On the steady state of a shadow system to the SKT competition model

We study a boundary value problem with an integral constraint that arises from the modelings of species competition proposed by Lou and Ni in [10]. Through local and global bifurcation theories, we obtain the existence of non-constant positive solutions to this problem, which are small perturbations from its positive constant solution, over a one-dimensional domain. Moreover, we investigate the stability of these bifurcating solutions. Finally, for the diffusion rate being sufficiently small, we construct infinitely many positive solutions with single transition layer, which is represented as an approximation of a step function. The transition-layer solution can be used to model the segregation phenomenon through interspecific competition.

1. Introduction. In this paper, we consider the following one-dimensional nonlocal boundary value problem, where v = v (x) is a positive function of x and λ = λ is a positive constant to be determined; a i , b i , c i , i = 1, 2 and are nonnegative constants.
The motivation for studying model (1.1) is that it is a limiting system or the so-called shadow system of the following Lotka-Volterra competition model with Ω = (0, L), ∂u ∂n = ∂v ∂n = 0, x ∈ ∂Ω, t > 0, (1.2) where d 1 , d 2 and ρ 12 , ρ 21 are positive constants, d i is referred as the diffusion rate and ρ ij as the cross-diffusion rate. System (1. al. [18] in 1979 to study the phenomenon of species segregation, where u and v represent the population densities of two competing species. A tremendous amount of work has been done on the dynamics of its positive solutions since the proposal of system (1.2). There are also various interesting results on its stationary problem that admits non-constant positive solutions, in particular over a one-dimensional domain. See [3,6,7,9,10,11,14,15,13,16], and the references therein. Great progress was made by Lou and Ni in [9,10] on the existence and asymptotic analysis of the steady states of (1.2) for Ω being a bounded domain in R N , 1 ≤ N ≤ 3. Roughly speaking, they showed that (1.2) admits only trivial steady states if one of the diffusion rates is large with the corresponding cross-diffusion fixed, and (1.2) allows nonconstant positive steady states if one of the cross-diffusion pressures is large with the corresponding diffusion rate being appropriately given. Moreover, they established the limiting profiles of non-constant positive solutions of (1.2) as ρ 12 → ∞ (and similarly as ρ 21 → ∞). For the sake of simplicity, we only state their results for ρ 21 = 0, while the similar results hold for the case when ρ 21 = 0. Moreover, we refer the reader to [19] and the references therein for recent developments in the analysis of the shadow systems to (1.2). Suppose that a1 a2 = b1 b2 = c1 c2 and d 2 = a 2 /µ j for any j ≥ 1, where µ j is the j−th eigenvalue of −∆ subject to homogenous Neumann boundary condition. Let (u i , v i ) be positive nonconstant steady states of (1.2) with (d 1 , ρ 12 ) = (d 1,i , ρ 12,i ). Suppose that ρ 12,i /d 1,i → r ∈ (0, ∞) as ρ 12,i → ∞, then (u i , ρ 12  We now denote d 2 = since the smallness of diffusion rate tends to create nonconstant solutions for (1.3). Putting c 2 /r =c 2 and assuming Ω = (0, L), we arrive at (1.1), where we have dropped the tilde over c 2 in (1.1) without causing any confusion. For a1 a2 > b1 b2 and if > 0 is small, Lou and Ni [10] established the existence of positive solutions v (λ , x) to (1.1) by degree theory. Moreover, v (x) has a single boundary spike at x = 0 if being sufficiently small. This paper is devoted to study the solutions of (1.1) that have a different structure, i.e., an interior transition layer. The remaining part of this paper is organized as follows. In Section 2, we carry out bifurcation analysis to establish nonconstant positive solutions to (1.1) for all small. The turning direction of each bifurcation branch is obtained. We also find the stability of the first bifurcation branch in Section 3 for b 1 = 0 in (1.1). In Section 4, we show that for any x 0 in a pre-determined subinterval of (0, L), there exists positive solutions to (1.1) that have a single interior transition layer at x 0 . Finally, we include discussions and propose some interesting questions in Section 5.
2. Nonconstant positive solutions to the shadow system. In this section, we establish the existence of nonconstant positive solutions to (1.1). First of all, we apply the following conventional notations as in [9,10] then we see that (1.1) has a constant solution and X is a Hilbert space defined by We first collect the following facts about the operator F before using the bifurcation theory.
