数理解析研究所講究録
第 1838 巻 2013 年 116-125

116

Multiplicity of solutions in the Lotka-Volterracompetition with cross-diffusion
四ツ谷晶二 (Shoji Yotsutani)
Fuculty of Science and Technology,

龍谷大学・理工学部

Ryukoku University

1

Introduction

This is ajoint project with Yuan Lou (Ohio State University), Wei-Ming Ni (University of Minnesota and East China Normal University) concerning mathematical analysis,
and Masaharu Nagayama (Hokkaido University), Tatsuki Mori (Ryukoku University)
concerning numerical computation.
In an attempt to model segregation phenomena in population dynamics, Shigesada,
Kawasaki and Teramoto [7] in 1979 incorporated the inter-competition system. In particular, the following system was proposed$[Matrix]$(1.1). Here andwhere is a bounded domain $R^{N}(N\geq 1)$ with smooth boundary
and $d_{j}(j=1,2)$
represent the densities of two competing species. The constants
denote the intrinsic growth rates of these two species,
are all positive, where
account for inter-specificaccount for intra-specific competitions whileandrepresent intraare their diffusion rates. The constants
competitions, andare thespecific population pressures, also known as self-diffusion rates, andcoefficients of inter-specific population pressures, also known as cross-diffusion rates.
$:=c_{1}/c_{2}$ .
$:=b_{1}/b_{2},$If $B<C$ , we call it
For convenience, we set $A:=a_{1}/a_{2},$$\partial\Omega$$u$$\Omega$$a_{j},$$a_{1},$$b_{1}$$b_{j},$$v$$c_{j}$$a_{2}$$b_{2},$$c_{1}$$c_{2}$$d_{1},$$\rho_{11},$$d_{2}$$\rho_{22}$$\rho_{12},$$\rho_{21}$

$C$

$B$

the strong competition case and $B>C$ the weak competition case.
If $\rho_{n}=\rho_{12}=\rho_{21}=\rho_{22}=0$ , no nonconstant steady state can exist for any diffusion. On the other hand, it seems not entirely reasonable to add just diffusions
rates
to models in population dynamics, since individuals do not move around completely
randomly. In particular, while modeling segregation phenomena for two competing
species one must take into account the cross-diffusion pressures.
Mimura and his collaborators started mathematical analysis around 1980 (see, e.g.
Mimura [4] . Considerable work has been done concerning the global existence of solupriori estimates are crucial to obtaintions to systems (1.1) under various hypotheses.
$d_{1},$$d_{2}$

$)$

$A$

117
the global existence. As for recent progress including stationary problems, see Ni [5], Ni[6], Yagi[9] and Yamada [10].

We are interested in the case $\rho_{11}=\rho_{21}=\rho_{22}=0$ to clarify the effect of the crossdiffusion. By putting, we have$r:=\rho_{12}/d_{1}$(1.2)$\{\begin{array}{l}u_{t}=d_{1}\Delta[(1+rv)u]+u(a_{1}-b_{1}u-c_{1}v) in \Omega\cross(0, \infty) ,v_{t}=d_{2}\Delta v +v(a_{2}-b_{2}u-c_{2}v) in \Omega\cross(0, \infty) ,\underline{\partial u}_{=}\underline{\partial v}_{=0}\end{array}$$\partial n$on$\partial n$$u(x, 0)=u_{0}(x),$ $v(x, 0)=v_{0}(x)$in$\partial\Omega\cross(0, \infty)$,$\Omega.$

The stationary problem for the above equation is$\{\begin{array}{ll}d_{1}\triangle[(1+rv)u]+u(a_{1}-b_{1}u-c_{1}v)=0 in \Omega,d_{2}\Delta v +v(a_{2}-b_{2}u-c_{2}v)=0 in \Omega,\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0 on \partial\Omega,u(x)>0, v(x)>0 in\Omega.\end{array}$(1.3)

Even for these special cases, it is not easy to understand the structure of stationary
solutions, and the stability of them.

