数理解析研究所講究録
第 1542 巻 2007 年 41-57

41

A Lotka-Volterra Cross-Diffusion Modelin Spatially Heterogeneous Environments
Kousuke KUTO*(久藤衡介・福岡工業大学)
Department of Intelligent Mechanical Engineering, Fukuoka Institute of Technology
3-30-1 Wajiro-Higashi, Higashi-ku, Fukuoka 811-0295, JAPANSKT Model in Heterogeneous Environments

1This article is concemed with the Lotka-Volterra reaction-diffusion system :(P)$\{\begin{array}{ll}u_{t}=\Delta[(1+kp(x)v)u]+u(a-u-c(x)u) in \Omega x(0, \infty),v_{l}=\Delta v+v(b+d(x)u-v) in \Omega x(0,\infty),\partial_{v}u=\partial_{v}v=0 on \partial\Omega x(0,\infty),u(\cdot,0)=u_{0}\geq 0, v(\cdot,0)=v_{0}\geq 0 in \Omega.\end{array}$is a bounded domain in $R^{N}(N\leq 3),$ and are positive constrts, is a realis a smooth positive
constant, $c(x)$ and $d(x)$ are smooth nonnegative functions, andfunction withon . From the ecological viewpoint, the unknown functions
and , respectively, represent the population densities of the prey and predator, whichinteract and migrate in . In the reaction terms, and denote the birth rates of theprey and predator, respectively. While and are spatially homogeneous, the preypredator interactions $c(x)$ and $d(x)$ are assumed to be spatially dependent nonnegative
functions. In the diffusion terms, the linear part represent. the natural dispersive forceof the movement of an individual. On the other hand, the nonlinear diffusion
Here,$\Omega$

$b$

$k$

$a$

$\rho(x)$$\partial_{v}p=0$$u$$\partial\Omega$$v$$\Omega$

$a$

$a$

$b$

$b$

$s$

$\Delta\psi(x)vu]=\nabla[u\nabla\phi(x)v)+p(x)v\nabla u]$(1.1)’Partially supported by a Grant-in-Aid for Scientific Research (No. 18740093), The Ministry of Education, Culture, $Spolts$ , Science and Technology, Japan.

42yields the most characteristic term in (P). In the homogeneous case when is a positive
constant, this term models the tendency such that the prey escapes to a low density region of the predator. In this case,is usually referred as the cmss-diffi.;ion term
whose nonlinear effect has attracted attention in the field of reaction diffusion systems(e.g., [1], $[6]-[17],$ $[21]$ ) after the research by Shigesada-Kawasaki-Teramoto [20]. (Inhonor of their pioneering research, such a class of Lotka-Volterra cross-diffusion systems is often called $SKT$ model.) In the heterogeneous case when $p(x)$ is spatiallyrepresents a type of the environment potential.
dependent on $(1.1.)$ reveals that
Further, $\Delta(p(x)vu)$ describes the spatially and density dependent diffusion such that the. We refer to [18] for a further ecologicalprey moves to the low value region of
backyound.
Our aim is to derive the spatially heterogeneous effects of $c(x),$ $d(x)$ and $p(x)$ on
the stationary solution set of (P). Then we study the foUowing strongly coupled elliptic$p$$\rho\Delta(vu)$$\rho(x)$$x,$$\rho(x)v$

system:(SP)$\{\begin{array}{ll}\Delta[(1+kp(x)v)u]+u(a-u-c(x)v)=0 in \Omega,\Delta v+v(b+d(x)u-v)=0 in \Omega,\partial_{v}u=d_{v}’v=0 on c?\Omega.\end{array}$

We are interested in positive solutions of (SP). Here it is said that $(u,v)$ is a positive$11y$ a positive solution correEcologica,’
solution if both of and are positive insponds to a coexistence steady-state of the prey and the predator. We study the positive
solution set of (SP) by considering to be the bifurcation parameter. In order to obtain
the bifurcation branch of the positive solution set, we define the semitrivial solution
sets with the bifurcation parameter by$u$$\Omega.$

$v$

$b$

$b$

$\Gamma_{u}:=\{(u,v,b)=(a,0,b) :b\in R\}$,$\Gamma_{v}:=\{(u,v,b)=(O,b,b) :b\in R\}$.

In [10], we prove that the positive solution branch which connectswith . Moreprecisely, for any fix $ed(a.k_{l)}(x),c(x),d(x))$ , we flnd a negative number $b=b_{*}<0$ anda positive number $b=b^{*}>0$ (both depend on $(a,k,p(x),c(x),d(x))$ ) such that the positive solution set contains a bounded continuum which bifurcates from $(a,0,b_{*})\in\Gamma_{u}$and joins. Hence, (SP) has at least one positive solution if $b_{*}<\cdot b<b^{*}$ .
and are constants, it is easily verified
In the spatially homogeneous case whenthatfoms a bounded line of the positive constant solutions (see Figure 1). This line
is expressed as$\Gamma_{u}$$\Gamma_{p}$$(0,b^{*},b^{*})\in\Gamma_{v}$$c,$ $d$$\rho$$\Gamma_{p}$$\Gamma_{p}=\{(\frac{a-bc}{1+cd},$ $\frac{b+ad}{1+cd},b)$: $-da<b< \frac{a}{c}\}$ .

$\Gamma_{v}$43,according to the spatial heterogeneity of$c(x),d(x))$ needs to be studied. Among other things, we prove that a spatial heterogeneity can cause
to form-shaped branch with respect to when the birth rate issmall and the cross-diffusion is large. As a rough sketch of the main result $(?beorem$$4.1.)$ , we can give the following theorem. Throughout this article, we denote the averageof $f(x)$ over by
dr $(:=|\Omega|^{-1}.f_{\Omega}f(x)dx)$ .

