数理解析研究所講究録
第 1881 巻 2014 年 66-70

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Dynamics of localized solutions for reaction-diffusion
systems on two dimensional domain-Spot dynamics on curved surfaceShin-Ichiro Ei
Institute of Math. for Industry, Kyushu University
Hiroki Yagisita
Faculty of Science, Kyoto Sangyo University

1Introduction
In nature, many kinds of spatial and $/or$ temporal patterns are observed, some of them

are simple and the others are complicated. To understand theoretically the dynamics ofsuch patterns, many model equations have been proposed and analyzed. Among them,some sort of reaction-diffusion systems are one of the most familiar classes.
In this report, we consider general type of reaction-diffusion systems in$R^{2}$

which possess

stable spot solutions. Spot patterns are one of the most typical and important patterns fromthe phenomenological point of view. In fact, spot patterns correspond to localized patterns
of materials which are observed in many phenomena such as biology and chemistry. Alsoin mathematical models, many model equations with spot solutions have been proposed asin the Gierer-Meinhardt model, Gray-Scott model (e.g. [5], [4], [7], [8]).

In this report, we assume the existence of a stable spot solution onof it on a curved surface in$R^{3}.$$A$$R^{2}$and the dynamics

curved surface is imaged as a cell membrane and a spotsolution is a localized structure with high density of materials on the surface. In fact,[1] reported the localization of the phosphatidylinositol lipids on the cell membrane ofDictyostelium cells. As the consequence, we show that the motion of spot solution along

the surface is essentially described as the gradient flow of the Gaussian curvature.

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Setting and results
In this section, we will give the several assumptions in the general framework.

Let us consider general type of reaction-diffusion systems written by$u_{t}=D\triangle u+F(u), t>0, x\in R^{2}, u\in R^{N}$ ,(2.1)

where $D$ is a diagonal matrix given by $D$ $:=diag\{d_{1}, d_{2}, \cdots, d_{N}\}$ with $d_{j}>0$ and $F$ is asmooth map fromLet
1$\mathcal{L}(u)$Let$R^{N}.$$:=D\triangle u+F(u)$and $X$ $:=\{L^{2}(R^{2})\}^{N}$ . The assumptions are as follows:$L$, where $r:=|x|.$be the linearized operator$D\triangle+tF’(S(r))$$)$

toThere exist a radially symmetric stationary pulse solution $S(r)$ of (2.1) such that$)$$\mathcal{L}(S(r))\equiv 0$

2$R^{N}$a positive constant$0$of (2.1) with respect to $S(r)$ , that is, $L;=$.

The spectral set ofRemark 2.1$\mathcal{L}’(S(r))$$L$consists of two sets$\sigma_{1}=\{0\}$and$\sigma_{2}\subset\{\lambda\in C;Re(\lambda)<-\gamma\}$for$\gamma.$is necessarily a spectrum of$L$because (2.1) has a translation invariance

and $LS_{x_{j}}=0(j=1,2)$ hold, where $x=(x_{1}, x_{2})$ .

Let

$Q$

and

That is,$Q$$R$

be projections corresponding to the spectral sets$:= \frac{1}{2\pi i}\int_{C}(\lambda-L)^{-1}d\lambda$

and $R;=Id-Q$ , where$C$$\sigma_{1}$and$\sigma_{2}$, respectively.

is a circle around$\{\lambda\in C;Re(\lambda)>-\gamma\}$and $Id$ denotes the identity. Define $E:=QX,$ $E^{\perp}:=RX.$

3holds.

$)$$E=span\{S_{x_{1}}, S_{x_{2}}\}$

Remark 2.2 The assumption 3) means the stability of $S(x)$ except translation.

$0$

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Letbe the adjoint operator of . Then
$L$$L^{*}$

such that$L^{*}\phi_{j}^{*}=0$

and$\langle S_{x_{j}},$$\phi_{j}^{*}\rangle_{X}=\pi$

by a radially symmetric function$L^{*}$

also has eigenfunctions $\phi_{j}^{*}(x)(j=1,2)$. Then we note that$\Phi^{*}=\Phi^{*}(r)$$\phi_{j}^{*}$

is represented ae$\phi^{*}=\partial_{x_{j}}\Phi^{*}$.