Lemma 2.1. The operator F(v, λ, ) defined in (2.2) satisfies the following properties: (1) F(v,λ, ) = 0 for any ∈ R + ; (2) F : (3) for any fixed (v 0 , λ 0 ) ∈ X × R + , the Fréchet derivative of F is given by Proof. Part (1)-(3) can be verified through direct calculations and we leave them to the reader. To prove part (4), we formally decompose the derivative in (2.3) as Obviously DF 2 : X ×R + → R×R is linear and compact. On the other hand, DF 1 is elliptic and according to Remark 2.5 of case 2, i.e., N = 1, in Shi and Wang [17], it is strongly elliptic and satisfies the Agmon's condition. Furthermore, by Theorem 3.3 and Remark 3.4 of [17], DF 1 is a Fredholm operator with zero index. Thus D (v,λ) F(v 0 , λ 0 , ) is in the form of Fredholm operator+Compact operator, and it follows from a well-known result, e.g, [8], that D (v,λ) F(v 0 , λ 0 , ) is also a Fredholm operator with zero index. Thus we have concluded the proof of this lemma.

QI WANG
Putting (v 0 , λ 0 ) = (v,λ) in (2.3), we have that To obtain candidates for bifurcation values, we need to check the following necessary condition on the null space of operator (2.4), Let (v, λ) be an element in this null space, then (v, λ) satisfies the following system First of all, we claim that λ = 0. To this end, we integrate the first equation in (2.6) over 0 to L and have that on the other hand, the second equation in (2.6) is equivalent to If λ = 0, we must have by equating the coefficients of the two identities above that then by comparing this with the formulā we conclude from a straightforward calculation that a 2 (A − B) = c 2 (B − C) and this implies thatv = −1 which is a contradiction. Therefore λ must be zero as claimed. Now put λ = 0 in (2.6) and we arrive at It is easy to see that (2.7) has nonzero solutions if and only if a2−c2−2c2v v is one of the Neumann eigenvalues for (0, L) and it gives rise to which is coupled with an eigenfunction v k (x) = cos(kπx/L). Moreover we can easily see that the zero integral condition is obviously satisfied. Then bifurcation might occur at (v,λ, k ) with provided that k is positive or equivalentlȳ v < a 2 − c 2 2c 2 , a 2 > c 2 . (2.10) We have shown that the null space in (2.5) is not trivial and in particular Remark 1. (0, a1 b1 ) is another trivial solution to (1.1) and local bifurcation does not occur at (0, a1 b1 ). Actually, putting and the null-space N (D (v,λ) F(0, a1 b1 , )) must be trivial. Having the potential bifurcation values k in (2.9), we can now proceed to establish non-constant positive solutions for (1.1) in the following theorem which guarantees that the local bifurcation occurs at (v,λ, k ).
Proof. To make use of the local bifurcation theory of Crandall and Rabinowtiz [1], we have verified all but the following transversality condition: If not and (2.12) fails, then there exists a nontrivial solution (v, λ) ∈ X × R + that satisfies the following problem By the same analysis that leads to the claim below (2.6), we can show that λ = 0 in (2.13), which then becomes (2.14) However, this reaches a contradiction to the Fredholm Alternative since cos kπx L is in the kernel of the operator on the left hand side of (2.14). Hence we have proved the transversality condition and this concludes the proof of Theorem 2.2.

Global bifurcation analysis.
We now proceed to extend the local bifurcation curves obtained in Theorem 2.2 by the global bifurcation theory of Rabinowitz in its version developed by Shi and Wang in [17]. In particular, we shall study the first bifurcation branch Γ 1 . Theorem 2.3. Assume that (2.1) and (2.10) hold. Then there exists a component S ⊂ X × R + × R + that satisfies, (iv) ∀ ∈ (0, 1 ), there exists (v, λ, ) ∈ S u and the same holds for S l .
Proof. Denote the solution set of (1.1) by and choose S to be the maximal connected subset ofD that contains (v,λ, 1 ). Then S is the desired closed set and (i) follows directly from (2.11) in Theorem 2.2.