Now, the change in the shape of$(\rho(x)$$\Gamma_{p}$$b$$a\subset$$\Gamma_{\rho}$

$a$

$k$

$\Omega$$f_{\Omega}f(x)$Theorem 1.1. If $a>0$ is sufficiently small and is sufficiently large, the positive
solution set of(SP) (with the $b\iota furcation$ parameter ) foms the bounded smooth curve
$k$

$b$

$\Gamma_{p}=\{(u(x;s),v(x;s),b(s)) :0<s<C\}$.

Here $(u(x;s),v(x;s),b(s))$ satisfies$(u(x;0),v(x;0),b(O))=(a,0,b_{*})$

for a negative number$b_{*}and$

and $(u(x;C),v(x;C),$ $b(C))=(O,b\cdot,b^{*})$

a positive number$b^{*}$. Furthemore, in the case when$f_{\Omega}\rho(x)dxf_{\Omega}d(x)dx>f_{\Omega}\rho(x)d(x)dx$

,

(1.2)

the following (i) and (ii) hold true for a small number $a^{*}=a^{*}(k,\rho(x),c(x),d(x))$ .

.

(i) $IfO<a\leq a^{*}/3$ then $b’(O)>0$, that is,$\Gamma_{p}$supereritically bifircatesfivm $(a,0,b.)$ .

subcritically $bifi\kappa atesfmm(a,0,b_{*})$ .
In this case,, (SP) possesses at least two positive solutions, at one pos itive solution} $\cup[b.,b^{*}$ ), no positive solution if
if(ii) $If2a^{*}/3\leq a\leq a^{*}$ , then $b’(O)<0$, that is,$\Gamma_{p}$$for \underline{b}:=\min_{0\leq s\leq C}b(s)$$b\in(\underline{b},b_{*})$$b\in(-\infty,\underline{b})\cup[b^{*}, \infty)$$lfb\in 1\underline{b}$.

Obviously, both sides of (1.2) are equal if either or is constant. If both $d(x)$andare spatially heterogeneous, (1.2) may hold. For any fixed positive number$\epsilon<f^{\rho(x)dx}$, if supp $( 1^{y-\epsilon})_{+}((p-\epsilon)_{+};=\max\{p-\epsilon,0\})$ and supp
are disjoint, (1.2)
holds true. This is because$d$$\rho$$\rho(x)$$d$$f_{\Omega}\rho(x)dxf_{\Omega}d(x)dx>\epsilon f_{\Omega}d(x)dx\geq f_{\Omega}\rho(x)d(x)dx$.

subcritially
For such a case, Theorem 1.1 asserts that if belongs to a certain range,
bifurcates from $(a,0,b,)$ . Then, $(SP)$ has at least two positive solutions even if isslightly lesser than $b.(<0)$ . It is observed that supp
provides a favorable domain$a$$\Gamma_{p}$$b$$(p-\epsilon)_{+}$

44

for the prey in which the cross-diffusion (escape) effect from the predator is comparatively strong, whereas supp gives a favorable domain for the predator in which theincrease of individuals due to preying is expected. From the ecological viewpoint, ourand supp can produce theresult implies that the spatial segregation of supp )coexistence steady-states even if the death rate of the predator is comparatively high.
$d$

$q$

$d$

$-\epsilon)_{+}$

Figure 1 Positive solution branch in spatially homogeneous case.

Figure2 Positive solution branch in case (1i) of Theorem 1.1.

In this article, the usual norms of the spaces
respectively, defined as$||u||_{p}$$:=( \int_{\Omega}|u(x)|^{p}dx)^{1/p}$

and$L^{p}(\Omega)$

for $p\in[1,\infty$ ) and$||u||_{\infty}$$:=m_{\frac{ax}{\Omega}}x\in|u(x)|$$C(\overline{\Omega})$

are,

.

As the functional $\theta amework$ for our analysis, we introduce the following Banach

spaces:$X;=W_{\nu}^{2,p}(\Omega)xW_{\nu}^{2,p}(\Omega),$$Y:=L^{p}(\Omega)xL^{p}(\Omega)(p>N)$ ,

(1.3)

45

where
. Here we note thatfor $p>N$ by the Sobolev embedding theorem.
$W_{v’}^{2p}(\Omega):=\{u\in W^{2,p}(\Omega) :\partial_{v}u|_{\partial\Omega}=0\}$$X\subset C^{1}(\overline{\Omega})xC^{1}(\overline{\Omega})$

2 Finite Dimensional Limiting System
2.1 Bounded Bifurcation Continuum
As a preliminary result, we obtain the following bounded bifurcation continuum of
positive solutions of (SP) with the bifurcation parameter :$b$Theorem 2.1. For any fixed $(a,k,\rho(x),c(x),d(x))$, there exist $b_{*}=b.(a,d)<0$ and$b”=b^{r}(a,k\rho, c)>0$ such that the positive solution set of (SP) (with the $bifi\kappa anon$, which bifurcates from $(u,v,b)=$parameter ) foms a boun&d continuum$b$$(a,0,b.)\in\Gamma_{u}$$\Gamma_{p}(\subset XxR)$

andjoins $(u,v,b)=(0,b^{*},b^{*})\in\Gamma_{v}$ .

Owing to Theorem 2.1, we find at least one positive solution when $b_{*}<b<b^{*}$ .
Theorem 2.1 can be proved by the combination of the local and global bifurcation
theorems ([2], [19]), the a priori estimates and the nonexistence region of the positive
solutions. We refer to [10] for the proof.

2.2 Lyapunov-Schmidt Reduction
We now study the spatial heterogeneous effect $of\rho(x),c(x)$ and $d(x)$ on the positive
solution branch
introduced in Theorem 2.1. More concretely, we derive a heterogeneous effect that enables to forn a c-shaped branch, when the cross-diffusion islarge and the birth (or death) rates andare small. For the derivation, we employ
the following scalings in (EP):$\Gamma_{p}$$k$$\Gamma_{\rho}$$a$$U=\epsilon w$, v=\epsilon 為$|b|$$a=\epsilon\alpha,$$b=\epsilon\beta,$$k= \frac{1}{\epsilon}$

.

