Traveling wave solutions for two species competitive chemotaxis systems

In this paper, we consider two species chemotaxis systems with Lotka-Volterra competition reaction terms. Under appropriate conditions on the parameters in such a system, we establish the existence of traveling wave solutions of the system connecting two spatially homogeneous equilibrium solutions with wave speed greater than some critical number c*. We also show the non-existence of such traveling waves with speed less than some critical number c*_0, which is independent of the chemotaxis. Moreover, under suitable hypotheses on the coefficients of the reaction terms, we obtain explicit range for the chemotaxis sensitivity coefficients ensuring c*= c*_0, which implies that the minimum wave speed exists and is not affected by the chemoattractant.


Introduction
The current work is concerned with the traveling wave solutions of two species competition chemotaxis systems of the form      where a, b, d, r, λ > 0 and µ i , χ i > 0 (i = 1, 2) are positive constants. In (1.1), u i (t, x), i = 1, 2 denote the density functions of two mobile species living together in the same habitat and competing for some limited resources available in their environment. These two competing species also produce some chemical substance which affects their reproduction dynamics in the sense that each mobile species have tendency to move toward its higher concentration. The density function of the chemical substance is denoted by v(t, x) and is being produced at the rates µ i by the species i, for each i = 1, 2. The positive constant χ i measures the sensitivity rate by the species i ∈ {1, 2} of the chemical substance. The chemical substance has a self degradation rate given by the positive constant λ. The positive constants a and b measure the interspecific competition between the mobile species. We assume that the first species diffuses at a rate equal to one while the second species diffuses at a rate d > 0. The positive constant r is the intrinsic growth rate of the second mobile species. When χ 1 = χ 2 = 0, the dynamics of the chemotaxis model (1.1) is governed by the following classical Lotka-Volterra diffusive competition system, u 1,t = u 1,xx + u 1 (1 − u 1 − au 2 ), x ∈ R u 2,t = du 2,xx + ru 2 (1 − bu 1 − u 2 ), x ∈ R. (1. 2) The asymptotic dynamics of (1.2) is of significant research interests and considerable results have been established in the literature. It is well known that the large time behavior of solutions to (1.2) is delicately related to the positive constants a and b. For example, consider the kinetic system of ODEs associated with (1.2), that is, It is easily seen that (1.3) has a trivial equilibrium solution e 0 = (0, 0), and two semi-trivial equilibrium solutions e 1 = (1, 0) and e 2 = (0, 1). When 0 < a, b < 1, or a, b > 1, (1.3) has a positive equilibrium given by e * = ( 1−a 1−ab , 1−b 1−ab ). The asymptotic dynamics of (1.3) depends on the strength of the competition coefficients a and b. We say that the competition on the first species u 1 (resp. on the second species u 2 ) is strong if a > 1 (resp. b > 1). We say that the competition on the first species u 1 (resp. on the second species u 2 ) is weak if 0 < a < 1 (resp. 0 < b < 1). Based on the magnitudes of a and b, the following four important cases arise.
(A1) 0 < b < 1 < a, which is referred to as the strong-weak competition case. (A2) 0 < a < 1 < b, which is referred to as the weak-strong competition case. (A3) 0 < a, b < 1, which is referred to as the weak-weak competition case. (A4) a, b > 1, which is referred to as the strong-strong competition case Due to biological applications, we are only interested in non-negative solutions of (1.1), (1.2), and (1.3). Let (u(t), v(t)) be a solution of (1.3) with u(0) > 0 and v(0) > 0. The following results are well known. In the case (A1), (u 1 (t), u 2 (t)) → e 2 as t → ∞ and hence the second population outcompetes the first. In the case (A2), (u 1 (t), u 2 (t)) → e 1 as t → ∞ and hence the first population outcompetes the second. In the case (A3), (u 1 (t), u 2 (t)) → e * as t → ∞ and hence both species coexist for all the time. In the case (A4), the limit of (u 1 (t), u 2 (t)) depends on the choice of the initial condition (u 1 (0), u 2 (0)).
