On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates

In 1979, Shigesada, Kawasaki and Teramoto [ 11 ] proposed a mathematical model with nonlinear diffusion, to study the segregation phenomenon in a two competing species community. In this paper, we discuss limiting systems of the model as the cross-diffusion rates included in the nonlinear diffusion tend to infinity. By formal calculation without rigorous proof, we obtain one limiting system which is a little different from that established in Lou and Ni [ 5 ].

1. Introduction. One main problem in population ecology is to understand the spatial distribution pattern of competing species. To theoretically investigate the problem, we firstly consider the two competing species community, and employ the nondimensional system of reaction-diffusion equations in Ω T , ∂ t z = d ε ∆z + z (a − b w − z) in Ω T , ∂ ν w = 0, ∂ ν z = 0 on ∂Ω T , w(x, 0) = w 0 (x), z(x, 0) = z 0 (x) in Cl Ω, (1.1) which describes the dynamics of two species' population at the position x and the time t, where Ω is a bounded smooth domain of R with ≥ 1; ∂Ω and Cl Ω are the boundary and the closure, respectively, of Ω; Ω T = Ω×(0, T ) and ∂Ω T = ∂Ω×(0, T ) for some T ∈ (0, +∞]; ν is the outward unit normal vector on ∂Ω; a, b and c are positive constants. The initial values w 0 (x) and z 0 (x) are nonnegative smooth functions which are not identically zero. In the diffusion terms, d and ε are positive, and ε ∆w and d ε ∆z represent the dispersive force associated with the random movement of each species. Kishimoto and Weinberger [2] showed that if Ω is convex and the strong competition condition 1 c < a < b holds true, then every nonconstant positive stationary solution of (1.1) is unstable, that is, the stable stationary solutions of (1.1) are given by (w, z) = (1, 0) and (w, z) = (0, a). This fact generically implies that when Ω is convex, the stable spatially inhomogeneous pattern of two competing species can never appear under strong competition condition. On the other hand, Matano and Mimura [8] showed the existence of stable nonconstant positive stationary solution of (1.1) under the strong competition condition, when Ω is suitable nonconvex domain. In ecological terms, the coexistence of two competing species crucially depends on the shape of habitat.
The spatial distribution pattern of two competing species in the nature may be realized as a result of various biological effects, for example, the shape of habitat stated above. In 1979, Shigesada, Kawasaki and Teramoto [11] proposed the reaction-diffusion system in order to show that the nonlinear dispersive force has an effect on the spatial segregation pattern of two competing species. In the diffusion terms, the crossdiffusion rates α and β are nonnegative, and the terms ε α ∆[w z] and d ε β ∆[w z] mean what the gradient in the concentration of one species induces the dispersive force of another species. Since then, many mathematicians have tried to study the system (1.2) from various viewpoints (for example, see Jüngel [1], Ni [10], and their references). Lou and Ni [4] obtained useful a priori estimates for any solution, and sufficient conditions for the existence/nonexistence of nonconstant positive solution. Moreover they in the subsequent paper [5] characterized the asymptotic behavior of solutions as α → +∞, and established the following two kinds of limiting systems: Proposition 1 (Theorem 1.4 in [5]). Suppose that ≤ 3, a = b and a = 1/c are satisfied, and that a/d is not equal to every eigenvalue of −∆ with homogeneous Neumann boundary condition on ∂Ω. Let { (w n , z n ) } be any sequence of nonconstant positive solutions of (1.2) with α = α n → +∞. Then there exists a small δ 1 = δ 1 (a, b, c, ε, d) > 0 such that if β ≤ δ 1 , then as α n → +∞, by passing to a subsequence if necessary, either (w n , α n z n ) converges uniformly to (w, Z), where (w, Z) is a positive solution of Since the appearance of the above proposition, there are many studies on the qualitative property of positive solution for (1.3) and (1.4) in various ranges of parameter (for instance, refer to Lou, Ni and Yotsutani [6,7], and Mori, Suzuki and Yotsutani [9] for (1.4), and Kuto [3] for (1.3)). From these studies, we can theoretically understand the global bifurcation structure of positive stationary solutions for (1.2). We note here that the above proposition is valid when β is not so large. Figure 1 and Figure 2 show the density distribution of numerical example for (1.2). Figure 1 says that a periodic solution with spatial segregation appears when α and β are large. Figure 2 is represented as the dynamics of (u, v)(x, t) with Figure 1, and shows that the u-component is a almost constant function in x ∈ Ω for each t > 0 when α and β are large.  In this paper, motivated by Proposition 1 and numerical examples, we address the following problem: When a, b, c, d and ε are suitably fixed, what kinds of limiting systems describing the dynamics of (1.2) can we derive as α → +∞ and β → +∞? To study the problem, we assume that β = β(α) is a continuous function in α ≥ 0 satisfying the condition (A) stated in the next section, and we shall discuss limiting systems in (1.2) as α → +∞.
2. Limiting systems. We set w = (w, z) and u = (u, v), and define the map Since it follows from det ∂ w ϕ(w) = det 1 + α z α w β z 1 + β w = 1 + β w + α z > 0 for any w ∈ R 2 + that u = ϕ(w) is a bijection from Cl R 2 + to Cl R 2 + , we find out that the inverse map w = Φ(u) = (Φ w , Φ z )(u) of u = ϕ(w) exists. Actually, by simple calculation, we can represent Φ w (u) and Φ z (u) as We should remark that and employing the change of variables u = ϕ(w), we represent (1.2) as with the Neumann boundary condition. Hereafter, we assume that β = β(α) is a nonnegative continuous function in α ≥ 0, and satisfies with constants κ and ω.