To prove that v(x) is positive on [0, L] and λ is positive for all (v, λ, ) ∈ Ψ with > 0, we introduce the following two connected sets: and , λ > 0}, then we want to show that P 0 = S + . First of all, we observe that P 0 is a subset of S + and P 0 is nonempty, since at least the part of S near (v,λ, 1 ) is contained in P 0 . Now we prove that P 0 is both open and closed in S + . The openness is trivial, since for any (v, λ, ) ∈ P 0 and the sequence (v k , λ k , k ) that converges in (v, λ, ) in Furthermore, the fact that λ k > 0 and k > 0 follows readily from λ > 0 and > 0.
We argue by contradiction. If It is well-known that (2.15) has only trivial solution, i.e, v ≡ 0, or v ≡ a2 c2 , hence v k converges to either 0 or a2 c2 uniformly on [0, L]. The case that v k converges to 0 can be treated by the same analysis that shows v k (x) > 0. If v k converges to a2 c2 , we apply the Lebesgue's Dominated Convergence Theorem to the integral constraint in (1.1) and send λ k → 0, then we have that and it implies that A = C, however this is a contradiction to (2.1). Therefore λ must be positive as desired. On the other hand, it is easy to see that We apply the Strong Maximum principle and Hopf's lemma to (1.1) and have that v ≡ 0 for all x ∈ [0, L] and λ = a1 b1 . However, we have from Remark 1 that bifurcation does not occur at (0, a1 b1 ). This is a contraction and we must have that v(x) > 0 on [0, L].
To prove (iii), we choose S u to be the component of Moreover, we introduce the following four subsets: and we want to show that We shall only prove the first part, while the latter one can be treated in the same way. We first note that S 0 is both open and closed with respect to the topology of S 0 u and we divide our proof into two parts.
First of all, it is easy to see thatλ k > 0 ,˜ k > 0 since both have positive limits as λ > 0 ,˜ > 0. On the other hand, we conclude fromṽ k →ṽ in X and the elliptic regularity theory thatṽ k →ṽ in C 2 ([0, L]). Differentiate the first equation in (1.1) and we have We have from Hopf's lemma and the factṽ (x) > 0 thatṽ (L) > 0 >ṽ (0), then this second order non-degeneracy implies thatṽ k (x) > 0, which is desired.
λ > 0 can be easily proved by the same argument as above and we now need to show thatṽ (x) > 0. Again we have from the elliptic regularity thatṽ k →ṽ in Applying the Strong Maximum Principle and Hopf's Lemma to (2.16), we have that eitherṽ > 0 orṽ ≡ 0 on (0, L). In the latter case, we must have (ṽ,λ) ≡ (v,λ) and this contradicts to the definition of S 0 u . Thus we have shown thatṽ > 0 on (0, L) and this finishes the proof of (iii).
If (a3) occurs, we can choose the complement to be However, for any (v, λ) ∈ Z, we have from the integration by parts that and this is also a contradiction. Therefore we have shown that only alternative (a1) occurs and S u is not compact in X × R + × R + . Now we will study the behavior of S u and that of S l can be obtained in the exact same way. First of all, we claim that the project of S u onto the -axis does not contain an interval in the form ( 0 , ∞) for any 0 > 0 and it is sufficient to show that there exist a positive constant¯ 0 such that (1.1) has only constant positive solution Multiplying both hand sides of (2.17) by w and then integrating over (0, L), we have that We can easily show from the Maximum Principle that both v(x) and λ are uniformly bounded in , then we have from the inequality above that Then we reach a contradiction for all > C0 (π/L) 2 unless w ≡ 0, where (π/L) 2 is the first positive eigenvalue of − d 2 dx 2 subject to Neumann boundary condition. If w ≡ 0, we have that v ≡v and this is a contradiction as we have shown in the case (a2). Therefore the claim is proved. Now we proceed to show that the projection S u onto the -axis is of the form (0,¯ ] for some¯ ≥ 1 . We argue by contradiction and assume that there exists > 0 such that ( ,¯ ) is contained in this projection, but this projection does not contain any < . Then we have from the uniformly boundedness of v (x) ∞ and sobolev embedding that, for each > 0, v C 3 ([0,L]) ≤ C, ∀(v , λ , ) ∈ S u and this implies that S u is compact in X × R + × R + . We reach a contradiction to alternative (a1). Therefore S u extends to infinity vertically in X × R + × R + . This finishes the proof of (iii) and Theorem 3.1.