Consider (1.2). It is also interesting to know the asymptotic behavior of the solutions (u(t, x), v(t, x)) with front like initial functions (u(0, x), v(0, x)), that is, initial functions connecting two equilibrium solutions of (1.3). This is strongly related to the so called traveling wave solutions. A traveling wave solution of (1.2) is a classical solution of the form (u 1 (t, x), u 2 (t, x)) = (U 1 (x − ct), U 2 (x − ct)) for some constant c ∈ R, which is called the speed of the traveling wave. A traveling wave solution is said to connect an equilibrium solution e + of (1. The existence of traveling wave solutions of (1.2) has been extensively studied (see [2-4, 11, 13-17, 28]). For example, assume 0 < a < 1. It is well known that there is a minimum wave speed c min ≥ c * 0 := 2 √ 1 − a such that (1.2) has a monotone traveling wave solution with speed c connecting the equilibrium solutions e and e 2 of (1.3) (at the left end and right end, respectively) if and only if c ≥ c min , where e = e 1 in the case (A2) and e = e * in the case (A3). There is no explicit formula available for c min in the literature and it is known (see [5]) that it is possible to have the strict inequality c min > c * 0 (= 2 √ 1 − a). When c min = c * 0 , it is said that the minimum wave speed is linearly determinate. The works [6,7,16] provide sufficient conditions on the parameters to ensure that c min is linearly determinate. For example, it is proved in [16,Theorem 2.1] that c min = c * 0 provided the following (A5) holds.
The objective of the present work is to investigate in how far the traveling wave theory for (1.2) can be extended to the two species chemotaxis system (1.1). It is clear that the space-independent solutions of the chemotaxis system (1.1) are the solutions of the ODE system (1.3). Note that case (A1) and case (A2) can be handled similarly. In the following, we focus on case (A2) and case (A3), and investigate the existence of traveling wave solutions of (1.1) connecting the unstable equilibrium e 2 of (1.3) at the right end and the stable equilibrium e of (1.3) at the left end, where e = e 1 in the case (A2) and e = e * in the case (A3) (see the following subsection for the definition of traveling wave solutions of (1.1)).

Definition of traveling wave solutions of (1.1)
Similarly to (1.2), a traveling wave solution of (1.1) is a classical solution of the form (u 6) and to connect e at the left end if lim As far as the chemotaxis model (1.1) is concerned, very little is known about the existence of traveling wave solutions. There are some recent works on the existence and non-existence of traveling wave solutions and spreading speeds of the single species chemotaxis model. In this regards, we refer the reader to the works in [9,[22][23][24][25][26] and references therein.
We note that (u 1 (t, x), u 2 (t, x), v(t, x)) = (U 1 (x − ct), U 2 (x − ct), V (x − ct)) is a traveling wave solution of (1.1) connecting e 2 at the right end and e at the left end if and only if (U 1 (x), U 2 (x), V (x)) is a steady state solution of the following system complemented with the boundary conditions (1.9) Observe that for any solution (u 1 (t, x), u 2 (t, x), v(t, x)) (respectively steady state (U 1 (x), U 2 (x), V (x))) of (1.1) (respectively (1.8)), the third component v(t, x) (respectively V (x)) is uniquely determined by the first two components. Hence for the sake of simplicity in the notations, we write u = (u 1 , u 2 ) and U = (U 1 , U 2 ) for vectors and denote by u(t, x) = U c (x − ct) traveling wave solutions of (1.1) with speed c ∈ R. In the following, we always assume that u i (0, ·) ∈ C b unif (R) with inf x∈R u i (0, x) ≥ 0 for i = 1, 2.

Standing assumptions and notations
To state the main results of this paper, we introduce some standing assumptions and notations in this subsection.
It is well known that the solutions (u 1 (t, x), u 2 (t, x)) of (1.2) with initials u i (0, ·) ∈ C b unif (R) and 0 ≤ u i (0, ·) (i = 1, 2) exist for all t ≥ 0 and are bounded. The first standing assumption is on the global existence and boundedness of classical solutions of (1.1).
It is known that, in the case (A3), that is, 0 < a, b < 1, e * is a stable equilibrium solution of (1.2). The third standing assumption is on the stability of e * for (1.1).
Consider (1.2). In the case (A2), it has traveling wave solutions connecting e 2 at the right end and connecting e 1 at the left end. In the case (A3), it has traveling wave solutions connecting e 2 at the right end and connecting e * at the left end. The next standing assumption is on the existence of traveling wave solutions of (1.1).
