数理解析研究所講究録 1475 巻 2006 年 139-148

138Singular Limit Analysis to Higher Dimensional
Patterns of a Chemotaxis Growth System

宮崎大学工学部

辻川

亨 (Tohru Tsujikawa)

Faculty of Engineering,Miyazaki University

Introduction

1

We consider the following model equations which describes the movement of the biological individuals by the diffusion and chemotaxis effects in $[4, 5]$ ;$\frac{\partial u}{\partial\tau}$$=d_{u}\triangle u-\nabla(u\nabla\chi(v))+f(u)$$\tau>0$$\{$$\frac{\partial v}{\partial\tau}$$=d_{v}\mathrm{b}\mathrm{v}$,$\mathrm{x}\in \mathrm{R}^{N}$

(1.1)

,

$+h(u, v)$

are respectively the population density and the concentration
and
where
are diffusion rates.
andof chemotactic substance at time and positionis the velocity of the direct movement of due to chemotaxis, whichof and .
generally satisfies $\chi(v)>0$ and $\chi’(v)\geq 0$ for $v>0$ . Here we specify the growth term $f(u)$as $f(u)=(g(u)-\alpha)u$ where $g(u)$ is the growth rate with cooperation and competition
effects and is the degradation rate due to exterior forces such as predation or intoxication.
Though the functional form of $f(u)$ is basically classified into several cases depending on$g(u)$ and, we consider the cubic-like form, which has three roots 0, and of $f(u)=0$ .
The term $h(u, v)$ in (1.1) is simply specified as $h(u, v)=\beta u-\gamma v$ with the production rate$\beta>0$ and the degradation rate $\gamma>0$ .
In $[4, 5]$ , vve studied (1.1) assuming the situation that the movement of individuals is
mainly due to chemotaxis and that the chemotactic substance diffuses so fast comparedwith the migration of individuals which move by diffusion and chemotaxis, so we introduce
a small parameter $>0$ . By using the suitable transformations [4], the equations (1.1$u(\tau, \mathrm{x})$$v(\tau, \mathrm{x})$$\mathrm{x}\in \mathrm{R}^{N}$$\tau$$u$$d_{v}$$u$$\nabla\chi(v)$$v$$d_{u}$$\alpha$$\overline{u}$$\underline{u}$$\alpha$$\epsilon$

140
can be rewritten as$\frac{\partial u}{\partial\tau}$$=\epsilon^{2}\Delta u-\epsilon k\nabla(u\nabla\chi(v))+f(u)$$\tau>0$$\{$$\frac{\partial v}{\partial\tau}$,$\mathrm{x}\in \mathrm{R}^{N}$

(1.2)

,

$=\triangle v+u-\gamma v$

is sutably normalized. As was stated above,$f(u)$ satisfies $f(0)=f(a)=f(1)=0$ for some $0<a<1$ , $f(u)<0$ for $0<u<a$ ,$f(u)>0$ for $a<u<1$ and $f’(0)<0$ , $f’(1)<0$ . Moreover, we assume $I_{0}^{1}f(u)du$ $>0$ . The
boundary and initial conditions are taken to be

where

$k$

is a positive constant such that$\lim$$(u(\tau, \mathrm{x})$$\chi(v)$, $v(\tau, \mathrm{x}))=(0,0)$(1.3)$\tau>0$$|\mathrm{x}\{arrow+\infty$and$/(0)\mathrm{x})$,$\mathrm{x}\in \mathrm{R}^{N}$$v(0, \mathrm{x}))=(u_{0}(\mathrm{x}), v_{0}(\mathrm{x}))$

.