(2.1)

Here $e>0$ is a small perturbation parameter. Furthermore, is a positive number andis a real number. Hereafter, will fUnction as the bifurcation parameter. From (2.1),
it is observed that the new unknown function $(w,z)$ satisfies th next perturbed elliptic$\alpha$$\beta$$\beta$$e$system:$\{\begin{array}{ll}\Delta w+\epsilon F(w,z)=0 in \Omega,\Delta z+\epsilon G(w,z,\beta)=0 in \Omega,\partial_{v}w=\partial_{v}z=0 on \partial\Omega,\end{array}$

(2.2)

46

where
(2.3)

$\{\begin{array}{l}F(w,z):=\frac{w}{1+\rho(x)z}(\alpha-\frac{w}{1.+\rho(x)z}-c(x)_{Z})G(w, z,\beta)\ddagger=z(\beta+\frac{d(x)w}{1+\rho(x)z}-z)\end{array}$Hence, (2.2) has the semitrivial solutions$(w,z)=(\alpha,0)$

and$(w,z)=(0,\beta)$ ,in addition to the trivial solution $w=z=0$ .
We solve (EP) by the Lyapunov-Schmidt reduction. For the Banach spaces andin (1.3), we $in\alpha oduce$ the linear operator $H$ : $Xarrow Y$ and the nonlinear operator$B:X\cross Rarrow Y$ by
$X$

$Y$

$H(w,z):=(\Delta w,\Delta z)$,$B(w,z,\beta):=(F(w,z),G(w,z,\beta))$

.

(2.4)

Hence, (2.2) is equivalent to the equation$H(w,z)+\epsilon B(w,z,\beta)=0$

By considering KerH $=R^{2},$

$X$

and

$Y$

.

(2.5)

can be decomposed as$X=R^{2}\oplus X_{1}$,$Y=R^{2}\oplus Y_{1}$,orthogonal space of in (resp. Y). Let : $Xarrow$: $Yarrow Y_{1}$ be the orthogonal projections. Hence, the unknown function$(w,z)\in X$ of (2.5) can also be decomposed as

where
and

$X_{1}$$X_{1}$. (resp.

$Y_{1}$) denotes the$L^{2}$$\bm{R}^{2}$

$P$

$X$

$Q$

$(w,z)=(r,s)+u$ ,$u=P(w,z)$ .

Since $H(r, s)=0$ and $(I-Q)H(X_{1}.)=0,$ $(2.5)$ is consequently reduced to(2.6)$QH(u)+\epsilon QB((r,s)+u,\beta)=0$

and$(I-Q)B((r, s)+u,\beta)=0$.

In order to solve (2.5), we first construct the solution set of (2.6) near$\epsilon=0$.

47

Lemma 2.2. For any $C>0$, there exist a small

and a neighborhood $N$ of$\{(w,z,\beta, \epsilon)=(r, s,\beta, 0)\in X\cross \mathbb{R}^{2} : |r|, |.;|, \beta|\leq C\}$ suchthat all solutions of (2.6)
contained in $N$ can be parameterized by$K:=[((r, s)+\epsilon U(r,s,\beta, \epsilon),\beta,\epsilon)$

Here, $U(r,s,\beta,e)$ is an$X_{1}$:$\epsilon_{0}>0$$|r|,$ $|s|,$ $\beta|\leq C+\epsilon_{0},$ $|\epsilon|\leq\epsilon_{0}$-valued smoothfunction defined $in|r|,$}.

$|.t|,\beta|\leq C+\epsilon_{0},$$|\epsilon|\leq\epsilon_{0}$.

Hence, an element $(w,z,\beta,\epsilon)=((r_{\iota};)+\epsilon U(r,s,\beta, \epsilon),\beta,\epsilon)\in K$ is a solution of (2.5) if
and only if$\Phi^{b^{\backslash }}(r, s,\beta):=(I-Q)B((r, s)+\epsilon U(r, s,\beta, e),\beta)=0$.

The proof of Lemma 2.2 can be carried out by using the implicit function theorem
and a compacmess argument. We refer to [11, Lemma 3.1] for the proof.

2.3 Exact Expression of the Limiting Solution Set
Lemma 2.2 asserts that for each, the solution set of (2.5) (equivalently(2.2)) in $N$ coincides with Kerdx), we obtain. Sinc$e(I-Q)(w,z)=$ ($\epsilon\in[-e_{0},e_{0}]$$f_{\Omega}wdx,$$\Phi^{\prime i}$$f_{\Omega}z$$\Phi^{0}(r, s,\beta)=(f_{\Omega}F(r,s)dx,\cdot f_{\Omega}G(r, s,\beta)dx)$(2.7)$=[_{s(\beta\frac{(\alpha-d(x)}{1.+s\rho}}rf_{\Omega} \frac{1}{-s+rf_{\Omega}1+s\rho(x)}\frac{r}{1+s\rho(x),(x)^{dx)}}-sc(x))dx)$

in the extreme case $e=0$ . Therefore, Ker
sets in ;$\Phi^{0}$comprises the union of the following four$R^{3}$

ム $=\{(0,0,\beta):\beta\in \mathbb{R}\}$ ,$\mathcal{L}_{z}=\{(0,\beta,\beta);\beta\in R\}$,

$L_{v}=\{(\alpha,0,\beta):\beta\in \mathbb{R}\}$,$\mathcal{L}_{p}=\{(f(s),s,g(s)):s\in \mathbb{R}\}$

,

where
$f(s \cdot);=f_{\Omega}\frac{\alpha-s\cdot c(x)}{1+s\rho(x)}dx/f_{\Omega}\frac{dx}{(1+sp(x))^{2}}$.