. Note that, when χ 1 = 0 and χ 2 = 0, (H4) becomes (1 − a) > r(ab − 1) + . We will prove that (H2)+(H4) (resp. (H3)+(H4)) implies the existence of traveling wave solutions of (1.1) connecting e 2 at the right end and connecting e 1 at the left end (resp. connecting e 2 at the right end and connecting e * at the left end) for speed c greater than some number c * (χ 1 , χ 2 )(≥ c * 0 ) (see Theorem 1.2). As it is mentioned in the above, when (A5) holds, the minimal wave speed c min of (1.2) is linearly determinate, that is, c min = c * 0 . The last standing assumption is on the existence and linear determinacy of the minimal wave speed of (1.1).
It is clear that (H5) implies (H4). When χ 1 = χ 2 = 0, the first two equations in (1.1) are independent of λ (hence λ can be chosen large enough such that 1 − a < λ) and (H5) reduces to (1.13) which holds trivially if d < 2 and ab ≤ 1. It will be shown that (H2)+ (H5) (resp. (H3)+(H5)) implies the existence and linear determinacy of the minimal wave speed of (1.1) connecting e 2 at the right end and e 1 at the left end (resp. connecting e 2 at the right end and e * at the left end) (see Theorem 1.4).

Statements of the main results
In this subsection, we state the main results of this paper. The first main result is on the stability of spatially homogeneous equilibrium solutions of (1.1) and is stated in the following theorem.   We have the following two theorems on the existence and nonexistence of traveling wave solutions of (1.1). Theorem 1.2. Suppose that (H4) holds. Then for every c > c * , (1.1) has a nontrivial traveling solution ) with speed c connecting e 2 at right end and satisfying that where κ ∈ (0, κ * χ1,χ2 ) satisfies c = c κ . Moreover, the following hold.
) of (1.1) also connects e * at the left end.
and connecting e 2 at the right end.
Observe that Theorem 1.3 provides a lower bound c * 0 (= 2 √ 1 − a) for the speeds of traveling wave solutions of (1.1). This lower bound is independent of the chemotaxis sensitivity coefficients χ 1 and χ 2 . The following theorem shows that this is the greatest lower bound for the speeds of traveling wave solutions of (1.1). (ii) If (H3) and (H5) hold, then for any c ≥ c * 0 , there is a traveling wave solution of (1.1) with speed c connecting e * and e 2 .
We remark that under the conditions of Theorem 1.4, the minimum wave speed of (1.1) exists, is linearly determinate, and is not affected by the chemotactic effect. In general, it is not known whether (1.1) has a minimal wave speed, and if so, whether both systems (1.1) and (1.2) have the same minimum wave speed. This question is related to whether the presence of the chemical substance slows down or speeds up the minimum wave speed. Note that Theorem 1.3 shows that the presence of the chemical substance doesn't slow down the minimum wave speed of (1.2) whenever it is linearly determinate. We plan to continue working on this problem in our future works. We see from (1.13) that Theorem 1.4 recovers [16, Theorem 2.1], which guarantees that the minimum wave speed of (1.2) is linearly determinate under hypothesis (1.13).
We also remark that there are also several interesting works on the dynamics of solutions to (1.1) when considered on bounded domains, see [1, 12, 18-21, 27, 29] and the references therein. For example, the works in [1,18,29] studied the stability of the equilibria of (1.1) on bounded domains with Neumann boundary conditions, while the works [12,19] considered (1.1) on bounded domains with some non-local term in the reaction terms.
The rest of the paper is organized as follows. In section 2 we present the proof of Theorem 1.1. In section 3 we construct some super and sub-solutions to be used in the proof of Theorem 1.2. Section 4 is devoted to the proof of Theorem 1.2. In section 5 we present the proof of Theorem 1.3. The proof of Theorem 1.4 is presented in section 6.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1. Throughout this section c ∈ R is an arbitrary fixed number. We first prove two lemmas, which are fundamental for the proofs of most of our results in the subsequent sections.
Lemma 2.1. Suppose that (H2) holds and let u(t, x; c) be a bounded classical solution of (1.8) defined for every x ∈ R and t ∈ R. If inf x,t∈R u 1 (t, x; c) > 0, then u(t, x; c) ≡ e 1 .
Proof. Let Let M 1 and M 2 be as in (1.11). Observe from the comparison principle for elliptic equations that Hence Then by the comparison principle for parabolic equations, we have This implies that 1 > χ 1 µ 1 l 1 and that Similarly, observe that Thus, as in the above, we conclude from the comparison principle for parabolic equations that Observe also that We conclude from the comparison principle for parabolic equations that From this point, we distinguish two cases and show that L 1 = l 1 = 1 and L 2 = 0.