(1.4)

-dimensional domain
For (1.2), we show the existence of the nonnegative global solution
and the exponential atractor with finite dimension [8].
In $[4, 5]$ , the existence and numerical stability of the radially symmetric stationary
solutions of (1.2) (1.4) in $\mathrm{R}^{N}(N=1, 2)$ are studied for small $>0$ . Moreover, we haveand show that by solving it the stability of thethe limiting system of (1.2) (1.4) as.
stationary solutions is suggested for small, weSince it seems to be different to do the numerical simulation of (1.2) (1.4) inconsider the limiting system and by solving this problem we suggest the existence of therealistic stationary patterns in this paper.
. In Section 3, the existence of
In Section 2, we introduce the limiting system as
is shown. In Section
radially symmetric stationary solutions of the limiting system inand show
4, we consider the stability of the radially symmetric stationary solutions in. Through this paper, we
the dependency of the parameter , the forms of $f(u)$ andand $f(u)$ , that is, $\chi(v)=sv^{2}/(s^{2}+v^{2})$ and $f(u)=$treat with the specified forms of$u($ 1- $u)(u$0.1 , $=0.05$ , $\gamma=1$ for the numerical simulations.
$\mathrm{i}\mathrm{n}2$$\epsilon$--$\epsilon$$\downarrow 0$$\epsilon>0$$\mathrm{R}^{3}$-$\epsilon$$\downarrow 0$$\mathrm{R}^{3}$$\mathrm{R}^{3}$$\chi(v)$$k$$\chi(v)$

$)$

-

2

$\epsilon$

Limiting System as$\epsilon$$\downarrow 0$

In order to study the pattern-dynamics arising in solutions to (1.2) (1.4) with small. To do it, we introduce the new, we derive the limiting system from (1.2) when
. Then (1.2) is rewritten as
time variable with-$\epsilon>0$$\epsilon\downarrow 0$$\tau=t/\epsilon$$t$$\epsilon u_{t}=$$\epsilon^{2}\Delta u-\epsilon k\nabla(u\nabla\chi(v))+f(u)$$t>0$ ,$\{$$\epsilon v_{t}=$$\Delta v+u-\gamma v$$\mathrm{x}\in \mathrm{R}^{N}$

.

(2.1)

141
Using the well known two-timing methods, one can intuitively understand that the time
evolution of the solution of (2.1) consists of two stages. In the first stage, the solution isapproximately described by the following system:$u_{t}=$$\frac{1}{\epsilon}f(u)$$t>0$ ,$\{$

$v_{t}=$$\mathrm{x}\in \mathrm{R}^{N}$

.

(2.2)

$\frac{1}{\epsilon}\{\Delta v+u-\gamma v\}$tends,
Since the system for is bistable from the assumption of $f(u)$ , the solution
where $0\leq u_{0}(\mathrm{x})<a$ , while it tends to 1 inin short time, to 0 in one region, say. This implies the occurrence of layer regions,wherethe other region, say
decomposes, that is,andsay , which is the boundary between two regions, the second variable
and. In these two subregions,
intoapproximately satisfies the following stationary problems:$u(t, \mathrm{x})$$u$$\Omega_{0\epsilon}$$a<u_{0}(\mathrm{x})$$\Omega_{1\epsilon}$$\mathrm{R}^{N}=\Omega_{0\epsilon}\mathrm{U}\Omega_{1\epsilon}\cup R_{\epsilon}$

in

$v$

$\Omega_{1\epsilon}$$\Omega_{0\epsilon}$$0=\Delta v+g_{i}(v)$$\mathrm{R}^{N}$$\Omega_{1\epsilon}$$\Omega_{0\epsilon}$$R_{\epsilon}$$\Omega_{i\epsilon}(\mathrm{i}=0,1)$

(2.3)

,

where $g_{0}(v)=-\gamma v$ and $g_{1}(v)=1-\gamma v$ .
In the second stage, the solution is no longer described by (2.2), (2.3) so that the layer
vary as time goes on. We now as,
andregions must change. This means thatsuch that there is an $(N-1)$ -dimensional hypersurface
sume the situation in the limit
holds as ) 0, that is,such that, which means the interface of , in
. Lettingwhere, we can derive the equation to describe thebe the normal velocity of the intrface
as follows ( see [10]):
dynamics of$\Omega_{0\epsilon}$$\Omega_{1_{\Xi}}$$R_{\epsilon}$$\epsilon\downarrow 0$$u$$\Gamma(t)$$\Omega_{i\epsilon}arrow\Omega_{i}(t)$$\mathrm{R}^{N}=\mathrm{Q}0(\mathrm{t})\cup{\rm Re}(\mathrm{t})\cup\Gamma(t)$$R_{\epsilon}(t)arrow\Gamma(t)$$\mathrm{R}^{N}$$\epsilon$$=\{\mathrm{x}\in \mathrm{R}^{N}, \mathrm{T}(\mathrm{t})\mathrm{x})=\mathrm{i}\}(\mathrm{i}=0, 1)$$\Gamma(t)$$V^{*}$$\Gamma(t)$,$V^{*}= \mathrm{c}^{*}+k\chi’(v)\frac{\partial v}{\partial n}-\epsilon(N-1)\kappa+\epsilon G$$t>0$ ,$\mathrm{x}\in\Gamma(t)$$0=\triangle v+g_{i}(v)$$t>0$ ,$\mathrm{x}\in\Omega_{i}(t)$