We have from Theorem 3.1 that there exist positive and monotone solutions is a decreasing solution. Then we can construct infinitely many non-monotone-solutions of (1.1) by reflecting and periodically extending v(x) and v(L − x) at ..., −L, 0, L, ...

3.
Stability of bifurcating solutions from (v,λ, k ). In this section, we proceed to investigate the stability or instability of the spatially inhomogeneous solution (v k (s, x), λ k (s)) that bifurcates from (v,λ) at = k . Here the stability refers to that of the inhomogeneous pattern taken as an equilibrium to the time-dependent system of (1.1). To this end, we apply the classical results from Crandall and Rabinowitz [2] on the linearized stability with an analysis of the spectrum of system (1.1).
First of all, we determine the direction to which the bifurcation curve Γ k (s) turns to around (v,λ, k ). According to Theorem 1.7 from [1], the bifurcating solutions (v k (s, x), λ k (s), k (s)) are smooth functions of s and they can be written into the following expansions where ϕ i ∈ H 2 (0, L) satisfies that L 0 ϕ i cos kπx L dx = 0 for i = 2, 3, andλ 2 , K 1 , K 2 are positive constants to be determined. We remind that o(s 3 ) in the v-equation of (3.1) is taken in the norm of H 2 (0, L). For notational simplicity, we denote in (1.1) Moreover, we introduce the notations and we can definef vλ ,f vvv ,ḡ v ,ḡ λ ,ḡ vλ ,ḡ vv , etc. in the same manner. Our analysis and calculations are heavily involved with these values and we also want to remind our reader that f (v,λ) = g(v,λ) = 0. By substituting (3.1) into (1.1) and collecting the s 2 -terms, we obtain that Multiplying (3.4) by cos kπx L and integrating it over (0, L) by parts, we see that therefore K 1 = 0 and the bifurcation at (v,λ, k ) is of pitch-fork type for all k , k ∈ N + . By collecting the s 3 -terms from (1.1), we have cos kπx L + f vv ϕ 2 +f vλλ2 cos kπx L + 1 6f vvv cos 3 kπx L = 0. (3.5)

QI WANG
Testing (3.5) by cos kπx L , we conclude through straightforward calculations that Therefore, we need to evaluate the integrals L 0 ϕ 2 cos 2kπx L dx and L 0 ϕ 2 dx as well asλ to find the value of K 2 .
Multiplying (3.4) by cos 2kπx L and then integrating it over (0, L) by parts, since K 1 = 0, we have from straightforward calculations that (3.7) Integrating (3.4) over (0, L) by parts, we have that . (3.10) By substituting (3.7) and (3.10) into (3.6), we obtain that On the other hand, we have from straightforward calculations that and Moreover, we can also have that For the simplicity of calculations, we assume that b 1 = 0 from now on. Substituting (3.12)-(3.14) into (3.15), we have that For the simplicity of notations, we introduce then one can easily see that (2.10) implies that bifurcation occurs at (v,λ, k ) if and only if t > c 2 and we shall assume that t > c 2 from now on. In terms of the new variable t, we observe that (3.16) becomes where in the last line of (3.17) we have used the notations Now we are ready to determine the sign of K 2 which is crucial in the stability analysis of (v k (x, s), λ k (s), k ) as we shall see later. To this end, we first have from straightforward calculations that F (c 2 ) = 2c 2 2v 2 > 0; moreover, ifv = 4 3 , the quadratic function F (t) has its determinant (144v 4 +33v 2 )c 2 2 and F (t) = 0 always have two roots which are ; (3.19) furthermore, we readily see that − β 2α − c 2 = (12v 2 +v)c2 2α and it implies that t * 1 < c 2 < t * 2 ifv ∈ (0, 4 3 ) and c 2 < t * 1 < t * 2 ifv ∈ ( 4 3 , ∞). In particular, ifv = 4 3 , we have that F (t) = βt + γ = − 68c2 3 t + and it has a unique positive root 59c2 51 . Then we have the following results on the signs of K 1 and K 2 . Proposition 1. Suppose that (2.10) holds and the bifurcation solutions takes the form (3.1). Then K 1 = 0 and the bifurcation branch is of pitchfork type at (v,λ, k ) for each k ∈ N + . Moreover, we assume that b 1 = 0 and denote t = a2−c2(1+v) 1+v , then we have that, ifv ∈ (0, 4 3 ), K 2 > 0 for t ∈ (c 2 , t * 2 ) and K 2 < 0 for t ∈ (t * 2 , ∞); ifv = 4 3 , K 2 > 0 for t ∈ (c 2 , 59c2 51 ) and K 2 < 0 for t ∈ ( 59c2 51 , ∞); ifv ∈ ( 4 3 , ∞), K 2 > 0 for t ∈ (c 2 , t * 1 ) ∪ (t * 2 , ∞) and K 2 < 0 for t ∈ (t * 1 , t * 2 ). The graphes of K 2 as a function t are illustrated in Figure 1. It should be observed that, K 2 > 0 for t slightly bigger than c 2 since the bifurcation value k 1 in this situation and we have that K 2 is always positive regardless ofv.