In this case, it follows from (2.2)-(2.4) that That is Using the fact that This implies that l 1 ≥ 1 since b ≥ 1 (see (H2) holds). Thus, we get from (2.4) that L 2 = 0, so we can proceed as in case 1 to show that l 1 = L 1 = 1 as well.
From both cases and the definition of l 1 , L 1 , and L 2 , we obtain u(t, x; c) ≡ (1, 0), which completes the proof of the lemma.
Proof of Theorem 1.1. Let c ∈ R be given and u(t, x; c) be a classical solution of (1.8) satisfying inf x∈R u 1 (0, x; c) > 0. It is easy to see that lim sup Hence without loss of generality we may suppose that u i (t, ·; c) ∞ ≤ M i for every t ≥ 0 and i = 1, 2.
(i) Suppose that hypothesis (H2) holds. Observe that u 1 (t, x; c) satisfies Hence, since (H2) holds, we may employ the comparison principle for parabolic equations to conclude that Now, if we suppose by contradiction that the statement of Theorem 1.1 (i) is false, then there exist a sequence t n → ∞ and x n such that inf By a priori estimates for parabolic equations, without loss of generality, we may suppose that there is some is an entire solution of (1.8) and it follows from (2.12) that Thus, by Lemma 2.1, we conclude that u * (t, x) ≡ e 1 . This contradicts with (2.13) since u * (0, 0) = lim n→∞ u(t n , x n ; c). Hence lim t→∞ u(t, ·; c) − e 1 ∞ = 0, which completes the proof of (i).
(ii) Suppose that hypothesis (H3) holds. The proof follows similar arguments as in (i). Indeed, note that in addition to (2.12), it also holds that Hence, since (H3) holds, we may employ the comparison principle for parabolic equations to conclude that (2.14) Therefore, similar arguments used to prove (i) together with Lemma 2.2 yield that lim t→∞ u(t, ·; c)−e * ∞ = 0. This completes the proof of the theorem.

Super-and sub-solutions
In this section, we construct some super-and sub-solutions to some elliptic equations related to (1.8). These super-and sub-solutions will be used in the proof of Theorem 1.2 in next section. We first introduce some notations. For fixed 0 < κ < κ max = min{ √ 1 − a, √ λ}, D 1 , D 2 ,D 2 > 0 and 0 < ε 1 ≪ 1, we define where (m) + = max{m, 0} for every real number m ∈ R, and M 1 and M 2 are as in (1.11), that is, M 1 = 1 1−χ1µ1 and M 2 = r r−χ2µ2 . We shall provide more information on how to choose the positive constants D 1 , D 2 ,D 2 and ε 1 in Lemma 3.3 below. Define the convex set is twice continuously differentiable and solves the elliptic equation Next, we present some lemmas.
is the positive solution of the algebraic equation (1.17). Then there is D 1 ≫ 1 such that for every u ∈ E(κ) and i ∈ {1, 2} the following hold.

Existence of traveling wave solutions for c > c *
In this section, we investigate the existence of traveling wave solutions of (1.1) and prove Theorem 1.2.
Proof. Let u ∈ E(κ) and y > 0 be given. We first show the uniqueness. Observe that system (4.1) is decoupled, hence the theory of elliptic scalar equations applies for each equation. Since the equations of (4.1) are of the same type for both i = 1 and i = 2, we shall only provide the arguments for the proof of U u,y 1 .
Observe from the choice of the positive constant R, see (3.8), and Lemma 3.1 that Whence for each x ∈ R fixed, the function is monotone decreasing. Thus by [8, Theorem 10.2, page 264 ] we deduce that a solution U u,y (x), if exists, is unique. Now, we show the existence of solution to (4.1). Again, we note from the choice of R (see (3.8)) and Lemma 3.1 that We also note, by Lemma 3.2, Since u κ 1 ∞ ≤ M 1 , it follows from [10, Theorem 5.1, Corollary 5.2, page 433] that there is at least one classical solution to the elliptic equation For reference later, we introduce the function Then (4.1) can be rewritten as with d 1 = 1 and d 2 = d. For every y > 0, 0 < κ < κ * , and u ∈ E(κ) we define U u,y (x) = (u κ 1 (x), u κ 2 (x)) for every |x| > y. With this extension, we have the following lemma. For convenience we let y 0 > 1 such that u κ i (−y) = M i for each i ∈ {1, 2} and y ≥ y 0 .