$\{$

,

, is theontomean curvature at the interface. Here, is the velocity of the traveling front solution of
the scalar bistable reaction diffusion equation ( see [3]). Although $G=O(1)$ for small in
general, we neglect this term in order to study the effect of the curvature to the motion of
the interface at the first step. Therefore, the equation is rewritten as

where

$n$

means the outerward unit normal vector from$\Omega_{1}(t)$$\Gamma(t)$$\Omega_{0}(t)$$\kappa$$c^{*}$$\epsilon$$V^{*}=c’+k \chi’(v)\frac{\partial v}{\partial n}-\epsilon(N-1)\kappa$$t>0$ ,$\mathrm{x}\in\Gamma(t)$

,

(2.4)

$\{$

$t>0$ ,

$0=\Delta v+g_{i}(v)$$\mathrm{x}\in\Omega_{i}(t)$,which we call the singular limit system or simply the interface equation of (2.1). The
sm oothness of on the interface is imposed to satisfy $v\in C^{1}$ , that is,$v$$\Gamma$$v(t, \cdot)\in C^{1}(\mathrm{R}^{N})$$t>0$ .

(2.5)

142
It clearly shows that the dynamics of the interface is determined by three effects; thevelocity of the 1-dimensional traveling front solution, the chemotactic effect due to theand the geometric effect of the interface. Moreover, from (1.3), we assume
gradient of
that
(2.6)
, $t>0$ .
$\chi(v)$$\lim_{|\mathrm{x}|arrow\infty}v(t, \mathrm{x})=0$

In the previous paper [4], we show the existence of radially symmetric stationary solutions
where$(u(r), v(r))$ of the interface equation (2.4)-(2.6) in $\mathrm{R}^{N}(N=1, 2,3)$ withthe center and the interface locate at the origin and $r=\eta$ , respectively. Moreover, the.
stability of these solutions was discussed for
Bonami et al. [1] treated with the case where the equation for is stationary and thealmost all same, that is, , effects ofpotentials of two equilibria $(0, 0)$ and $(1, 1/\gamma)$
chemotaxis and curvature are same of order with respect to . In this situation, the solutionof the interface equation is good approximation to one of the original reaction-diffusion
equation.
$|\mathrm{x}|=r$$N=1_{\dot{J}}2$

$v$

.are

$c^{*}$$\epsilon$

3

Existence of the radially symmetric stationary solutions in$\mathrm{R}^{3}$

In this section, we consider the existence of a radially symmetric stationary solution of
the interface eqaution (2.4)-(2.6). In order to show that, we first treat with the following
problem:$0=c’+k \chi’(v)v_{r}-\frac{(N-1)\epsilon}{r}$$\{$$0=v_{rr}+ \frac{N-1}{r}v_{r}+g_{i}(v)$$v_{r}(0)=0$ ,,where
Then, the solutions
scribed by$|\mathrm{x}|=r$$\Omega_{1}=(0, \eta)$,,$\lim_{farrow\infty}v(r)=0$$r=\eta$$T$$\in\Omega_{i}$

and

,

(3.1)