$g(s):=s-f(s)f_{\Omega} \frac{d(x)}{1+sp(x)}dx$(2.8)

It should be remarked that
contains the limiting set of the positive solution set of(2.2) as $earrow 0$ . Then it is importunt to study the profiles of $f(s)$ and $g(s)$ .
$l_{p}$

48

Lemma 2.3. Let $f(s)$ and $g(s)$ be the functions defined by (2.8). The following profiles

of $f(s)$ and $g(s)$ hold true:(i) There exists $s_{0}=s_{0}(\alpha,c(x),\rho(x))>0$ such that$f(.;)>0$ for $s\in[0,s_{0}$ ),$\{$$f(s)<0$ for$s\in(.\iota_{t)}’, \infty)$

(2.9)

.

Additionally, $f(O)=\alpha$ also holds true.
(ii) It follows that $g(O)=-\alpha f_{\Omega}d(x)dx<0$ and$g’(O)=1+f_{\Omega}c(x)dxf_{\Omega}d(x)dx-\alpha(f_{\Omega}\rho(x)dxf_{\Omega}d(x)dx-f_{\Omega}\rho(x)d(x)dx)$.

(2.10)For the zero point$s_{0}$

of $f(s),$ $g(s)$ satisfies$\max_{0\leq s\leq s_{0}}g(s)=g(s_{0})=s_{0}>0$

and

Proof. In view of (2.8), it is easy to verify $f(O)=\alpha$ and$g’(s_{0})>0$.

$\lim_{1arrow\infty}f(s)=-\infty$. Then,$f$possesses at least one zero point. After direct calculations, we know$f’(.;)= \{2f_{\Omega}\frac{\rho(x)}{(1+s\rho(x))^{3}}dxf_{\Omega}\frac{\alpha-s\cdot c(x)}{1+s\cdot\rho(x)}dx$(2.11)
ム $\}/(f_{\Omega}\frac{dx}{(1+s\rho(x))^{2}})^{2}$$-f_{\Omega} \frac{dx}{(1+s\rho(x))^{2}}f_{\Omega}\frac{c(x)+\alpha\rho(x)}{(1+s\rho(x))^{2}}$For any zero point$s_{0}$.

of $f,$ $(2.8)$ yields.

(2.12)$f_{\Omega} \frac{\alpha-s_{0}c(x)}{1+s_{0}p(x)}dx=0$

Then, letting $s=s_{0}$ in (2.11) implies.

$f’(s_{0})=-f_{\Omega} \frac{c(x)+\alpha p(x)}{(1+s_{0}\rho(x))^{2}}dx/f_{\Omega}\frac{dx}{(1+s_{0}p(x))^{2}}<0$

Consequently, we know that is a unique zero point of $f(s)$ , which gives (2.9). It
immediately follows Rom (2.8) that $g(O)=-\alpha;_{\Omega}d(x)dx<0$ and $g(s_{0})=\cdot l_{0}’>0$ . Since$s_{0}$$g’(s)=1-f’(s)f_{\Omega} \frac{d(x)}{1+s\rho(x)}dx+f(s)f_{\Omega}\frac{\rho(x)d(x)}{(1+sp(x))^{2}}dx$,(2.13)

letting $s=s_{0}$ in (2.13) yields $g’(s_{()})>0$ . Furthermore, (2.10) can be obtained bysubstituting (2.8) and (2.11) into (2.13) and letting $s=s_{0}$ in the resulting expression.
ロ
Thus the proof of Lemma 2.3 is complete.

49

It follows from Lemma 2.3 that$(f(0),0,g(0))=(\alpha,0,$ $-\alpha f_{\Omega}d(x)dx)\in \mathcal{L}_{w},$$(f(s_{0}),s_{0},g(s_{0}))=(0, s_{0},s_{0})\in \mathcal{L}_{z}$Hence, the bounded curve $\{(f(s),s,g(s)) : 0<s<s_{0}\}\subset \mathcal{L}_{p}$ implies the limiting

solution set of the positive solution set of (2.2) as$\epsilonarrow 0$.

3 Construction of the Perturbed Solution Branch
3.1 Perturbed Solution Branch of (2.2), we construct the positive solution set of (2.2) by
In this section, for small
perturbing the limiting solution set $\{(f(s), s,g(s)):0<s<s_{0}\}$ obtained in Lemma2.3.
More concretely, we aim to prove the following theorem.
$\epsilon\succ 0$

Theorem 3.1. For anyfixed $(\alpha,p(x),c(x),d(x))$, there exist a smallof bounded$\epsilon_{0}>0$

and a family$cu’\gamma es$$\{S(\xi,\epsilon)=(r(\xi,\epsilon), s(\xi,\epsilon),\beta(\xi,\epsilon))\in \mathbb{R}^{3} :such that for each$\epsilon\in(0,\epsilon_{0}J$(\xi,\epsilon)\in[0,C_{\epsilon}]\cross[0,\epsilon_{0}]\}$, all positive solutions of (2.2) are parameterized as$(w(\xi,e),z(\xi,\epsilon),\beta(\xi,e))=((r,s)+\epsilon U(r,s,\beta,e),\beta)$:$\}$$\Gamma_{j}=\{$$(r, s\cdot,\beta)=(r(\xi,\epsilon),s(\xi, \epsilon),\beta(\xi,\epsilon)),$$\xi\in(0,C_{i})$where $U(r, s,\beta, e)$ is the -valued smooth $fi\ell nction$ defined in Lemma 2.2 andis a cenain smoothfznction which satisfies$X_{1}$$S(0,e)=(\alpha,0,\beta_{*}(\epsilon)),$$S(\xi,0)=(f(\xi),\xi,g(\xi)),$for thefimctions and defined by (2.8). Here,
$f$

$g$

$\beta.(e)=\frac{b_{l}(\epsilon\alpha)}{\epsilon}<0$

for the$fi\ell nctionsb_{*}$

smooth fiznction in

and

$b$

’

a

(3.1)

