U (U * ) correspond respectively to the matrices: Similar to the proof of Theorem 4.1, we find the right most points of the corresponding parabolas are on the horizontal axis given by A simple calculation shows the number above is negative by the choice of σ 1 and σ 2 in (4.18). Thus by the theory in , the essential spectrum ofL is contained in a closed sector in the left complex plane with vertex on the horizontal axis left of the origin. Moreover, we may choose this sector with the further property that outside it there is a finite number of eigenvalues ofL. Proof. The conclusion follows immediately from Lemma 4.2 and relation (4.16).
Having established the location of the essential spectrum of the operator L in the space C σ1,σ2 , we next study the location of its eigenvalues. We first note that from Corollary 3.4, for c > 2 √ α, we have (U * (ξ)) ′ (e σ1ξ + e −σ2ξ ) is unbounded as ξ → −∞, which is different from the situations met in [02-Bates], , therefore their methods can not be carried over to our case.
Lemma 4.4. Let σ 1 and σ 2 satisfy (4.18). Then 0 is not an eigenvalue of the operator L in the space C σ1σ2 (R).
In summary, both case A and Case B show that for anyr ∈ (−δ + r,δ + r), |rV (ξ)| < (U * ) ′ (ξ), ξ ∈ R, i.e., S is open. Now the set S is a non-empty, open and closed subset of R, hence S ≡ R. However, this is impossible by the definition of S, since (U * ) ′ (ξ) is bounded. Therefore the equation LV = 0 cannot have a nontrivial solution in C σ1σ2 .
The next lemma shows that there is no eigenvalue of the operator L in C σ1,σ2 with positive real part.
Lemma 4.5. Let C C 0 be the complexified space of C 0 (R) and λ be an eigenvalue of the operatorL, given by (4.16), with corresponding eigenfunction U ∈ C C 0 , then Re λ < 0.
From (4.27), we obtain the follwing inequality The last inequality is true because M ij (U * ) ≥ 0 if i = j. From the positivity theorem for parabolic equations we deduce that X i (ξ, t) > 0 for ξ ∈ R and t >t (cf p.14 in ). However by the t−periodicity of V , we have that X i (ξ, t) > 0 for all ξ ∈ R and t > 0. Contradiction with the existence ofξ. This shows that (4.35) V (ξ, t) < r 0 T (U * ) ′ (ξ) for ξ ∈ R, t ≥ 0.
We are then in the same situation as in the proof of the claim at the beginning of this lemma. Using similar arguments as in the proof of the claim, we extend the above inequality to: V (ξ, t) ≥ (r 0 − δ)T (U * ) ′ (ξ) for ξ ∈ R, t > 0.
It then follows that r 0 − δ ∈ S. Contradiction with the definition of r 0 . Hence r 0 = 0. However, this contradicts the assumption that at least one component of V assume positive value. Thus we must have λ 1 < 0. This concludes the proof of the lemma.
Theorem 4.6. Assume [H1] to [H3] and that σ 1 and σ 2 satisfy (4.18), the operator L in C σ1,σ2 has a dense domain of definition. For any complext number with Re λ > 0 large enough, (λ − L) −1 exists and is defined on all of C σ1,σ2 , and satisfies the following estimate Proof. The proof follows the same idea as in  but with resolvent estimates in C 0,τ replaced by in the space C σ1,σ2 . We skip the proof.
Proof. The conclusion follows from Theorem (4.7)and Hille-Yoshida Theorem.
Proof. The stability of U * leads to the consideration of the stability of the trivial solution for system (4.2), and the analysis of the spectrum of the operator L in (4.3). Corollary 4.3, Lemma 4.4 and Lemma 4.5 show that the spectrum of the operator L in the space C σ1,σ2 is contained in a closed angular region in the left open complex plane. Thus, following the methods in Theorem 2.1 on p.227 in , we obtain the conclusion of this theorem.