$(i=0,1)$$v\in C^{1}(\mathrm{R}_{+})$,

and $0_{0}=(\eta, \infty)$ .
except for the first equations of (3.1) for $N=3$ is de-$(\eta, v(r;\eta))$$\frac{1}{\gamma}+$(a$- \frac{1}{\gamma}$)$\frac{\eta\sinh\sqrt{\gamma}r}{r\sinh\sqrt{\gamma}\eta}$$r\in(0, \eta)$(3.2)$v(r;\eta)\equiv\{$$\frac{\alpha\eta}{r}e^{-\sqrt{\gamma}(r-\eta\}}$with(3.1), we obtain$\alpha=v(\eta;\eta)=\eta \mathrm{K}_{\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{I}_{\frac{3}{2}}(\sqrt{\gamma}\eta)/\backslash \Gamma\gamma$$r\in(\eta, \infty)$. Substituting (3.2) into the first equation in$c’-k \chi’(\frac{\eta \mathrm{I}_{\frac{3}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{1}{2}}(\sqrt{\gamma}\eta)}{\sqrt{\gamma}})\eta \mathrm{I}_{\frac{3}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{3}{2}}(\sqrt{\gamma}\eta)-\frac{2\epsilon}{\eta}\equiv H(\eta, k, \epsilon)=0$

.

(3.3)

143
By using the solution ny of $H(\eta, k, \epsilon)=0$ , one easily finds that the solution of (3.1) isrepresented by $(\eta, v(r;\eta))$ .
. For fixed small
Theorem 1. [4] Let $k^{*}>0$ be a constant to satisfy$(>k^{*})$ such that for$>0_{7}$ there exists a constant
there are at least, for $0<k<k^{*}$ ,and y7such thatandtwo solutions
withthere are at least one solution
for $N=3$ , respectively.
$c^{*}- \frac{k^{*}}{2\sqrt{\gamma}}\chi’(\frac{1}{2\gamma})=0$$k^{*}<k<\overline{k}(\epsilon)$$\overline{k}(\epsilon)$$\epsilon$$(\overline{\eta},v(r;\overline{\eta}))$$(\underline{\eta}, v(r,\underline{\eta}))$$=O(\epsilon)$$\overline{\eta}=O(1)$$(\underline{\eta}, v(r;\underline{\eta}))$$\underline{\eta}=O(\epsilon)$

Letting $\eta=\eta(k)$ be a solution of (3.2), we define the pair of functions $(u^{0}(r), v^{0}(r))$ by$u^{0}(r)=\{\begin{array}{l}1r\in(0,\eta)0r\in(\eta,\infty)\end{array}$

(3.4)

$\{$

$v^{0}(r)=v(r;\eta)$$r\in(0, \infty)$

and call it a radially symmetric stationary solution of the interface equation (2.4)-(2.6) for$N=3$ , respectively.
Next, as $\chi(v)=sv^{2}/(s^{2}+v^{2})$ , we draw numerically the global picture of radially symmetric stationary solutions of (3.1) for $N=3$ when is varied in Figure 1, In this case,such that for (i) $0<s$ $<s^{*}$ , there
there is a critical value $s^{*}>0$ of a parameter ofare three branches, while for (ii) $s^{*}<s$ , there are two ones when is varied. In Figure 2,
the existence region of the solution is shown in the $(k, s)$ plane for $N=3$ .
On the other hands, by the numerical simulations it does not able to suggest whichis realistic till now. Therefore, from the theoretical view point,
stationary solution in
we consider the stability of the radially symmetric stationary solutions of the interface
equation in the 3-dimensional domain in the next section. Moreover, it is shown that theare at least unstable with respect to the disturbances ofstationary solutions
radial direction.
$k$

$s$

$\chi(v)$$k$-$\mathrm{R}^{3}$$(\underline{\eta}, v(r;\underline{\eta}))$4

Stability of the radially symmetric stationary solutions in$\mathrm{R}^{3}$

In this section, we show the stability of the radially symmetric stationary solution of.
(2.4)-(2.6) for $N=3$ , which satisfies $\eta=O(1)$ for small
To study the stability, we represent deformations of the interface $r=\eta$ by the polar, where takeswith the azimuthal angle
coordinate$(0,2\pi)$ , whiletakes 0 for
1 for. For $\Gamma=(r \sin ? \cos\varphi, r\sin\theta\sin\varphi, r\cos\theta)$ , it follows from,the first equation in (2.4) that$\epsilon>0$$(\theta, \varphi)$$(r, \theta, \varphi)=(\eta+\zeta(t, \theta, \varphi), \theta, \varphi)$$(r, \ , \varphi)\in(0, \eta+\zeta(t, \theta, \varphi))\mathrm{x}$$\zeta(t, \theta, \varphi)$$(0, \pi)\mathrm{x}$