,

$S(\xi,\epsilon)$$S(C_{P_{\wedge}},\epsilon)=(0,\beta^{*}(\epsilon),\beta^{*}(\epsilon))$$\beta.(\epsilon)and\beta^{*}(e)$are defined by(3.2)$nd \beta^{*}(\epsilon)=\frac{b^{*}(\epsilon\alpha)}{\epsilon}>0$

is a certain
obtained in Theorem 2.1. Additionally,, which is obtained in (2.9).
such that$\epsilon\in[0,e_{0}]$$C_{f^{\backslash }}$,$C_{()}=;_{0}$The sketch of the proof of Theorem 3.1 will be given in the next subsection.
$(2.10)$ revealsin (3.1). Since $\beta(\xi,0)=g(\xi)$ when
We now observethat$\beta(\xi,\epsilon)$$\epsilon=0,$$I\omega,d);=f_{\Omega}^{p(x)dxf_{\Omega}d(x)dx-f_{\Omega}^{\rho(x)d(x)dx}}$

50

when $>0$
is an important term for the direction of the bifurcation of atis sufficiently small and $I()d)>0$ , theis sufficiently small. In the c&\’e when
direction changes according to the value of .
$(\alpha,0,\beta_{*}(\epsilon))$$\Gamma_{\epsilon}$$P,$$\epsilon>0$$\alpha$Theorem 3.2. Let $I(p,d)>0$ and$\alpha$.

$:=(1+f_{\Omega}c(x)dxf_{\Omega}d(x)dx)I(\rho,d)^{-1}\succ 0$.

such that
For any small positive number , there exists a small$(0,\alpha_{*}-\eta]x[0,e_{0}]$ , then, that is, theof atsupercritical. On the other hand, if $(\alpha,\epsilon)\in[\alpha_{*}+\eta,\eta^{-1}]x[0,a]$, thenthat is, the $bifi\kappa ation$ ofis subcritical. Funhemore,$\epsilon_{0}>0$$\eta$$bifi\kappa a\dot{n}on$$\beta_{\xi}(0,\epsilon)>0$$\Gamma_{e}$$\iota f(\alpha,e)\in$$(\alpha,0,\beta_{*}(\epsilon))$is,$\beta_{\xi}(0,\epsilon)<0$$\Gamma_{b^{\backslash }}at$$[\alpha_{*}+\eta,\eta^{-1}]x[0,\epsilon_{t)}],$ $then\beta(\xi,\epsilon)$$(\alpha,0,\beta_{*}(e))$$\iota f(\alpha,e)\in$

satisfies$\underline{\beta}(\epsilon):=..\min_{\xi\epsilon|0,C_{\wedge}\cdot]}\beta(\xi,e)<\beta_{*}(\epsilon)$,

and thefollowing properties hold true:
.

(i)

(ii)
(iii)

$lf\beta<\underline{\beta}(e)or\beta\geq\beta\cdot(\epsilon),$$t2.2$ )has no positive solution.

$If\beta=\underline{\beta}(\epsilon)or\beta_{*}(\epsilon)\leq\beta<\beta\cdot(e),$$If\underline{\beta}(\epsilon)<\beta<\beta_{*}(\epsilon),$$(2.2)$$(2.2)$

has at least one positive solution.

has at least two positive solutions.

Proof. (Assuming Theorem 3.1) Let

be the smooth
curve defined by (3.1). We observe that $S(\xi,0)=Cf(\xi),\xi,g(\xi))$. Additionally, it ispossible to prove that
$S(\xi,\epsilon)=(r(\xi,e),s(\xi,\epsilon),\beta(\xi,\epsilon))$$t^{\backslash }, \lim_{arrow 0}(r(\xi,\epsilon),\beta(\xi, e))=(f(\xi),g(\xi))$

where$s_{0}’$

in $C^{1}([0,s_{0}])xC^{1}([0, s_{0}])$ ,(3.3)

is the number obtained in (2.9). When $I(\rho,d)>0$ , it follows from (2.10) that
(3.4)$\{\begin{array}{ll}g’(0)>0 if 0<\alpha<\alpha_{*},\ovalbox{\tt\small REJECT}(0)<0 if \alpha>\alpha..\end{array}$

Let be any fixed small positive number. Hence, (3.3) and (3.4) enable us to find a], that is,, thensmall
such that if
supercritically bifurcates from. On the other hand, if, then $\beta_{\xi}(0,e)<0$, that is,. In thesubcritically bifurcates fromand attains
latter case,satisfies$\eta$$\epsilon_{0}\succ 0$$(\alpha,\epsilon)\in(0,\alpha_{r}-\eta$$x[0,\epsilon_{0}]$$\Gamma_{r}$$g_{j}(\xi)(:=\beta(\xi,\epsilon))$$g_{\epsilon}’(0)<0$$\Gamma_{\epsilon}$$(\alpha,e)\in[\alpha_{*}+\eta,\eta^{-1}]x$$(\alpha,0,\beta_{*}(e))$$[0_{\Phi}]$$\beta_{\xi}(0,\epsilon)>0$.

$(\alpha,O,\beta_{*}(e))$$g(\xi)<g(C_{\epsilon})(0\leq\xi\leq C_{t^{\backslash }})$

51

its minimum value at a certain
and set. We denote the minimum value by$\xi=\underline{\xi}(e)\in(0,C_{b^{\backslash }})$$\underline{\beta}(\epsilon)(:=g_{\epsilon}(\underline{\xi}(\epsilon)))$$K_{F,}(\beta):=\{\xi\in(0,C_{p,}) ; g_{Pj}(\xi)=\beta\}$Then, in the case when $e>0$ is sufficiently small,$\beta\geq\beta^{*}(\epsilon)$, contains at least one element$if\beta=\underline{\beta}(\epsilon)$or.