2

Limiting equations

Let us consider about a time-dependent limiting equation for (1.2) as $rarrow\infty.$
It seems from numerical computations for (1.2) and limiting equations for (1.3)
derived in Lou-Ni [1], [2] that we suspect that limiting equations with $rarrow\infty$ for (1.2)
is the following equation.
$v(x, t)$ , andUnknown functions are$\tau(t),$$\{\begin{array}{l}\frac{d}{dt}(\int_{\Omega}\frac{\tau}{v}dx)=\int_{\Omega}\frac{\tau}{v}(a_{1}-b_{1}\frac{\tau}{v}-c_{1}v)dx in (0, \infty) ,\frac{\partial v}{\partial t}=d_{2}\triangle v+v(a_{2}-c_{2}v)-b_{2}\tau in \Omega\cross(0, \infty) ,\frac{\partial v}{\partial n}=0 on \partial\Omega\cross(0, \infty) ,\tau(0)=\tau_{0}>0, v(x, O)=v_{0}(x)>0 in \Omega.\end{array}$This is formally derived by rewriting the first equation in (1.2) as$u_{t}=d_{1}r \Delta[(\frac{1}{r}+v)u]+u(a_{1}-b_{1}u-c_{1}v)$

,

(2.1)

118
after dividing by . In fact, this implies that $u(x, t)v(x,t)$ is equaland letting
depending only on . On the other hand, by integrating the above
to a constantequation in , we get$rarrow\infty$

$r$

$t$

$\tau(t)$$\Omega$$\frac{d}{dt}(\int_{\Omega}udx)=\int_{\Omega}u(a_{1}-b_{1}u-c_{1}v)dx.$

Thus, we obtain (3.1).
The stationary limiting equation for (3.1) becomes as follows:$\{\begin{array}{l}\int_{\Omega}\frac{\tau}{v}(a_{1}-b_{1}\frac{\tau}{v}-c_{1}v)dx=0,d_{2}\Delta v+v(a_{2}-c_{2}v)-b_{2}\tau=0 in \Omega,\frac{\partial v}{\partial n}=0 on\partial\Omega,\tau>0, v(x)>0 in\Omega.\end{array}$

3

(2.2)

1-dimensional caseNow, we consider the 1-dimensional case with $\Omega=(0,1)$ . The stationary limiting
equation (2.2) becomes as follows:$\{\begin{array}{l}\int_{0}^{1}\frac{1}{v}(a_{1}-b_{1^{\frac{\tau}{v}}})dx-c_{1}=0,d_{2}v_{xx}+v(a_{2}-b_{2}\frac{\tau}{v}-c_{2}v)=0, in (0,1),v_{x}(0)=v_{x}(1)=0,v>0, in (0,1) .\end{array}$Due to the scaling and reflection properties of solutions to autonomous ordinary
differential equations, all solutions to the (3.1) are obtained by several reflections and asuitable re-scaling from solutions of the following system:$\{\begin{array}{l}\int_{0}^{1}\frac{1}{v}(a_{1}-b_{1}\frac{\tau}{v})dx-c_{1}=0,d_{2}v_{xx}+v(a_{2}-b_{2}\frac{\tau}{v}-c_{2}v)=0 in (0,1) ,v_{x}(0)=v_{x}(1)=0,\tau>0, v>0, v_{x}>0, in(O, l).\end{array}$(3.1)

Now, we will discuss about the structure of stationary solutions and their stability.
This system (3.1) consists of a nonlinear elliptic equation and an integral constraint.
As far as existence and non-existence in one dimensional domain are concerned, LouNi-Yotsutani [3] obtained nearly complete knowledge. They also obtained the precise
qualitative behavior of solutions to this limiting system as the diffusion rate varies.

119
Their basic approach is to convert the problem of solving the system to a problem
of solving its “representation” in a different parameter space. This is first done without
the integral constraint, and then they use the integral constraint to find the “solution
curve” in the new parameter space. This turns out to be a powerful method as it gives
fairly precise information about the solutions.
We have recently made clear the remained delicate parts due to the explicit representation by elliptic functions.
We summarized the structure of solutions of (3.1). We concentrate on the case
$B<C$(strong competition case).

The following two theorem are due to [3].

Theorem 3.1 (Existence) Suppose that $B<C$ . If$\max\{0, \frac{B+C-2A}{C-B}\}\frac{a_{2}}{\pi^{2}}<d_{2}<\frac{a_{2}}{\pi^{2}},$then there exists a solution $(v(x), \tau)$ of (3.1).