$u$

$u$

$(r, \theta, \varphi)\in(\eta+$$\infty)\mathrm{x}(0, \pi)\mathrm{x}(0, 2\pi)$$\frac{r\sin\theta}{\sqrt{r^{2}\sin^{2}\theta+\zeta_{\varphi}^{2}+\zeta_{\theta}^{2}\sin^{2}\theta}}\zeta_{t}=c+k\chi’(v)\frac{v_{r}r^{2}\sin^{2}\theta-v_{\theta}\sin^{2}\theta\zeta_{\theta}-v_{\varphi}\sin^{3}\theta\zeta_{\varphi}}{r\sin\theta\sqrt{r^{2}\sin^{2}\theta+\zeta_{\varphi}^{2}+\zeta_{\theta}^{2}\sin^{2}\theta}}-2\epsilon\kappa+O(\zeta^{2})$.

144
By using the balance of the above equation with respect to lower parts ofderivatives, it holds that$\zeta_{t}=$$\langle$

and their$k\{\chi’(v_{0})(v_{r}^{(1)}+v_{r}^{(2)})+\chi’(v_{0})v_{0r}(v^{\langle 1)}+v^{(2)})\}$(4.1)$+2 \epsilon(\frac{\zeta}{\eta^{2}}+\frac{\zeta_{\theta\theta}}{2\eta^{2}}+\frac{(_{\varphi\varphi}}{2\eta^{2}\sin^{2}\theta}+\frac{\zeta_{\theta}\cos\theta}{2\eta^{2}\sin\theta})+O(\zeta^{2})$Defining the completely orthonormal system$\{Y_{\ell,m}(\theta, \varphi)\}$.

on the sphere by$Y_{\ell,m}(\theta, \varphi)=$

where$P_{\ell}^{m}(x)=(1-x^{2})^{\bigcup_{2}} \frac{d^{|m|}}{dx^{|m|}}P_{t}(x)m$and$P_{t}(x)= \frac{1}{2^{\ell}\ell!}\frac{d^{\ell}}{dx^{\ell}}(x^{2}-1)^{\ell}$, we have$\frac{1}{4\pi}\oint_{0}^{2\pi}\int_{0}^{\pi}\epsilon\{\frac{2\zeta}{\eta_{0}^{2}}+\frac{\zeta_{\theta\theta}}{\eta^{2}}+\frac{\zeta_{\varphi\varphi}}{\eta_{0}^{2}\sin^{2}\theta}+\frac{\zeta_{\theta}\cos\theta}{\eta_{0}^{2}\sin\theta}\}Y_{\ell,m}(\theta, \varphi)\sin\theta d\theta d\varphi$$=- \frac{\epsilon}{\eta^{2}}(f+2)(\ell-1)(_{\ell,m}(t)$$\frac{1}{4\pi}I_{0}^{2\pi}\int_{0}^{\pi}v^{(1)}Y_{\ell,m}(\theta, \varphi)\sin$