$K_{b}(\beta)$

is empty if$\beta<\underline{\beta}(\epsilon)$or, and contains$\beta.(\epsilon)\leq\beta<\beta^{*}(\epsilon)$. In vi $ew$ of (3.1), we know that for any fixedat least two elements) is equal to that of positive solutions of (2.2). Thus
, the number of elements ofthe proof of Theorem 3.2 is complete.
$if\underline{\beta}(e)<\beta<\beta_{*}(e)$$K_{r}\phi$$\beta$$o$

3.2 Sketch of the Proof of Theorem 3.1
As the first step of the proof of Theorem 3.1, we construct the local branches of
positive solutions of (2.5) near the bifurcation points.
Lemma 3.3.
tive number
Ker

where

mere exist a neighborhood$\delta_{*}such$that for any$U_{*}(\subset \mathbb{R}^{3})$$e\in[0,\delta_{*}]$$(\alpha,0,-\alpha+_{\Omega}d(x)dx)$:$\xi\in[0,\delta_{*}]|\cup\{(\alpha,O,\beta)\in u.\}$$s_{0}$

where

(3.5)

$(r(0,\epsilon),$ $s(0,\epsilon),\beta(0,e))=(\alpha,0,\beta.(e))$

Lemma 3.4. Let be the positive number in (2.9). Thereand a positive number such that for any
of
Ker

,

is a certain smoothfimction, which satisfies$(r(\xi,0),$ $s(\xi,0),\beta(\xi,0))=(f(\xi),\xi,g(\xi)),$$R^{3})$

and a posi-,$\Phi\cap u_{*}\cap\overline{R}_{+}^{3}=\{(r(\xi,\epsilon),s(\xi,\epsilon),\beta(\xi,\epsilon))$$(r(\xi,\epsilon),$ $s(\xi,\epsilon),\beta(\xi, e))$of$(0, s_{0}, s_{0})$$\delta$“$\Phi^{\epsilon}\cap u^{*}\cap\overline{R}_{+}^{3}=\{(\hat{r}(\xi,e),\hat{s}(\xi,e),\hat{\beta}(\xi,e))\in \mathbb{R}^{3} :$\hat{S}(\xi,\epsilon):=(\hat{r}(\xi,e),\hat{s}(\xi,e),\hat{\beta}(\xi,\epsilon))$$\alpha ist$a neighborhood$\epsilon\in[0,\delta^{*}]$.

$u\cdot(\subset$,,\xi\in[0,\delta\cdot]\}\cup\{(0,\beta,\beta)\in u\cdot\}$

is a smooth $fi\ell ncnon$ with$\hat{S}(\xi,0)=(f(s_{0}-\xi), s_{0}-\xi,g(s_{0}-\xi)),\hat{S}(0,\epsilon)=(0,\beta\cdot(e),\beta^{*}(\epsilon))$.

With the aid of the local bifurcation theorem [2], we can prove Lemmas 3.3 and 3.4(see [10]). The next lemma is the most crucial part of the proof of Theorem 3.1.

Lemma 3.5. There exists a neighborhoodof $\{(f(s),s,g(s)):0\leq s\leq\_{0}\}$such that
iently small, then all positive solutions of (3.1) contained inis$u$ can be parameterized by (3.1).
$u(\subset Xx\mathbb{R})$$\iota f\epsilon>0$$suff\dot{\kappa}$

52

Proof We use the perturbation theory of Du-Lou [4, Appendix] in the proof. For the

introduced in Lemmas 3.3 and 3.4, we put$\delta^{*}/2]):=\{(\vee f(s),s,g(s)) : s\in[\delta_{*}/2,s_{0}-\delta\cdot/2J\}$ . Here,
represents the zero point of $f$obtained in Lemma 2.3. From (2.7) and (2.8), some direct calculations yieldpositive numbers

and$\delta_{*}$$\delta^{*}$$\mathcal{L}_{p}([\delta_{*}/2,$$s_{0}-$$s_{t)}$

det$\Phi_{(r,s)}^{0}(f(s), s,g(s))=s^{\backslash }f(s)g’(s)f_{\Omega}\frac{dx}{(1+s\rho(x))^{2}}$

.

(3.6)

.
From (2.9), we know that $f\cap s>0$ for any
Therefore, (3.6) reveals thatis invertible if and only if $g’\cap s\neq 0$ . In
such a case, the implicit function theorem ensures a certain positive numberof ($fOs,s\gamma$ such that for each $e\in[0,\delta]$ ,and a neighborhood$(f\subset s),\overline{s},g(s\gamma)\in l_{p}([\delta_{*}/2, s_{0}-\delta^{*}/2])$$\Phi_{(,.r,s)}^{0}(f\cap s,\overline{s},g\cap s)$$\delta=\delta\cap s$$\prime W_{\overline{s}}$

Ker
Here,

$\Phi^{P,}\cap u_{\overline{s}}=\{(r\phi,\epsilon),s(\beta,\epsilon),\beta):\beta\in(g\cap s-\delta,g\cap s+\delta)\}$$u_{\overline{s^{\backslash }}};=\prime W_{\overline{s}}x(g\cap s-\delta,g(\overline{s})+\delta)$$(r(g\cap s,0),s(g(s\gamma,o))=(f(3S$

and$(r\phi,\epsilon),s\varphi,e))$(3.7)

is a smooth function with.

On the other hand, if $g’\cap s=0,$ $(3.6)$ implies
dim Ker.

$\Phi_{(r,s\cdot)}^{0}(f\cap s,\overline{s},g(s\gamma)=co\dim$$r\bm{t}k\Phi_{tr.s’)}^{0}(f\cap s,\overline{s},g\cap s)=1$

Range$\Phi_{(r,s)}^{0}(f\cap s,\overline{s},g\cap s)=1$; therefore

.

(3.8)

After some computations, we can verify(3.9)$\Phi_{\beta}^{0}(f\cap s,\overline{s},g\cap s)=(\begin{array}{l}\overline{s}0\end{array})S$

By virtue of (3.8) and (3.9), we can use the spontaneous bifurcation theory of Crandalland
Rabinowitz [3, Theorem 3.2 and Remark 3.3] to obtain a positive numbera neighborhoodofsuch that, for any $e\in[0,\delta]$ ,$\delta=\delta\cap s$$u_{\overline{s}}$

Ker
Here,

$(fCs),\overline{s},g\cap s)$.