Remark 2.
We want to note that the assumption b 1 = 0 is made only for the sake of mathematical simplicity since K 2 becomes extremely complicated to calculate if b 1 > 0 in (1.1). On the other hand, we will shall see in Section 4 that, system (1.1)

QI WANG
with b 1 = 0 admits solutions with single transition layer for being sufficiently small. Moreover, this limiting condition is also necessary in our analysis of the transition-layer solutions in Section 4.
We are ready to present the following result on the stability of the bifurcation solution (v 1 (s, x), λ 1 (s)) on the first branch established in Theorem 2.2. We want to point out that all the rest branches are unstable-see our Remark 3. Here the stability refers to the stability of the inhomogeneous solutions taken as an equilibrium to the time-dependent counterpart to (1.1).
The bifurcation branches described in Theorem 3.1 are formally presented in Figure 2. The solid curve means stable bifurcation solutions and the dashed means the unstable solution.
To study the stability of the bifurcation solution from (v,λ, 1 ), we linearize (1.1) at (v 1 (s, x), λ 1 (s), 1 (s)). By the principle of the linearized stability in Theorem 8.6 [1], to show that they are asymptotically stable, we need to prove that the each eigenvalue η of the following elliptic problem has negative real part: We readily see that this eigenvalue problem is equivalent to (3.20) where v 1 (s, x), λ 1 (s) and 1 (s) are as established in Theorem 2.2. On the other hand, we observe that 0 is a simple eigenvalue of D (v,λ) F(v,λ, 1 ) with an eigenspace span{(cos πx L , 0)}. It follows from Corollary 1.13 in [1] that, there exists an internal I with 1 ∈ I and continuously differentiable functions ∈ I → µ( ), s ∈ (−δ, δ) → η(s) with η(0) = 0 and µ( 1 ) = 0 such that, η(s) is an eigenvalue of (4.25) and µ( ) is an eigenvalue of the following eigenvalue problem (3.21) moreover, η(s) is the only eigenvalue of (3.20) in any fixed neighbourhood of the origin of the complex plane (the same assertion can be made on µ( )). We also know from [1] that the eigenfunctions of (3.21) can be represented by (v( , x), λ( )) which depend on smoothly and are uniquely determined through v( 1 , x), λ( 1 ) = cos πx L , 0 , together with v( , x) − cos πx L , λ( , x) ∈ Z.
Proof of Theorem 3.1. Differentiating (3.21) with respect to and setting = 1 , we arrive at the following system since µ( 1 ) = 0 (3.22) where the dot-sign means the differentiation with respect to evaluated at = 1 and in particularv = ∂v( ,x) . Multiplying the first equation of (3.22) by cos πx L and integrating it over (0, L) by parts, we obtain thatμ According to Theorem 1.16 in [1], the functions η(s) and −s 1 (s)μ( 1 ) have the same zeros and the same signs for s ∈ (−δ, δ). Moreover η(s) = L π 2 and we readily see that sgn(η(s)) = sgn(K 2 ) for s ∈ (−δ, δ), s = 0. Moreover, we can show that the real part of any eigenvalue of (3.20) is negative over the complex plane. Therefore, we have proved Theorem 3.1 according to the discussions above.
Remark 3. We can show that the non-monotone bifurcating solutions (v k (s, x), λ k (s)), k ≥ 2, are always unstable. Actually, by the same calculations as above, one can easily show that the Moser index of (v k (s, x), λ k (s)) is always greater than 1 for all k ≥ 2. It is shown that the project of the continuum of the first branch Γ 1 (s) onto the -axis is an interval of the form (0, 0 ). Therefore, if Γ 1 (s) turns to the right around (v,λ, 1 ), it will eventually switches its direction and then turns to the left. This is illustrated in Figure 2. However, we do not know at what value will this occur.