,

Odfld

$\varphi=\frac{d}{dr}v_{0}(\eta)\zeta_{\ell,m}(t)$,$\frac{1}{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}v^{(2)}Y_{\ell,m}(\theta, \varphi)\sin\theta d\theta d\varphi=\eta \mathrm{I}_{\ell+\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\ell+\frac{1}{2}}(\sqrt{\gamma}\eta)\zeta_{\ell,m}(t)$,$\frac{1}{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}\frac{\partial}{\partial r}v^{(1)}Y_{l,m}(\theta, \varphi)\sin\theta d\theta d\varphi$,$=- \frac{1}{3}[\frac{1}{2}+\sqrt{\gamma}\eta(\mathrm{I}_{\frac{\iota}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{3}{2}}(\sqrt{\gamma}\eta)-2\mathrm{I}_{\frac{3}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{5}{2}}(\sqrt{\gamma}\eta))]\zeta_{\ell,m}(t)$$\frac{1}{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}\frac{\partial}{\partial r}v^{(2)}Y_{\ell,m}(\theta, \varphi)\sin\theta d\theta d\varphi$$= \frac{1}{2l+1}[\frac{1}{2}+\sqrt{\gamma}\eta(\ell \mathrm{I}_{\ell-\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\ell+\frac{1}{2}}(\sqrt{\gamma}\eta)-(\ell +1)\mathrm{I}_{t+\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\ell+\frac{3}{2}}(\sqrt{\gamma}\eta)]\zeta_{t,m}(t)$where$\zeta_{\ell,m}(t)=\frac{1}{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}\zeta(\theta, \varphi, t)Y_{\ell,m}(\theta, \varphi)$$\sin\theta d\theta d\varphi$.

145
It follows from (4.1) that$\frac{d}{dt}\zeta_{t,m}=$$\{k\chi’(v_{0})v_{0\mathrm{r}}$$[v_{0r}+\eta \mathrm{I}_{\ell+\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\ell+\frac{1}{2}}(\sqrt{\gamma}\eta)]$$-k \chi’(v_{0})[\frac{1}{3}[$ $\frac{1}{2}+\sqrt{\gamma}\eta(\mathrm{I}_{\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{3}{2}}(\sqrt{\gamma}\eta)-2\mathrm{I}_{\frac{3}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{6}{2}}(\sqrt{\gamma}\eta))]$(4.2)$- \frac{1}{2\ell+1}[\frac{1}{2}+\sqrt{\gamma}\eta(\ell \mathrm{I}_{\ell-\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\ell+\frac{1}{2}}(\sqrt{\gamma}\eta)$$-( \ell+1)\mathrm{I}_{l+\frac{1}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{t+\frac{3}{2}}(\sqrt{\gamma}\eta))]]-\frac{\epsilon}{\eta^{2}}(\ell+2)(\ell-1)\}\zeta_{\ell,m}(t)$$\equiv F(\ell, k, \epsilon)\zeta_{\ell,m}(t)$

where

,

$v_{0r}=-\eta \mathrm{I}_{\frac{3}{2}}(\sqrt{\gamma}\eta)\mathrm{K}_{\frac{3}{2}}(\sqrt{\gamma}\eta)$.

for all
Definition ( Linearlized stability of the stationary solution ) If
is stable. If not, the solution is$\ell\in \mathrm{N}(\ell>1)$ , then the stationary solution
unstable.
, that is, the stability of the stationary solution
Remark 1. (i).
under the radially symmetric disturbances is determined by the sign of(ii) $F(1, k, \epsilon)=0_{1}$ that is, the stationary solution has phase shift free in (2.4).
. In Figure
Next, we numerically treat with the functional form of
3, the curves of $F(\ell, k, \epsilon, s)=0$ for $\ell=2,3,4$ is shown in the $(k, s)$ -plane. For small $s>0$ ,the solution is stable and with any fixed $s>0$ , the solution is so for large $k>0$ . Figure 4
show the form of $F(l, k, \epsilon, s)$ for $s=0.3,0.5$ , 0.6, 1.0 and $\ell=2,3,4$ . It is known that theseabove results are similar as that of the case for $N=2$ in [4].
$F(\ell, k_{7}\epsilon)<0$$(\eta, v(r;\eta))$$F(0, k, \epsilon)=\frac{\partial}{\partial\eta}H(\eta;k, \epsilon)$$\frac{\partial}{\partial\eta}H(\eta;k, \epsilon)$$F(\ell_{?}k, \epsilon)=F(\ell, k, \epsilon, s)$

Proposition 1. ( Asymptotic behavior of$F$( , , )) It holds that$\ell$$k$$\epsilon$$\lim_{karrow k^{\mathrm{c}}}\{F(\ell, k, \epsilon)\overline{\eta}^{2}+(P+2)(\ell-1)F^{*}(\epsilon)\}=0$

