Global bifurcation sheet and diagrams ofwave-pinning in a reaction-diffusion modelfor cell polarization

We are interested in wave-pinning in a reaction-diffusion model for cell polarization 
proposed by Y.Mori, A.Jilkine and L.Edelstein-Keshet. 
They showed interesting bifurcation diagrams and stability results 
for stationary solutions for a limiting equation by numerical computations. 
Kuto and Tsujikawa showed several mathematical bifurcation results 
of stationary solutions of this problem. 
We show exact expressions of all the solution 
by using the Jacobi elliptic functions and complete elliptic integrals. 
Moreover, we construct a bifurcation sheet 
which gives bifurcation diagram. 
Furthermore, we show numerical results of the stability of stationary solutions.

The model is where W = W (x, t) denotes the density of an active protein, V = V (x, t) denotes the density of an inactive protein, ε, D are diffusion coefficients, W 0 (x) denotes the initial density of the active protein, and V 0 (x) denotes initial density of the inactive protein.
It is easy to see that the mass conservation holds, where m is the total mass determined by the mass of the initial densities W 0 (x) and V 0 (x). Letting D → ∞ in (TP), we formally obtain the following time dependent limiting equation: where W = W (x, t),Ṽ =Ṽ (t) is the density depending only on t. W 0 (x) denotes the initial density, andṼ 0 denotes an initial constant density.
Owing to the mass conservation, the stationary problem of (TP) can be reduced to the following Neumann problem with a nonlocal constraint: where W = W (x), V = V (x), and m is a given initial total mass determined by initial densities.
Straight understanding of a stationary limiting problem for (TLP) is The second equation automatically holds from the first and third equation. Hence the above system is equivalent to For simplicity we concentrate on monotone increasing solutions, since we can obtain other solutions by reflecting this kind of solutions. Thus, we get Here it should be noted that we may omit the condition W (0) > 0, since this condition follows from other conditions. Thus we obtain a stationary limiting problem as where m, ε are given positive constants, W = W (x) is an unknown function, andṼ is an unknown nonnegative constant.
Interesting bifurcation diagrams are obtained in [7] by numerical computations. Kuto and Tsujikawa [4] obtained several mathematical results for (SLP) with suitable change of variables.
The main purpose of this paper is to show exact expressions of all the solutions for (SLP) by using the Jacob's elliptic functions and complete elliptic integrals, and construct a global bifurcation sheet in the space (Ṽ , ε 2 , m). Furthermore, we show numerical results on the stability of stationary solutions.
Each level curve with the height m of the sheet corresponds to the bifurcation diagram in the plane (Ṽ , ε 2 ) for (SLP) with given m. Thus, we can obtain all bifurcation diagrams including all, for instance, even secondary bifurcation branches.
Our method to obtain all the exact solutions essentially based on the method which started in Lou, Ni and Yotsutani [6]. It is developed by Kosugi, Morita and Yotsutani [5] to investigate the Cahn-Hilliard equation treated in Carr, Gurtin and Semrod [1], although we need some extra steps.
This paper is organized as follows. In Section 2 we state main theorems, and show figures of the global bifurcation sheet, bifurcation diagrams and stability results by numerical computations. In Section 3 we give proofs of main theorems by using Propositions 3.1, 3.
The following formulas for the complete elliptic integrals are important.
Now, let us introduce an auxiliary problem to investigate (SLP). LetṼ > 0 be given, let us consider the problem We note that (AP;Ṽ ) is equivalent to for givenṼ > 0, since it is easy to see that a condition 0 < W (0) <Ṽ + 1 holds for any solution of (AP;Ṽ ). The existence and the uniqueness of the solution W (x) of (AP;Ṽ ) is well-known (see, e.g. Smoller and Wasserman [2], and Smoller [3]). However, we need to know more precise information to investigate (SLP). The following theorem gives the representation formula for all solutions of (AP;Ṽ ).
Moreover, the solution is unique. The solution W (x;Ṽ , ε 2 ) has properties The solution W (x;Ṽ , ε 2 ) is represented by is the unique solution of the following system of transcendental equations (2.14) Here, sn(·, ·), cn(·, ·) are Jacobi's elliptic function. K(·) is complete elliptic integral of the first kind.
We show the graph of A(h, s) and E(h, s) in Figures 1 and 2.
Let us define the global bifurcation sheet S by We obtain exact representation of the global bifurcation sheet S as by Theorem 2.2. For each m, we can obtain the bifurcation diagram by directly from the global bifurcation sheet S. We will mathematically investigate precise properties of the global bifurcation sheet and bifurcation diagrams in a forth-coming paper. For instance, we see the following facts: · For m ∈ (0, 1], bifurcation diagrams are the empty set. · For m ∈ (1, ∞), bifurcation diagram given by (2.21) are graphs withṼ axis (smooth single-valued function inṼ ) except m = 2 withṼ = 1.
We show figures of bifurcation sheet, bifurcation diagrams, and profiles of W (x;Ṽ , ε 2 ). Figure 3 shows the global bifurcation sheet S by using the expression (2.18).      where M(h, s) is defined by (2.19).
We use results in Kosugi, Morita and Yotsutani [5] for proofs of Propositions 3.1 -3.4. We see from Proposition 1.1 and its proof in [5] that the following lemma holds.
are represented by two parameters (h, s) with 0 < h < 1 and 0 < s < 1 as follows. where E(h, s) and A(h, s) are defined by (2.13) and (2.14) respectively. Moreover, Proposition 3.2 immediately follows from the above lemma. Proposition 3.4 follows from the above lemma and Proposition 3.3. We will give proofs of Propositions 3.1 and 3.3 in Sections 4 and 5, respectively. Now, we give a proofs of Theorems 2.1 and 2.2.
Proof of Theorem 2.1. We see from Proposition 3.2 and Proposition 3.3 that conclusions hold except (2.6). We see thatṼ is a solution of (AP;Ṽ ). Thus, we obtain (2.6) by the uniqueness of solutions of (AP ;Ṽ ).
We get We have and obtain Hence, we get Therefore, we obtain We prepare several lemmas. We see from Lemma 3.2 and the proof of Lemma 3.4 of [5] that the following lemma holds. and and In addition, Then, r(v) is monotone decreasing in (0, ∞) and as v → ∞.
Proof. It is easy to see that (5.10).
We have Let us put We get By virtue of a tool of obtaining Groebner basis, we see that a system of algebraic equation We can see from Strum's theorem concerning zeros for a single algebraic equation that (5.2) does not have real zero h with 0 < h < 1. Hence the system (5.2) for (s, h) has no root in (0, 1) × (0, 1).
Lemma 5.4. LetṼ > 0 be fixed. There exists a unique curve We can obtain the solution exactly, and get .
Thus, we obtain (5.18) by Let us show that E(h, s(h;Ṽ )) is decreasing with respect to h.

Thus, we have
which is easy to prove by differentiation. On the other hand, we show that We show that F 3 (H, s) > 0 in (0, 1) × (0, 1) (5.24) in Appendix. Thus we complete the proof.
Thus we complete the proof by Lemmas 5.4 and 5.5.