Theorem 3.2 (Nonexistence) Suppose that $B<C.$(i) If, then there exists no solution of (3.1).
(iii) If $A<B$ , there exists no solution of (3.1).
$d_{2} \geq\frac{a_{2}}{\pi^{2}}$(iii) If $B \leq A<\frac{B+C}{2}$ , then there exists a $d_{2}^{*}=d_{2}^{*}(A, B, C, a_{2})>0$ such that
there exists no solution of (3.1) for$d_{2}\in(0, d_{2}^{*}].$We see that the above theorem is sharp by the following theorems. The existence
region depending on the the ratio $C/B$ . The situation drastically changes at $C/B=7/3.$
Theorem 3.3 Suppose that $B<C\leq 7B/3$ . (3.1) has a solution $(v(x), \tau)$ if and onlyif$d_{2}$

satisfies$\max\{0, \frac{B+C-2A}{C-B}\}\frac{a_{2}}{\pi^{2}}<d_{2}<\frac{a_{2}}{\pi^{2}}.$Moreover, the solution is unique.
Theorem 3.4 Suppose that $7B/3<C.$ $(3.1)$ has the unique solution $(v(x), \tau)$ if$\max\{0, \frac{B+C-2A}{C-B}\}\frac{a_{2}}{\pi^{2}}<d_{2}<\frac{a_{2}}{\pi^{2}}$Moreover, there exists the only one connected non-empty open set $D$ with$D \subset\{(A, d_{2}):B<A<\frac{B+C}{2}, 0<d_{2}<\{\frac{B+C-2A}{C-B}\}\frac{a_{2}}{\pi^{2}}\}$

such that (3.1) has exactly two solutions $(v(x), \tau)$ if and only if $d_{2}\in D.$

120

Figure 3.1: Case $B<C\leq 7B/3$

Figure 3.2: Case $7B/3<C$
The following theorems in [3] give the shape of solutions to (3.1) as) Suppose that $B<C.$
Theorem 3.5 (Shape of solutions as
Let $(v(x, d_{2}), \tau(d_{2}))$ be solutions of (3.1). If $A\geq B$ , then$d_{2}\uparrow a_{2}/\pi^{2}.$$d_{2}\uparrow a_{2}/\pi^{2}$$v(x;d_{2})arrow 0,$$\frac{v(x;d_{2})-v(0;d_{2})}{v(1;d_{2})-v(0;d_{2})}arrow\frac{1-\cos(\pi x)}{2},$$\frac{\tau(d_{2})}{v(x;d_{2})}arrow\frac{a_{2}}{b_{2}}\cdot\frac{1}{1-\sqrt{1-\frac{B}{A}}\cos(\pi x)}$uniformly on $[0,1]$ as$d_{2}\uparrow a_{2}/\pi^{2}.$$1=$$w_{*}$

Figure 3.3:

$u$

as

$d_{2}\uparrow a_{2}/\pi^{2}$

Figure 3.4:

$v$

as

$d_{2}\uparrow a_{2}/\pi^{2}$

121
The following theorems in [3] give the shape of solutions to (3.1) as
newnumber $(B+3C)/4$ appears. The shape is drastically change at $A=(B+3C)/4.$$d_{2}\downarrow 0.$

Theorem 3.6 (Shape of solutions as$(v(x, d_{2}), \tau(d_{2}))$ be
solutions of (3.1). If$d_{2}arrow 0$$v( O;d_{2})arrow 2\cdot\frac{a_{2}}{c_{2}}\cdot\frac{\frac{B+3C}{4}-A}{C-B},$$\frac{\tau(d_{2})}{v(0;d_{2})}arrow\frac{a_{2}}{2c_{2}}\cdot\frac{C-A}{C-B}.$

Figure 3.5:

$u$$\frac{A-B}{\frac{B+3C}{4}-A},$for) Suppose that $B\neq C$ . Let
and $B<C$ , then$A< \frac{B+3C}{4}$$A< \frac{B+3C}{4}$$v(x;d_{2}) arrow\frac{a_{2}}{c_{2}}\cdot\frac{A-B}{C-B}$

for $x>0,$$\frac{\tau(d_{2})}{v(x;d_{2})}arrow\frac{a_{2}}{b_{2}}\cdot\frac{C-A}{C-B}$

for $x>0.$

for $A\leq(B+3C)/4$

Figure 3.6:

Theorem 3.7 (Shape of solutions asfor$(v(x, d_{2}), \tau(d_{2}))$ be solutionsof (3.1). If $B<C$ and$d_{2}arrow 0$$A \geq\frac{B+3C}{4}$$v(x;d_{2}) arrow\frac{3a_{2}}{4c_{2}}$for $x>0,$$\frac{\tau(d_{2})}{v(0;d_{2})}arrow\infty,$$\frac{\tau(d_{2})}{v(x;d_{2})}arrow\frac{a_{2}}{4c_{2}}$for $x>0$ , as$u$

for $(B+3C)/4<A$$v$

for $A\leq(B+3C)/4$) Suppose that $B\neq C$ . Let, then$A \geq\frac{B+3C}{4}$$v(O;d_{2})arrow 0,$

