Representation formulas for stationary solutions of a cell polarization model

Abstract

We are interested in the global bifurcation structure of solutions for a nonlinear boundary value problem with nonlocal constraint (SLP) that appears in a cell polarization model with mass conservation proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. We obtained primitive representation formulas of all solutions and investigated a surface ${\mathcal {S}}$ consisting of all bifurcation diagrams with heights. However, we could not find any parameterization of the surface ${\mathcal {S}}$. In this paper, we show parameterizations of the surface ${\mathcal {S}}$ and concrete representation formulas of all global bifurcation diagrams of (SLP). These results are also beneficial for numerical computations to get all bifurcation diagrams. We propose new methods in view of them.

Appendix Sample code to obtain figures of bifurcation diagrams

We give a sample code written by Maple to obtain figures of the surface ${\mathcal {S}}$ parameterized by (2.40) in Fig. 10. We note that

            EllipticK$\left(\sqrt{h}\right)$, EllipticE$\left(\sqrt{h}\right)$, EllipticPi$\left(hp/(1+\sqrt{1-hp^2}),\sqrt{h}\right)$ correspond to

            K$\left(\sqrt{h}\right)$, E$\left(\sqrt{h}\right)$, $\mathrm{\Pi }$$\left(- hp/(1+\sqrt{1-hp^2}), \sqrt{h}\right)$, respectively.

Moreover, $d : V_*/\pi ^2 \sim 0$ corresponds to $h : 0 \sim 1$ for each fixed $V_*$ in ${\mathcal {G}}_{\infty }$. It is better to use fine mesh with high digits to obtain clear figures, especially for $h \approx 1$ ($d \approx 0$). Options         view = [0 .. 4, 0 .. 0.4, $m-10^{-5}$ .. $m + 10^{-5}]$, orientation = $[-90, 0, 0]$

in display($\cdots$) give a top view of ${\mathcal {S}}$ with height $m-10^{-5}$ and $m+10^{-5}$.

This program give the left figure of Fig. 6. We get the right figure of Fig. 6 by replacing

$$\begin{aligned} \left[ V\_inf, d\_inf, m\_inf \right] , \ \ \left[ \frac{1}{subs(p = 2-p, V\_inf)}, \frac{subs(p = 2-p, d\_inf)}{subs(p = 2-p, V\_inf)^2}, m\_inf \right] \end{aligned}$$

to

$$\begin{aligned} \left[ p, 1-h, m\_inf \right] , \ \ \left[ p, 1-h, m\_inf \right] , \end{aligned}$$

respectively, and erasing grG.

Fig. 10

Sample code in Maple