$\Phi^{\epsilon}\cap u_{\overline{s}}=\{(r(\xi,\epsilon),s(\xi,\epsilon),\beta(\xi,\epsilon)):\xi\in(-\delta,\delta)\}$$(r(\xi,\epsilon),$ $s(\xi,\epsilon),\beta(\xi,\epsilon))$is a smooth function in $(\xi,e)\in[-\delta,\delta]x[0,\delta]$ with$(r(O,0),s(O,0),\beta(0,O))=(g\cap s,\overline{s},f\cap s)$

Since (3.7) or (3.10) holds true for each

2

.

,

$u_{\overline{s}}.$$p([\delta./2, s_{0}-\delta^{*}/2])c\cup fu_{\overline{s}}$:$\overline{s\cdot}\in[\delta_{*}/2, s_{0}-\delta\cdot/2]$}.

(3.10)

53

Furthermore, the compactness ofpoints$\mathcal{L}_{p}([\delta_{*}/2, s_{0}-\delta^{*}/2])$

enables us to find finitely manysuch that$\{s_{j}\}_{j=1}^{n}$$\{\begin{array}{ll}(f(s_{j}),s_{j},g(s_{j}))\in \mathcal{L}_{\rho}([\delta_{*}/2, s_{0}- /2]) for1\leq j\leq n,\mathcal{L}_{p}([\delta_{*}/2, s_{0}-\delta^{*}/2])\subset\bigcup_{j=1}^{n}u_{\dot{\text{ノ}}}, (where u_{j}:=u_{s_{j}}).\end{array}$Wth regard to Lemmas 3.3 and 3.4, we put $u_{0};=u_{*\bm{t}}du_{n+1};=u*$ . Hence, we
can a.ssume $u_{j}\cap u_{j+1}\neq\emptyset(j=0,1,2, \ldots,n)$ without lo $ss$ of generality. By virtue of, there exists a(3.7) and (3.10), we put. Then, for any
such thatsmooth function$\delta_{j}:=\delta(s_{1})$$\epsilon\in[0,\delta_{j}](1\leq j\leq n)$$(r_{j}(\xi,\epsilon),$ $s_{j}(\xi,e),\beta_{j}(\xi,e))$

Ker

Here

$\Phi^{r,}\cap u_{j}=\{(r_{j}(\xi,\epsilon),s_{j}(\xi,\epsilon),\beta_{j}(\xi,\epsilon)):\xi\in(-\delta_{j},\delta_{j})\}=:J_{j}^{b}$$(r_{j}(\xi,\epsilon),$ $s_{j}(\xi,\epsilon),\beta_{j}(\xi,\epsilon))$.

(3.11)

satisfies$(r_{j}(0,0),$ $s_{j}(0,0),\beta_{j}(0,0))=(f(s_{j}), s_{j},g(s_{j}))$for each $1\leq j\leq n$ . Furthermore, by considering Lemmas 3.3 and 3.4, we put$J_{\acute{0}}^{j}:=\{(r(\xi,\epsilon),s(\xi,\epsilon),\beta(\xi,\epsilon)) :\xi\in(0,\delta_{*}]\}$$J_{n+1}^{\prime j}:=\{(\hat{r}(\xi,\epsilon^{\backslash }),\hat{s}(\xi,\epsilon),\hat{\beta}(\xi,\epsilon)) :and,\xi\in(0,\delta^{*}]\}$,. Consequently, it follows from (3.11) and Lemmas 3.3 and 3.4 that
(3.12).
Ker
for any$u:= \bigcup_{j-\triangleleft}^{n+1}u_{j}$$\Phi^{1i}\cap u\cap \mathbb{R}_{+}^{3}=\bigcup_{j=0}^{n+1}J_{j}^{b}$$\epsilon\in[.0,\min_{()\leq j\leq n+1}\delta_{j}]$. Hence, (3.12) implies that Ker
andforms a one-dimensional sub-manifold. Indeed, with the aid of the perturbation theoryof Du-Lou [4, Proposition A3], we can construct a bounded smooth curve $S(\xi,e)=$which satisfies
Here, we put$\delta_{0}$$:=\delta_{*}$$\delta_{n+1}$$:=\delta’$$\Phi^{b}\cap u\cap R_{+}^{3}$$(r(\xi,\epsilon),$ $s(\xi,\epsilon),\beta(\xi, e))$$\{\begin{array}{l}\bigcup_{j=0}^{n+1}J_{j}^{P_{d}}=S((0,C_{\epsilon}),\epsilon)(r(\xi,0),s(\xi,0),\beta(\xi,0))=(f(\xi),\xi,g(\xi))(r(0,\epsilon),s(0,\epsilon),\beta(0,\epsilon))=(\alpha,0,\beta_{*}(\epsilon))(r(C_{i},\epsilon)e\beta(C_{6^{\backslash }},\epsilon))=(0,\beta\cdot(\epsilon),\beta\cdot(\epsilon))\end{array}$Thus, we have proved Lemma 3.5.

$o$

In order to prove Theorem 3.1, we have to show that (2.2) does not admit any
positive solution outside of $u$ obtained in Lemma 3.5. The next lemma can be shownby a contradiction argument. We refer [10] for the proof.

54

be any fixed neighborhood of $\{(f(s), s,g(s)) : 0\leq s\leq s_{0}\}$ .
If $e>0$ is sufficiently small, then for any positive solution $(w,z)$ of (2.2), there exists$(r,s,\beta)\in\gamma$ such that

Lemma 3.6. Let$V(\subset \mathbb{R}^{3})$$(w,z)=(r,s)+\epsilon U(r, .:\cdot,\beta, \epsilon)$

Here, $U(r,s,\beta,\epsilon)$ is the$X_{1}$.

valuedfunction defined in Lemma 2.2.

