EXISTENCE OF TRAVELING WAVEFRONT FOR DISCRETE BISTABLE COMPETITION MODEL

We study traveling wavefront solutions for a two-component competition system on a one-dimensional lattice. We combine the monotonic iteration method with a truncation to obtain the existence of the traveling wavefront solution.

1. Introduction. In this paper, we study the following two-component lattice dynamical system (LDS): where u j = u j (t), v j = v j (t), t ∈ R, j ∈ Z, d > 0, h > 1, k > 1, b > 0 and a > 0. This model arises in the study of strong competition of two species in a habitat which is divided into discrete niches. Here the unknowns u j , v j are the populations of species u, v at niches j, respectively, constants a, b are the birth rates and h, k are the competition coefficients of species u, v. We assume that h, k > 1 so that these two species are of strong competition. By a renormalization, the carrying capacities are taken to be 1 for both species. Also, the diffusion (or migration) rates are taken to be 1 for u and d for v. Moreover, the system also can be regarded as a spatial discrete version of the following reaction-diffusion system: where u = u(x, t), v = v(x, t), x ∈ R and t ∈ R. We are interested in the traveling fronts of (1.1) connecting (0, 1) and (1, 0), i.e., a solution of (1.1) in the form u j (t) = U (j + ct) and v j (t) = V (j + ct) for all (j, t) ∈ Z × R for some wave speed c ∈ R and wave profile (U, V ) with the boundary condition (u j , v j )(t) → (0, 1) as j → −∞, (u j , v j )(t) → (1, 0) as j → ∞, (1.3) Note that the speed c is an unknown to be determined. To study the existence of traveling front, it is more convenient to work on (U, W ), where W := 1 − V . We also let Then 0 ≤ U, W ≤ 1 on R and (1.4) is equivalent to with (U, W )(−∞) = (0, 0) and (U, W )(+∞) = (1, 1), where h, k > 1 and a, b, d > 0 are always assumed. Note that as we discuss the relative work for the traveling front, we have four situations determined by the competition coefficient h and k. As 0 < k < 1 < h (or 0 < h < 1 < k), the species u is superior than the species v. We expect the species u would take over the territory of the species v. Here we refer the work of Okubo, Maini, Williamson and Marry [10], Hosono [6,7] and Kan-on [8] for the continuum problem (1.2). For the discrete version, we refer the reader the work of Guo and Wu [4]. As h > 1, k > 1, both equilibria (0, 1) and (1, 0) are stable and so we have the bistable nonlinearity. Then the chance of who can take whose territory depends on their initial data. The reader can see the work of Gardner [3] and Conley and Gardner [2] for the continuum case. As 0 < h, k < 1, we call this case as co-existence with weak competition which means the two species can be happy to live together. Please see the work of Tang and Fife [11] for more detail.
In this paper, we shall always assume that h, k > 1 and the aim is to study the existence of a traveling wavefront. Motivated by the work [1] considering a discrete periodic media for bistable dynamics, we extend the method from a single equation to a system. In the beginning, we transfer the system to an integral system and by choosing an appropriate integral factor, we have two monotonic operators. Using the iterated monotone method, we first obtain the existence of the monotone solution for the truncated problem. Extending to the whole domain, we have the theorem as follows. or it is a semi-trivial solution with the following two alternatives: Moreover, c < 0 for the case (1.8) and c > 0 for the case (1.9).
Hereafter we denote that (a 1 , a 2 ) ≤ (b 1 , b 2 ) means a 1 ≤ b 1 and a 2 ≤ b 2 . Also, (a 1 , a 2 ) < (b 1 , b 2 ) means a 1 < b 1 and a 2 < b 2 . The proof of Theorem 1.1 is based on the method of [1]. Applying this method to our system may produce the existence of semi-trivial solution as indicated in Theorem 1.1. The main difficulty here is due to the coupling of two equations so that there are more cases to be analyzed in the proof of Theorem 1.1. Moreover, unlike the situation of a single equation, we were unable to determine the sign of the speed. To the author's knowledge, the question of determining the sign of the speed for PDE case (1.2) is still open. However the phenomenon of propagation failure occurs when we replace the diffusion coefficient of the first equation in (1.1) byd > 0 and assume that both d andd are sufficiently small. For this aspect, we refer the reader to [5] (and [9] for the case of a single equation).
When the former case in Theorem 1.1 occurs, we have the exact tail behavior (U, W )(−∞) = (0, 0) and (U, W )(+∞) = (1, 1) so that we have the existence of a traveling front for our problem. We state the theorem as follows.
The plan of this paper is as follows. In the next section, we give the proof of Theorem 1.1. Then we prove the existence theorem (Theorem 1.2) in section 3.

2.
Proof of Theorem 1.1. As c = 0, given a constant µ > 0 large enough, we define Then (1.5) is reduced to solving the following integral equations: where µ > 0 is chosen sufficiently large so that the above integrals are well-defined in R and the following monotonic property holds, i.e., Following [1], for each n ∈ N, we consider the following truncated problem: with the exterior conditions: Since the solution of the above truncated problem is discontinuous at x = ±n, it is more convenient to consider the following system of integral equations Remark 1. When c = 0, the problem (1.5) with c = 0 is reduced to solving the following equations: Moreover, the integral system associated with the truncated problem of (2.1)-(2.2) with c = 0 and (2.3)-(2.4) is the following integral equations: . We now prove the following existence result for the truncated problem.
Proof. We divide the proof into two parts.
From the monotonic property, we have (1, 1). It follows from the monotonicity property that there exist (U n * , W n * )(ξ) and (U * n , W * n )(ξ) such that Using Lebesque's Dominated Convergence Theorem, it is easy to see that (U n * ,W n * )(ξ) and (U * n , W * n )(ξ) are solutions of (2.5). It is also easy to see that they are in C 1 (−n, n).

Uniqueness. Let
− W * n (ξ) ≥ 0 and from the construction of µ, we have This contradicts with the definition of h 1 and so h 1 = 0 = h 2 . Here we can set U n (ξ) := U n * (ξ) = U * n (ξ) and W n (ξ) := W n * (ξ) = W * n (ξ). The case of h 2 ≥ h 1 > 0 is similar. Hence we have the uniqueness. The proof is completed.
Remark 2. When c = 0, (2.5) has a minimal solution (U n * , W n * ) and a maximal solution (U * n , W * n ). Notice that (U n * , W n * ) and (U * n , W * n ) are nondecreasing and are constant in (b, b + 1) for each b ∈ Z.
Next, we prove the following monotonicity in c of solutions of (2.5).
With these properties of solutions to truncated problem, we first prove the existence and monotonicity property of solutions to (1.5).
Proof of Theorem 1. There are three possibilities.
When both U c ≡ 0 and W c ≡ 0, (1.6) follows from the strong maximum principle. If U c ≡ 0 (so that U c (0) = 0), then W c (0) = a and W c satisfies the equation (2.6) Hence (1.8) follows by applying the strong maximum principle to (2.6). Note that by the monotonicity of W c , we have W c (±∞) exist such that W c (−∞) = 0 and W c (∞) = 1. Moreover, by integrating the equation (2.6) from −∞ and ∞, we obtain that If W c ≡ 0 (so that W c (0) = 0), then U c (0) = a and U c satisfies the equation