Proof. (i) Let i ∈ {1, 2}. Since for every x ∈ R fixed, the function U i → A u i (U i ) is monotone decreasing, then it follows from Lemma 3.3 and the comparison principle for elliptic equations (see [8,Theorem 10 , ∀ − y < x < y, which together with the fact that U u,y (x) = u κ 1 (x) for every |x| ≥ y, completes the proof of (i). (ii) Suppose that (H2) holds and that U u,y 1 (x) = u 1 (x) for every |x| ≤ y. Hence u 1 (x) satisfies Since u 1 (x) ≥ u κ 1 (x) ≥ 0 and u 1 (±y) > 0, then the Harnack's inequality for elliptic equations implies that u 1 (x) > 0 for every |x| ≤ y. Observe that with and that 1 > η 1 : Now we take m * 1 := min{η 1 M 1 , m 1 }. We claim that for every y > max{y 0 , x 1 } it holds that m * 1 ≤ u 1 (x) for every x ∈ [−y, x 1 ]. Indeed, let y > max{x 1 , y 0 } and suppose that u 1 (x) attains its minimum at some point, sayx 1 ∈ [−y, x 1 ]. Ifx 1 is a boundary point, then the claim easily follows. On the other hand, ifx 1 is an interior point then u 1,x (x 1 ) = 0 and u 1,xx (x 1 ) ≥ 0. This along with (4.5) and the fact that 0 ≤ u i ≤ M i for each i = 1, 2 imply that This clearly implies that u 1 (x 1 ) ≥ η1 1−χ1µ1 = η 1 M 1 ≥ m * 1 since we have shown in the above that u 1 (x 1 ) = min |x|≤y u 1 (x) > 0. So, the claim holds and the result follows.
(iii) Suppose that (H3) holds and U u,y 2 (x) = u 2 (x) for |x| ≤ y. The proof follows similar arguments as in (ii) by observing that Hence we can take 0 < ε 2 ≪ 1 such that we take m * 2 = m 2 = min{m 2 , η 2 M 2 , M 2 } and then proceed as in the proof of (ii) to show that u 2 (x) ≥ m * 2 for every x ∈ [−y, x 2 ] with y > max{x 2 , y 0 }. Theorem 4.3. Let 0 < κ < κ * and y ≥ y 0 be given. Then U u,y ∈ E(κ) for every u ∈ E(κ). Moreover, the mapping E(κ) ∋ u → U u,y ∈ E(κ) is continuous and compact with respect to the compact open topology. Therefore, by Schauder's fixed point theorem, it has a fixed point, say u * ,y .
Proof. Let y > y 0 be fixed. It follows from Lemma 4.2 (i) that U u,y ∈ E(κ) for every u ∈ E(κ). Since U u,y i ∞ ≤ M i for every i = 1, 2 and u ∈ E(κ), by a priori estimates for elliptic equations and the uniqueness of solution to (4.1) guaranteed by Lemma 4.1 and the Arzela-Ascot's theorem, it follows that the mapping E(κ) ∋ u → U u,y ∈ E(κ) is continuous and compact with respect to the compact open topology. Therefore, by Schauder's fixed point theorem, it has a fixed point, say u * ,y .

Lemma 4.4. It holds that
where B λ,κ is as in (1.15).
Thanks to Lemma 4.4, to complete the proof of Theorem 1.2, it remains to study the asymptotic behavior of u * (x) and x → −∞, which we complete in next two lemmas. Proof. We prove this result by contradiction. Suppose that there is a sequence {x n } with x n → −∞ such that inf n≥1 |u * (x n ) − e 1 | > 0. (4.17) Consider the sequence u * ,n (x) = u * (x + x n ) for every x ∈ R and n ≥ 1. By a priori estimates for elliptic equations and Arzela-Ascoli's theorem, without loss of generality, we may suppose that there is someũ ∈ C 2 such that u * ,n →ũ as n → ∞ locally uniformly in C 2 (R). Moreover,ũ also satisfies (4.6). Recalling m * 1 and x 1 given by Lemma 4.2 (ii), we deduce that since x n → −∞ as n → ∞. Therefore by Lemma 2.1, we obtainũ(x) ≡ (1, 0). In particular,ũ(0) = e 1 , which contradicts with (4.17), sinceũ(0) = lim n→∞ u * (x n ).