2.1.
Case where κ = 0 and ω = 0. Suppose the case where U = u/α and V = v are positive and bounded as α → +∞. After simple calculations, as α → +∞, we have and then we obtain

From the Neumann boundary condition and
as α → +∞, we see that U = U (x, t) as α → +∞ converges to a constant function U * = U * (t) in x ∈ Ω for each t > 0, and U * (t) satisfies By ∂ t [Φ z (α U, V )] = d ε ∆V + g(α U, V ), we have the following limiting system: The stationary problem of (2.2) is the same as (1.4) established by Lou and Ni [5], and the positive solution of (2.2) corresponds to the bounded positive solution of (1.2) as α → +∞, because

2.2.
Case where κ = 0 and ω ≥ 0. Suppose the case where U = u and V = α v are positive and bounded as α → +∞. We put After simple calculations, as α → +∞, we obtain for the case where ω > 0, and for the case where ω = 0. Hence we find out that as α → +∞, we obtain the following limiting system: The stationary problem of (2.3) is the same as (1.3) established by Lou and Ni [5], and the positive solution of (2.3) corresponds to the bounded positive solution of (1.2) such that the z-component is close to 0 as α → +∞, because are satisfied as α → +∞.

2.3.
Case where κ > 0. Suppose the case where U = u α , V = α v − β(α) u α are positive and bounded as α → +∞. We set Defining P * (U ) = lim α→+∞ P (U ) for each function P (U ) depending on α, we easily obtain with the Neumann boundary condition. From the above equations, we have From the Neumann boundary condition and we see that the solution of (2.4) as α → +∞ tends to a constant function in x ∈ Ω for each t > 0. We denote U * = U * (t) and V * = V * (x, t) by the limit functions of U = U (x, t) and V = V (x, t), respectively, as α → +∞. By we derive the equation Moreover the limiting equation of (2.5) as α → +∞ becomes Summarizing the above argument, we have the following limit system: are satisfied as κ → 0, we should note here that (2.6) becomes (2.2) as κ → 0, which implies that (2.6) is an extension of (2.2).

2.4.
Estimate of positive stationary solution. In this subsection, we consider the estimate of positive solution for the stationary problem of (2.1). Setting   3. Concluding remarks. In this paper, we discussed the limiting system of (1.2) when the cross diffusion rate β = β(α) has the asymptotic behavior (A) as α → +∞. The limiting system (2.6) obtained in this paper is a little different from that established in Lou and Ni [5]. Although the numerical examples show the validity of the limiting system (2.6), we can not give the rigorous proof on the validity because the uniform boundedness of nonnegative solution for the limiting system (2.6) is not obtained. Moreover we have not yet studied the property of positive solution for the limiting system (2.6). Since the limiting system (2.6) has just been obtained, many problems such as the above remain.