Figure 3.7:$A$$d_{2}arrow 0.$

Figure 3.8:$v$

for $(B+3C)/4<A$122
Let us comment on the stability of stationary solutions in $1-\dim$ case.
The following Figure 3.9 shows numerical results for$d_{1}=1, d_{2}=*, r=700,000$$a_{2}=*, b2=1, c2=2.$$a_{2}=1, b2=1, c2=1.$

We note that $C<7B/3,$ $(B+C)/2=1.5$ and $(B+3C)/4=1.75.$

Figure 3.9: Stability and instability
Yaping Wu[8] gave a proof of instability forsufficiently small with $(B+C)/2<A<(B+3C)/4.$
Recently, she has also given proofs of the asymptotically stability forwith $(B+C)/2<A<(B+3C)/4,$
sufficiently close to
sufficiently small with $(B+3C)/4<A.$$d_{2}$$d_{2}(<a_{2}/\pi^{2})$$a_{2}/\pi^{2}$$d_{2}$

4

Multi-dimensional problem

We have done various numerical computations for the case is rectangles in 2dimensional space. It seems that the structure of stable stationary solutions is essentiallyvery similar to 1-dimensional case, though there are much varieties of shape of solutionsin 2-dimensional case than in one-dimensional case.
$\Omega$

Figure 4.1: $2D$ global

123
Now, we will state some mathematical results. We prepare notations. Let$\lambda_{0}=0<\lambda_{1}\leq\lambda_{2}\leq\cdots$$\varphi_{0}=$const.

$\varphi_{1},$$\varphi_{2},$$\cdots$$,$

be eigen values and corresponding eigen functions of-

$A$

in

$\Omega\subset R^{N}$

with Neumannboundary.
Theorem 4.1 Suppose that $N\leq 3$ andbe a simple eigen values with an eigen
andfunction . Then, there exists exactly two positive non-constant solutionswithof (2.2) for sufficiently close toMoreover,$\lambda_{1}$$(v_{-}, \tau_{-})$$\varphi_{1}$$(v_{+}, \tau_{+})$$d_{2}$$a_{2}/\lambda_{1}$$d_{2}<a_{2}/\lambda_{1}$$\tauarrow 0,$$\frac{\tau_{\pm}(d_{2})}{v_{\pm}(x;d_{2})}arrow\frac{a_{2}}{b_{2}}\cdot\frac{1}{1+\mu\pm\varphi_{1}(x)}$as$d_{2}\uparrow a_{2}/\lambda_{1}$, where$\mu-,$ $\mu+(\mu_{-}<0<\mu_{+})$

are solutions of$\frac{\int_{\Omega}(1+\mu\varphi_{1}(x))^{-2}dx}{\int_{\Omega}(1+\mu\varphi_{1}(x))^{-1}dx}=\frac{A}{B}.$

Remark. The set
is uniquely determined though there is a freedom
to pick up . The condition $N\leq 3$ comes from Harnack’s inequality in our proof.
$\{(v_{-}, \tau_{-}), (v_{+}, \tau_{+})\}$$\varphi_{1}$Remark. For $N=1,$ $\Omega=(0,1)$ , it is easy to see that$\lambda_{1}=\pi^{2}, \varphi_{1}(x)=\cos\pi x, \frac{1}{1-\mu^{2}}=\frac{A}{B}, \mu\pm=\pm\sqrt{1-\frac{B}{A}}.$

Remark. For $N=2,$ $\Omega=(0,1)\cross(0, \ell)$ with $0<\ell<1$ , it is easy to see that$\lambda_{1}=\pi^{2}, \varphi_{1}(x, y)=\cos\pi x, \frac{1}{1-\mu^{2}}=\frac{A}{B}, \mu\pm=\pm\sqrt{1-\frac{B}{A}}.$

Remark.

$A$

similar theorem holds for the equation (1.3) with sufficiently large

Remark.

Suppose that $N\leq 3$ andbe a simple eigen values. Then,andbydefined Theorem 4.1 are asymptotically stable for sufficiently close towith$(v_{-}, \tau_{-})$$\lambda_{1}$$(v_{+}, \tau_{+})$$a_{2}/\lambda_{1}$$r.$$d_{2}$$d_{2}<a_{2}/\lambda_{1}.$The following general lemma plays crucial role to prove Theorems 4.1.