Lemma 4.6. Suppose that hypothesis (H3) holds. Then Proof. We proceed also by contradictions. The ideas are similar to that of the of the proof of Lemma 4.5.
Now we complete the proof of Theorem 1.2.
Proof of Theorem 1.2. For every c > c κ * , there is a unique 0 < κ c < κ * satisfying c = c κc = . This implies that u(t, x) = u * (x − ct) is a traveling solution of (1.1) with speed c. Moreover, it follows from (4.8) that u * connect e 2 at right end. Since u * ∈ E(κ), then u * 1 (x) > 0 by comparison principle for elliptic equations. This in turn implies that u * 2 − 1 ∞ > 0. Thus u * is not a trivial solution of (1.1). Assertion (i) and (ii) of the theorem follows from Lemmas 4.5 and 4.6 respectively.

Proof of Theorem 1.3
In this section, we present the proof of nonexistence of nontrivial traveling wave solutions of (1.1) with speed c < c * 0 (= 2 √ 1 − a) connecting e 2 at right end. Our first step toward the proof of the non-existence is to show that, for any nontrivial traveling solution u(x − ct) of (1.1) connecting u(∞) = e 2 at the right end, there holds u 1,x < 0 for x ≫ 1.
Lemma 5.1. Let u(t, x) = u(x − ct) be a nontrivial traveling wave solution of (1.1) connecting e 2 at the right end. Then there is X 0 ≫ 1 such that u 1,x (x) ≤ 0 for every x > X 0 .
Proof. We proceed by contraction. Suppose that the statement of the lemma is false. Then, since u 1 (∞) = 0 and u 1 (x) > 0 for every x ∈ R, then there is sequence of local minimum points {x n } n≥1 of u 1 (x) satisfying x n → ∞ as n → ∞. Since u 2 (∞) = 1, then lim n→∞ u 2 (x n ) = 1. From the representation formula v(x; u) = 1 it follows from the dominated convergence theorem that Hence lim Thus there is n 0 ≫ 1 such that Since {x n } is a sequence of local minimum points, we have that u 1,xx (x n ) ≥ 0 and u 1,x (x n ) = 0 for every n ≥ 1, which clearly contradicts with (5.1). Thus the statement of the Lemma must hold. Now, we present the proof of Theorem 1.3.
6 Proof of Theorem 1.4 In this section, we present the proof of Theorem 1.4. To this end, we first recall some results on the spreading speeds and stability for single species chemotaxis model. Lemma 6.1. [9,24] Consider the single species chemotaxis model where all the parameters are positive, and let (u(t, x; u 0 ), w(t, x; u 0 )) denote the unique nonnegative classical of (6.1) for every u 0 ∈ C b unif (R) with u 0 ≥ 0 defined on a maximal interval of existence [0, t max,u0 ). Then the following hold.
(i) If χµ <b, then t max,u0 = +∞ and u ∞ ≤ max{ u 0 ,ã b−χµ } for every t ≥ 0. Moreover, if u 0 ∞ > 0 then lim inf (ii) If 2χµ <b and inf x∈R u 0 (x) > 0 then Throughout the rest of this section, we assume that (H5) holds. Note that (H5) implies (H4). By the definition of the function F 2 (κ, χ 1 , χ 2 ), we have which means that inequality (3.11) also holds for every κ ∈ (0, √ 1 − a). Hence c * = c * 0 . As a result, to complete the proof of Theorem 1.4, it remains to show the existence of a non-trivial traveling wave connecting e 2 at the right end with minimum speed c * 0 = 2 √ 1 − a.
Proof of Theorem 1.4. (i) Suppose that hypotheses (H2) and (H5) hold and r > 2χ 2 µ 2 . Chose a decreasing sequence {c n } n≥1 such that c n → c * 0 as n → ∞. For every n ≥ 1, letũ n =Ũ cn (x − c n t) be a traveling wave solution of (1.1) connecting e 1 and e 2 given by Theorem 1.2. Let and define U cn =Ũ cn (x +x n ) for every x ∈ R and n ≥ 1. Note that U cn 1 satisfies for every n ≥ 1. Recall that U cn i ∞ < M i for every n ≥ 1 and i = 1, 2. Hence, by a priori estimates for elliptic equations, if possible after passing to a subsequence, we may suppose that there is some U ∈ C 2,b (R) such that (U cn , V (·; U cn )) → (U, V (·; U)) as n → ∞ in C 2,b loc (R). Moreover, U(x) satisfies x ∈ R. (6.4) We note that U 1 (·) also satisfies properties (6.3). From this point, we complete the proof of the spatial asymptotic behavior of U(x) in the following six steps.