Thanks to (2.9), there always exist nonconstant positive solutions to (1.1) for each being small However, according to Proposition 1 and Theorem 3.1, the smallamplitude bifurcating solutions (v 1 (s, x), λ 1 (s), 1 (s)) are unstable for 1 being sufficiently small. Moreover, by the same arguments above, we can show that, given a small neighbourhood of the origin of the complex plane, the real part of each eigenvalue of (3.20) inside and out side of this neighbourhood is negative. Therefore, we are motivated to find positive solutions to (1.1) that have large amplitude.

QI WANG
An illustration of the bifurcation branch Γ 1 (s) for K 2 < 0 An illustration of the bifurcation branch Γ 1 (s) for K 2 > 0 4. Existence of transition-layer solutions. In this section, we show that, for being sufficiently small, system (1.1) always admits solutions with a single transition layer, which is an approximation of a step-function over [0, L]. For the simplicity of calculations, we assume that b 1 = 0 and consider the following system throughout the section.
Our first approach is to construct the transition-layer solution v (λ, x) of (4.1) without the integral constraint, with λ being fixed and being sufficiently small. We then proceed to find λ = λ and v (λ , x) such that the integral condition is satisfied. In particular, we are concerned with v (x) that has a single transition layer over (0, L), and we can construct solutions with multiple layers by reflection and periodic extensions of v (x) at x = ..., −2L, −L, 0, L, 2L, ...
To this end, we first study the following equation and λ is a positive constant independent of .
The following properties of H also follows from straightforward calculations.
We also need the following properties of the linear operator L defined in (4.10).
Proof. To show that L is invertible, it is sufficient to show that L defined on L p (0, L) with the domain W 2,p (0, L) has only trivial kernel. Our proof is quite similar to that of Lemma 5.4 presented by Lou and Ni in [10]. We argue by contradiction.
Without loss of our generality, we assume that there exists Φ i ∈ W 2,p (0, L) i , therefore we have from the elliptic regularity and a diagonal argument that, after passing to a subsequence if necessary as i → ∞,Φ i converges to someΦ 0 in C 1 (R c ) for any compact subset R c of R; moreover,Φ 0 is a C ∞ -smooth and bounded function and it satisfies d 2Φ where V 0 (λ, z) is the unique solution of (4.5). Assume that Φ i (x i ) = 1 for x i ∈ [0, L], then we have that f v (λ i , V i (λ, x i )) ≥ 0 according to the Maximum Principle. We claim that V that |z i | = | xi−x0 √ i | ≤ C 0 for some C 0 independent of i . If not, there exists a sequence z i → ±∞ as i → 0, then we easily see thatṼ i λ, z i →v 2 and 0 respectively. On the other hand, since < 0 for all i small, this reaches a contradiction and the claim is proved. Now that z i is bounded for all i small, we can always find some z 0 ∈ R such that z i → z 0 andΦ On the other hand, we differentiate equation (4.5) with respect to z and obtain that where V 0 = dV0 dz . Multiplying (4.18) byΦ 0 and (4.19) by V 0 and then integrating them over (−∞, z 0 ) by parts, we obtain that QI WANG then we can easily show thatΦ 0 (z 0 ) = 0 and this is a contradiction. Therefore, we have proved the invertibility of L and we denote it inverse by L −1 .
To show that L −1 is uniformly bounded for all p ∈ [1, ∞], it suffices to prove it for p = 2 thanks to the Marcinkiewicz interpolation Theorem. We consider the following eigen-value problem By applying the same analysis as above, we can show that for each λ ∈ a2 b2 , , there exists a constant C(λ) > 0 independent of such that µ i, ≥ C(λ) for all sufficiently small. Therefore where < ·, · > denotes the inner product in L 2 (0, L). This finishes the proof of Lemma 4.3.