124
Lemma 4.2 Suppose that $N\geq 1$ andbe defined by$\varphi_{1}$be eigen values corresponding to$\lambda_{1}$. Let $g(\mu)$$g( \mu):=\frac{\int_{\Omega}(1+\mu\varphi_{1}(x))^{-2}dx}{\int_{\Omega}(1+\mu\varphi_{1}(x))^{-1}dx}$$for \mu\in(-1/\max_{\overline{\Omega}}\varphi_{1}, -1/\min_{\overline{\Omega}}\varphi_{1})$. Then$\frac{dg(\mu)}{d\mu}=\{\begin{array}{ll}+ for \mu>0,0 for \mu=0,- f\sigma r\mu<0.\end{array}$Moreover, for $N\leq 4,$$\{\begin{array}{l}g(\mu)arrow\infty as \mu\uparrow-1/\min_{\overline{\Omega}}\varphi_{1},g(\mu)arrow\infty as \mu\downarrow-1/\max_{\overline{\Omega}}\varphi_{1}.\end{array}$Idea of a proof of Theorem 4.1.
Step 1:$\{\tau(d_{2})\}$

is bounded, and $\{v(x;d_{2})\}$ is

Step 2: There exists a sequence
Step 3:$\hat{v}(x)\equiv 0$

and$\{d_{2,j}\}$$L^{\infty}$

such that-bounded as$d_{2}\uparrow a_{2}/\lambda_{1}.$and $v(x;d_{2,j})arrow\hat{v}(x)$ .

$\tau(d_{2,j})arrow\hat{\tau}$$\hat{\tau}=0.$-bounded. We use theStep 4: $\{\tau(d_{2})/v(x;d_{2})\}$ is -bounded. $\{v(x;d_{2})/\tau(d_{2})\}$ isassumption $N\leq 3$ to apply Harnach’s inequality to $\{v(x;d_{2})/\tau(d_{2})\}.$$L^{\infty}$$L^{2}$

such that $v(x;d_{2,j})/\tau(d_{2,j})arrow w(x)$ , and $w(x)=$Step 5: There exists a sequence
for some . Moreover, $w(x)>0.$$\{d_{2,j}\}$$a_{2}^{-1}b_{2}\cdot(1+\mu\varphi_{1}(x))$$\mu$

Step 6: Substitute $w(x)$ into the first equation, we obtain the equation for
Step 7: Apply the implicit function theorem to $w:=v/\tau$ as follows. Let us take$p>N$ . Define$\mu.$$p$$F:W_{\nu}^{2,p}\cross R^{1}\cross(0, \infty)arrow L^{p}\cross R^{1}$by$F(w, \tau, d_{2}):=(d_{2}\Delta w+w(a_{2}-c_{2}\tau w)-b_{2}, \int_{\Omega}\frac{1}{w}(a_{1}-\frac{b_{1}}{w})dx-c_{1}\tau|\Omega|)$

We see that
$F( \frac{a_{2}}{b_{2}}\cdot(1+\mu\pm\varphi_{1}(x)), 0, \frac{a_{2}}{\lambda})=(0,0)$

,

and
$Ker$

of

$DF_{(w,\tau)}( \frac{a_{2}}{b_{2}}\cdot(1+\mu\pm\varphi_{1}(x)),$ $0,$ $\frac{a_{2}}{\lambda_{1}})=(0,0)$.

Thus we can apply the implicit function theorem. Moreover, we obtain$\tau’(\frac{a_{2}}{\lambda_{1}})<0, \tau(\frac{a_{2}}{\lambda_{1}})=0.$

with

125

References[1] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential
Equations, 131 (1996),, 79-131.
[2] Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J.
Differential Equations, 154 (1999), 157-190.
[3] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterracompetition with cross-diffusion, Discrete Contin. Dyn. Syst, 10 (2004), 435-458.

[4] M. Mimura Stationary pattern of some density-dependent system with competitive

dynamics, Hiroshima Math. J. 8 (1981), 621-635.
[5] W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices
Amer. Math. Soc., 45 (1998), 9-18.
[6] W. M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference
Series in Applied Mathematics, 82 (2011), SIAM.
[7] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting
species, J. Theor. Biol., 79 (1979), 83-99.

[8] Y. Wu, The instability of spiky steady states for a competing species model withcross diffusion, J. Differential Equations, 213 (2005), 289-340.
[9] A. Yagi, Abstract Pambolic Evolution Equations and their Applications, 2009,
Springer.

[10] Y. Yamada, Nonlinear diffusion equation with cross-diffusion -Reaction diffusion equation arising in mathematical biology, Sugaku, 64 (2012), 384-406.