Step 3. In this step, we prove that lim sup x→∞ U 1 (x) = 0. Suppose not. According to Step 2, there would exist a sequence of minimum points {x n } n≥1 of the function U 1 satisfying x n → ∞ and U 1 (x n ) → 0 as n → ∞ with U 1,x (x n ) = 0 and U 1,xx (x n ) ≤ 0 for every n ≥ 0. Hence, we deduce from (6.4) that In particular, we obtain that since U 1 (x n ) > 0 for n ≥ 1. Letting n → ∞ in the last inequality and using the facts that U 2 ∞ ≤ M 2 and V (·, U) ∞ ≤ 1 λ (µ 1 M 1 + µ 2 M 2 ), we obtain that This clearly contradicts with hypothesis (H2). Thus we must have that lim sup x→∞ U 1 (x) = 0.
Step 5. In this step, we prove that lim inf x→∞ U 2 (x) > 0. Suppose not. In this case, since U 1 (∞) = 0 and lim sup x→∞ U 2 (x) > 0, we can repeat the arguments used in Step 3 for the equation satisfies by U 2 (x) to end up with the inequality r ≤ χ 2 µ 2 M 2 . This clearly contradicts the fact that r > 2χ 2 µ 2 .
Finally, we can employ Lemma 4.5 together with the fact that U 1 (x) ≥ 1 2 for every x ≤ 0 to conclude that lim x→−∞ U(x) = e 1 . This completes the proof of (i) .
(ii) Suppose that (H3) and H(5) hold. Note that hypothesis (H3) implies that r > 2χ 2 µ 2 . The proof of the minimal wave in this case follows similar arguments as in (i). So, we shall provide general ideas of the proof. As in (i), chose a sequence {c n } n≥1 such that c n → c * 0 + as n → ∞. For every n ≥ 1, let u n =Ũ cn (x − c n t) be a traveling wave solution of (1.1) connecting e 1 and e 2 given by Theorem 1.2. Next, we letx i n = min{x ∈ R :Ũ cn i (x) = min{1 − a, 1 − b} 2(1 − ab) }, i = 1, 2 andx n = min{x 1 n ,x 2 n }, (6.6) and define U cn (x) =Ũ cn (x +x n ) for every x ∈ R and n ≥ 1. Note that U cn 1 satisfies By a priori estimates for elliptic equations, if possible after passing to a subsequence, we may suppose that there is some U ∈ C 2,b (R) such that (U cn , V (·; U cn )) → (U, V (·; U)) as n → ∞ in C 2,b loc (R). Moreover U(x) satisfies (6.4). It is clear from (6.7) that U i (x) ≥ min{1−a,1−b} 2(1−ab) for every x ≤ 0 and j = 1, 2. Hence we may employ Lemma 4.6 to conclude that lim x→−∞ U(x) = e * . Since c * 0 = 2 √ 1 − a < 2, we can proceed as in the proof of Step 5 to conclude that lim sup x→∞ U 2 (x) > 0. Now, we can proceed as in the proof of Step 6 by using the fact that (H3) to conclude that lim inf x→∞ U 2 (x) > 0. Next, observe from (6.7) that there is some i 0 ∈ {1, 2} such that U i0 (0) = min{1−a,1−b} 2 (1−ab) . Hence U(x) ≡ e * . Hence, we can proceed as in the proof of Step 2 using the stability of e * to conclude that lim inf x→∞ U 1 (x) = 0. This in turn, as is Step 3 yield that U 1 (∞) = 0. As result, since lim inf x→∞ U 2 (x) > 0 and r > 2χ 2 µ 3 , we may use Lemma 6.1 (ii) to conclude that lim x→∞ U 2 (x) = 1. This completes the proof of (ii).

Appendix
In this section we present the proof of Lemmas 3.1 and 3.2. Hence, since R e − √ λ|x−y| u κ 1 (y)dy > 0 for every x ∈ R, we deduce that The statement of Lemma 3.2 follows from (7.2)-(7.4).