Under these three assumptions, we consider the reaction-diffusion systems (2.1) on asmoothly curved surface$\mathcal{M}\subset R^{3}$:$u_{t}=\delta^{2}\triangle_{\mathcal{M}}u+F(u), t>0, x\in \mathcal{M}$where $\delta>0$ is a sufficiently small constant and$\Delta_{\mathcal{M}}$(2.2),

is the Laplace-Beltrami operator on$\mathcal{M}.$

For simplicity, we assume the surface$\mathcal{M}$is expressed by the isothermal coodinate as$\mathcal{M}=\{\Gamma(u, v);(u, v)\in R^{2}\}$with $ds^{2}=E(u, v)(du^{2}+dv^{2})$ . Let $K(u, v)$ be the Gaussian

curvature of. Then we have:$\mathcal{M}$andTheorem 2.1 For$T:=\delta^{2}t$$H\in \mathcal{M}$, there exists a map $\Psi=\Psi(w;H)\in R^{2}$ defined for $|w|<<1$

such that $\Psi(0;H)=0$ and the solution $u(t, x)$ of (2.2) is expressed $os$$u(t, x)=S(|\Psi(x-H(T);H(T))|/\delta)+O(\delta)$

and $H(T)$ satisfies$\frac{dH}{dT}=\frac{M_{0}}{E(h(T))}\{K_{u}(h(T))\Gamma_{u}(h(T))+K_{v}(h(T))\Gamma_{v}(h(T))\}+O(\delta)$

where $H=\Gamma(h)\in \mathcal{M},$ $h=(h_{1}, h_{2})\in R^{2},$$M_{0}$$:= \frac{3}{4}\int_{0}^{\infty}r^{3}\langle DS_{r},$$\Phi_{r}^{*}\rangle$,(2.3)dr.

(2.3) implies that the spot solution moves toward points with maximal Gaussian curvature if$M_{0}$is positive and toward points with minimal Gaussian curvature if$M_{0}$

is negative

while the $sign$ is depending on each model. The proofs will be basically given by similarmanner to [2].
Finally, we give the explicit value of the constant$M_{0}$for the Gierer-Mienhardt model.

We consider the following Gierer-Meinhardt model for sufficiently small $\epsilon>0,0\leq\tau<<1$
and $a>0$ :$\{\begin{array}{l}u_{t} = \epsilon^{2}\Delta u-u+\frac{u^{p_{1}}}{(v+a)^{p_{2}}},\tau v_{t} = d\Delta v-v+\underline{u^{p_{3}}}(v+a)^{p_{4}}’\end{array}$

(2.4)

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where

$p_{j}$

are the positive constants satisfying $p_{1}>1$ and$0< \frac{p_{1}-1}{p_{2}}<\frac{p_{3}}{p_{4}+1}.$$(2.4)$

has astable radially symmetric spiky solution in the form of $S(r)=(U(r/\epsilon), V(r))$ for $r=|x|$and functions $U(r),$ $V(r)$ ([3], [5], [6]). In particular, when $p_{3}=2$ and $1<p_{1}<3$ or$p_{3}=p_{1}+1$

and $1<p_{1}<\infty,$$\Phi^{*}(r)$is given by $\Phi^{*}(r)=c(U(r/\epsilon), 0)+0(1)$ as$\epsilon\downarrow 0$

a constant . Then we have the following theorem on the value of the constant$c$([3]) for$M_{0}$:

Theorem 2.2 If $p_{3}=2$ and $1<p_{1}<3$ or $p_{3}=p_{1}+1$ and $1<p_{1}<\infty$ are satisfied for(2.4), then the constant$M_{0}$

is given by$3^{3} \int_{0}^{\infty}r^{3}(U’(r))^{2}dr$$M_{0}=\epsilon\overline{4\int_{0}^{\infty}r(U’(r))^{2}dr}+o(\epsilon^{3})>0.$

The above theorem says that the spot solution moves toward points with maximal

Gaussian curvature.

References[1] Yoshiyuki Arai, Tatsuo Shibata, Satomi Matsuoka, Masayuki J. Satoa, Toshio

Yanagida and Masahiro Ueda,
lipidsSelf-organization of the phosphatidylinositol

signaling system for random

cell migration,

PNAS Early Edition,

doi $/10.1073/pnas$ .0908278107.
[2] S.-I. Ei, M. Mimura and M. Nagayama, Interacting Spots in reaction diffusion systems,

DCDS 14 (2006), 31-62.
[3] S.-I. Ei and J. Wei, Dynamics of metastable localized patterns and its application

to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial
dimension, Japan J. Indust. Appl. Math. 19(2) (2002), 181-226.

[4] P. Gray and S. K. Scott, Autocatalytic reaction in the isothermal continuous stirredtank reactor, Chem. Engng. Sci. 39, 1087-1097 (1984).
[5] W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear

Neumann problem, Duke Math. J. 70 (1993), pp. 247-281.

70[6] W.-M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow
system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18, (2001),

259-272.
[7] J.E. Pearson, Complex patterns in a simple system, Science Vol. $216(1993),189-192.$[8] J. Wei, Pattern formation two-dimensional Gray-Scott model: existence of single spot

solutions and their stability, Phy. D, 148(2001), 20-48.

Institute of Math. for Industry
Kyushu University

Fukuoka 819-0395
JAPAN$E$-mail address ichiro@imi.kyushu-u.ac.jp,

Faculty of ScienceKyoto Sangyo University
Kyot $0603-8555$

JAPAN-mail address hrkOygst@cc.kyoto-su.ac.jp.

$E-$