Consequently, together with Lemmas 3.5 and 3.6, we obtain Theorem 3.1.

4 Stationary Solution Set of the SKT Model
In view of (2.1), it is convenient to state our result on the positive solution set ofthe following system, which is obtained from (SP) with $(a,k)=(e\alpha,e^{-1})$ :$(SP)_{\epsilon}\{\begin{array}{ll}\Delta[(1+e^{-1}\rho(x)v)u]+u(\epsilon\alpha-u-c(x)v)=0 in \Omega,\Delta v+v(b+d(x)u-v)=0 in \Omega,(_{\gamma}u=\partial_{v}v=0 on \partial\Omega.\end{array}$

The following theorem is the main result obtained in this paper.
ientlyTheorem 4.1. Let $(\alpha,\rho(x),c(x),d(x))$ be fixed arbitrarily. Then, $\iota fe>0$ issmall, the positive solution set of (SP),, (with the $b_{l}furcation$ parameter b) form abounded smooth curve$suff\dot{\kappa}$$\Gamma_{p}=\{(u,v,b)=(u(\xi,\epsilon),v(\xi,e).b(\xi,e))\in Xx\mathbb{R}:\xi\in(0,C_{\epsilon})\}$

where$(u(\xi,\epsilon),v(\xi,\epsilon),b(\xi,\epsilon))$,satisfies$(u(O,\epsilon),v(O,\epsilon),b(O,\epsilon))=(\epsilon\alpha,0,b_{*}(\epsilon\alpha))\in\Gamma_{u}$,$(u(Cj’\epsilon),v(C_{i},\epsilon),b(C_{f,},\epsilon))=(0,b^{*}(\epsilon\alpha),b^{*}(\epsilon\alpha))\in\Gamma_{v}$,are thefrnctions obtained
andand $b(\xi,e)<b\cdot(e\alpha)(0\leq\xi<C_{\epsilon}\cdot)$ . Here,in Theorem 2.1 and is the positive finction intrcoduced in Theorem 3.1.
In the case when$b_{*}(\epsilon\alpha)$$b^{*}(\epsilon\alpha)$$C_{\epsilon}$$I(p,d);=f_{\Omega}\rho(x)dxf_{\Omega}d(x)dx-f_{\Omega}\rho(x)d(x)dx>0$ ,for the positive number$\alpha$.

$:=(1+f_{\Omega}c(x)dxf_{\Omega}d(x)dx)I(p,d)^{-1}$55

and any small positive number , there exists a small positive number $a$)such that if $0<\alpha\leq\alpha_{*}-\eta$ and $0<\epsilon\leq e_{0}$ , then $b_{\xi}(0,\epsilon)>0$ and the $bihrc\cdot ation$) is supercritical. On the other hand,of from (sa,is, then $b_{\xi}(0, \epsilon)<0$ and the bifercation of
andfromsubcritical. In such a case, for the minimum value of ;$=\epsilon_{0}(a_{*}, \eta)$$\eta$$lf\alpha_{*}+\eta\leq\alpha\leq\eta^{-1}$$0,$ $b_{*}(\epsilon a)$$\Gamma_{p}$$0<\epsilon\leq\epsilon_{0}$$\Gamma_{p}$$(e\alpha,0,b_{*}(e\alpha))$$b$,$\underline{b}:=.\min_{\epsilon\epsilon 1)l}.b(\xi,e)<b_{*}(\epsilon\alpha)$

the following properties hold $tme$ :(i) If$b<\underline{b}$or$b\geq b^{*}(\epsilon\alpha)$, then $(SP)_{i}$ possesses no positive solution.

(ii) $lfb=\underline{b}$ or $b.(ea)\leq b<b^{*}(e\alpha)$ , then $(SP)_{b}$ possesses at least one positive

solution.
(iii) $If\underline{b}<b<b_{*}(e\alpha)$, then$(SP)_{\epsilon}$possesses at least lwo positive solutions.

Pvoof It follows Rom (2.1) that $(w,z)$ is a posi.tive solution of (2.2) if and only if(4.1)$(u,v,b)=e( \frac{w}{1+\rho(x)z},z,\beta)$

is a positive solution of$(SP)_{\epsilon}$. Therefore, Theorem 3.1 implies that for any fixed

there exists a small positive number
all positive solutions of
can be parameterized by$(\alpha,\rho(x),c(x),d(x))$ ,$r_{0}$

such that if$\epsilon\in(0,\epsilon_{0}$],$(SP)_{\epsilon}$$r_{p}=\{(u(\xi,\epsilon),v(\xi,e),b(\xi,\epsilon))\in X\cross \mathbb{R}:(\xi,\epsilon)\in[0,C_{1j}J\cross[o_{a)}]\}$

Hence by (4.1),$(u(\xi,e),v(\xi,\epsilon),b(\xi,\epsilon))$.

satisfies$(u( \xi,\epsilon),v(\xi,\epsilon),b(\xi,\epsilon))=\epsilon(\frac{w(\xi,\epsilon)}{1+p(x)z(\xi,e)},z(\xi, e),\beta(\xi,\epsilon))$for the function$(w(\xi,\epsilon),z(\xi,\epsilon),\beta(\xi,\epsilon))$obtained in (3.1). In view of (3.2), we knowthat$b(O,\epsilon)=\epsilon\beta.(e)=b_{*}(\epsilon\alpha)<0,$.

$b(C_{\epsilon},\epsilon)=\epsilon\beta^{*}(e)=b^{*}(\epsilon\alpha)>0$

In the case when $I(p,d)>0$ , ffom Theorem 3.2 and the one-to-one relationships of. Thus the(4.1), we obviously obtain the desired c-shaped bifurcation curve ofロ
proof of Theorem 4.1 is compl $ete$ .
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