,

.
where
, we can prove this proposition from (4.2) by using theProof. Because of$(z)$ astends toandasymptotic behavior of the modified Bessel functions
infinity.
$\ell>1$ , it
Remark 2. If $F^{*}(\epsilon)>0$ , it follws from the proposition that for any integer, that is, the stationary solution becomes stable as tends to .
holds
, thenAs tends to , it holds that if$F^{*}(\epsilon)$$=\epsilon$$-k’ \chi’(\frac{1}{2\gamma})/(8\gamma^{2})$$\lim_{karrow k^{*}}$$\mathrm{y}=\infty$$\mathrm{I}_{\ell+\frac{1}{2}}(z)$$z$$\mathrm{K}_{\ell+\frac{1}{2}}$$k^{*}$$k$$F(\ell_{r}k, \epsilon)\overline{\eta}^{2}<0$$k$$k^{*}$$F^{*}(\epsilon)>0$$0>F(2, k, \epsilon)>F(3, k, \epsilon)>\cdots>F(\ell, k, \epsilon)>F(\ell+1, k, \epsilon)>$

. ..,

14G
if

$F^{*}(\epsilon)<0$, then$0<F(2, k, \epsilon)<F(3, k,\epsilon)<\cdots<F(P, k, \epsilon)<F(\ell+1, k, \epsilon)’<\cdots$.

For the numerical simulation, it holds thatfor 0.98 $\cdots<s<5.45\cdots$ .
in
for $\ell>1$ as tends toOn the other hands, it is suggested thatis the turning point of the global branch of the stationary solution,
Figure 4. Sincefrom Theorem
we may assume that
becomes of order for small as tends to
1. Then, we have
(4.3)
$F^{*}(\epsilon)<0$$k$$F_{3}(\ell, k, \epsilon)<0$$\overline{k}(\epsilon)$$\overline{k}(\epsilon)$$\overline{\eta}(\epsilon)$$\epsilon$$\epsilon$$k$$\overline{k}(\epsilon)$$\frac{F(\ell,k,\epsilon)}{\overline{\eta}(\in)}=-\frac{(\ell+2)(\ell-1)\mu}{\epsilon}+O(1)$for some positive constant . Therefore, as
$\mu$

$k$

tends to$\overline{k}(\epsilon)$, it follows from the (4.3) that$0>F(2, k, \epsilon)>F(3, k, \epsilon)>\cdots>F(P, k, \epsilon)>F(\ell+1, k, \in)>\cdots$.

In this paper, we do not discuss the relation of the solutions between the interface
equation (2.4) and the original reaction-diffusion equation (1.2). That is, the solution of(2.4) becomes the good approximation of the solution of (1.2). Moreover, there is the
problem such that the asymptotic behavior of the critical eigenvalues of the linearized( see [9]).
eigenvalue problem of (1.2) is represented by using
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dimension, submitted to ASPM
$F(\ell, k, \epsilon)$

147

2

$s$

1

0

$k$
$\mathrm{k}$

20

10

0

30

3-dim.. syrrvnetr ic stat i onary solut i on (s $=0.$ 42)

Radial I$\mathrm{y}$symmetr$i\mathrm{c}$stati onary solutions

Figure 2

Figure 1$\mathrm{s}$

$J‘$

$k$

Figure 3, 2

Figure 3. 1$\mathrm{F}(0.

3, k, l)$$k$

Figure 4. 1

.

148

$\mathrm{F}(0.5.k.

$\mathrm{F}(0.5.k, l)$\mathit{1})$

0.2
0
-0.2
-0.4
-0.6

-0.8
$k$

$k$

- $\{$

$0$

2

4

6

8

10

Figure 4. 2$\mathrm{F}$$(0.6.k, l)$$\mathrm{F}(0.6.k, l)$

0.15
0.1

o.os
0
$k$

.0.05
1.6

$k$

1.8

22.2

2.4

2.6

Figure 4. 3$\mathrm{F}(1.0.k, \mathit{1})$$\mathrm{F}\mathrm{t}1$$k$. 0. , )
$k$

$l$

$k$

Figure 4. 4