Proposition 2. Let x 0 ∈ (0, L) be arbitrary. Suppose that λ ∈ a2 b2 +δ, (a2+c2) 2 4b2c2 −δ for δ > 0 small. Then there exists a small 4 = 4 (δ) > 0 such that for all ∈ (0, 4 ), (4.2) has a family of solutions v (λ, x) such that where C 4 is a positive constant independent of and V (λ, x) is given by (4.8). In particular, By Lemma 4.1 and 4.3, we have L −1 G ∞ ≤ C 1 C 3 . Therefore, it follows from (4.14) and Lemma 4.2 that, for any Ψ ∈ B, provided that is small. Moreover, it follows through (4.15) and some simple calculation that, if is sufficiently small, for any Ψ 1 and Ψ 2 in B. Hence T is a contraction mapping from B to B and the contraction mapping theory implies that T has a fixed point Ψ in B for being sufficiently small. Therefore, v constructed above is a smooth solution of (4.2). Finally, we can easily verify that v satisfies (4.20) and this completes the proof of Proposition 2.
We proceed to employ the solution v (λ, x) of (4.2) obtained in Proposition 2 to constructed solutions of (4.1). Therefore, we want to show that there exists λ = λ and (v (λ , x), λ ) such that the integral condition in (4.1) is satisfied. Now we are ready to present another main result of this paper. and b 1 → 0 as → 0. Denote Then there exists 0 > 0 small such that for each x 0 ∈ (x 1 , x 2 ) and ∈ (0, 0 ), system (1.1) admits positive solutions (v (λ , x), λ ) such that compact uniformly on (x 0 , L], is exactly the same as (2.9) when b 1 = 0 and it is required for the existence of small amplitude bifurcating solutions. In particular, we have that x 1 = 0 if c 2 < a 2 ≤ 2a1 C + c 2 and x 1 = (a2−c2)c1−2a1c2 (a1+c1)(a2−c2) L if a 2 > 2a1 C + c 2 ; moreover x 2 < L for all a 2 > c 2 . Similar as the stability analysis in Section 3, the limit assumption on b 1 is only made for the sake of mathematical simplicity.
Proof. We shall apply the Implicit Function Theorem for the proof. To this end, we define for all ∈ (−δ, δ) for δ being sufficiently small, where λ ∈ λ 0 − δ,λ 0 + δ andλ 0 is a positive constant to be determined.

5.
Conclusion and discussion. In this paper, we carry out local and global bifurcation analysis for the nonconstant positive solutions v (λ , x) to the nonlinear boundary value problem (1.1). It is shown that bifurcating solutions exist for all > 0 being small-see (2.9). Though it might be well-known to some people and it may hold even for general reaction-diffusion systems, we show that all the local branches must be of pitch-fork type. For the simplicity of calculations, we assume that b 1 = 0 and then determine the stability of these bifurcating solutions. In particular, we have that the bifurcating solutions are always unstable as long as bifurcating value k is sufficiently small. Finally, we constructed positive solutions to (1.1) that have a single transition layer, where we have also assumed that b 1 = 0 in our calculations. Our results complement [10] on the structures of the nonconstant positive steady states of (1.1) and help to improve our understandings about the original SKT competition system (1.2).
We want to note that, though the assumption b 1 = 0 in Section 3 and Section 4 is made for the sake of mathematical simplicity, it is interesting question to answer whether or not (1.1) admits solutions for all B < A < C or C < A < B. It is also an interesting and important question to probe on the global structure of all the bifurcation branches. It is proved in [17] that the continuum of each bifurcation branch must satisfy one of three alternatives, and new techniques need to be developed in order to rule out or establish the compact global branches. Moreover, more information on the limiting behavior of v not only as approaches to zero, but some positive critical value which may also generates nontrivial patterns. See [12] for the work on a similar system. The stability of the transition-layer solutions is yet another important and mathematically challenging problem that worths attention. To this end, one needs to construct approximating solutions to (1.1) of at least -order. Therefore, more information is required on the operator L , for example, the limiting behavior of its second eigenvalue.
Our mathematical results are coherent with the phenomenon of competitioninduced species segregation. We see from the limiting profile analysis of (1.2) in [10] that u(1 + v) converges to the positive constant λ as ρ 12 → ∞ provided that ρ 12 and d 1 are comparable. Then the existence of the transition layer in v implies that u = λ 1+v must be in the form of an inverted transition layer for being small. These transition-layers solution can be useful in mathematical modelings of species segregation. Therefore, the species segregation is formed through a mechanism cooperated by the diffusion rates d 1 , d 2 and the cross-diffusion pressure ρ 12 . Eventually, the structure of v (x) in (1.1) provides essential understandings about the original system (1.2).