Journal of

J. Math. Biology 9, 49--64 (1980)

mathematical

61olo9y

t~ by Springer-Verlag1980

Spatial Segregation in Competitive Interactlon-Diffusion Equations Masayasu Mimura ~ and Kohkichi KawasakiZ 1 Department of Applied Mathematics, Konan University, Kobe 658, Japan 2 Department of Biophysics, Kyoto University, Kyoto 606, Japan

Smnmary. The effect of cross-population pressure on the Volterra type dynamics for two competing species is investigated. On the basis of cross-diffusioninduced instability, spatial segregation is studied. Spatially discrete models are also discussed. It is shown that this effect has a tendency to enhance the stability assuring coexistence of species. In continuous and discrete eases, time-dependent segregation processes are studied numerically.

Key words: Spatial s e g r e g a t i o n - Nonlinear diffusion.

1. Introduction

The Volterra type dynamics for two competing species, say N~ and N2, is described by

lil= (R1 - alnl - bins)n1, (1) /~z : (R2 --

b2nl

--

aanz)nz,

where ni is the population density of the species N,, as is the coefficient of intraspecific competition and b~ is that of interspecitic competition for i = 1, 2. The system (1) has four stationary points

) (~) (0, 0), \(1{1'a~0 , O,

(Rla~-Rzbl Rzal-Rlbz~ b~b2"axaz ~ 1"

and (~, n2) = \ alas

When al/b2 > RI[R z > bl[az, it is seen that the point (~1, ha) is positive and stable and others are all unstable, which indicates, from an ecological viewpoint, that the coexistence of two species is realizable. On the other hand, when ax/bz < R~/R2 < bl/a2, (nl, nz) is unstable and alternately the other two (R~/a~, 0) and (0, Rz[aa) are stable, that is, only one species can survive in competition.

0303-6812/80/0009/0049/$03.20

50

M. Mimura and K. Kawasaki

Now we consider a one dimensional space in which two competing species migrate under self- and cross-diffusion. The equation describing the situation takes the form 8

82

~-~ul = ~-~ [(% + y l u l +/31u2)ul] + (R1 - alul - blu2)ul,

(2) 8 82 a-~ u~ = ~ [(~,~. +/3~u~ + ~,~u~)u~] + (R~ - b~u~ - a~u2)u~, which was originally proposed by Shigesada et al. (1979). The diffusion terms such as

~2 Ox----~ [D~(u~, u2)u~]

(i = 1, 2)

are introduced to describe the situation that individuals are randomly walking and disperse repulsively (Okubo, 1975). The biological interpretation is based on Morishita's phenomenological theory of'environmental density' (Shigesada et al., 1979). This system (2) incorporates the movement of individuals subject to the dispersive force characteristic of self- and cross-population pressures. Here ~ is a postitive constant, ~,~ and 3~ are both nonnegative constants for i = 1, 2. When y~ =/35 = 0 for i = 1, 2, the system (2) is reduced to a normal reaction-diffusion equation that has been discussed by many authors. The effects of self-population pressure on one species dynamics were discussed by Gurney and Nisbet (1975) and by Gurtin and MacCamy (1977). The cross-population pressure causes a kind of cross-diffusion for migration, the idea of which is found in the works of Kerner (1959), Gurtin (1972) and Rosen (1977). We note that the competitive interaction in (2) does not fall within the category of activator-inhibitor relationship such as a prey-predator interaction in ecology (see, for instance, Segel and Levin, 1976). That is, in the absence of cross-population pressures, in other words, when/31 =/32 = 0, the system (2) does not exhibit diffusion-induced instability in Turing's sense. The main purpose of this paper is to answer the question, 'under zero flux boundary conditions, does (2) exhibit any spatial segregation for large time?'. For segregation problems, we must mention two results related to our work. One is shown by Levin (1974) who treated competing species in patchy habitats in the absence of self- and cross-population pressures. For the completely symmetric case with two patches, his model is given by a~

,~(u~

u~) + (R - au~

~'u'~u ~

(i,j = 1, 2, i ~ j). fl~

o~(u~ -

u~) + ( R -

bu~ -

au~)u~,

(3)

Competitive Interaction-Diffusion Equations

51

This system is considered to be a spatially discrete version of (2) under zero flux boundary conditions. Under the condition 0 < c~ < (R/2)(b - a)/(b + a), Levin found the stationary point t~,l,t~~.2, -1,~2~g) given by ~ = ~

=

R-2a 2a

1 J ( R _ 2~) ( R -

2eb + a'~ b - a]

R " --' - ~ -2a ~ + ~ a a1 2 ( R - 2 ~ ) ( R -

2 bb - +a ] "a]

2a

and

This result implies the segregation in two patches, although two species cannot coexist for c~ = 0. Levin also proved that this equilibrium is stable for 0 < ~ < (R/2)(b - a)/(2b + a). The other result is given by Shigesada et al., (1979), who introduced an environmental potential, that is, a heterogeneous environment into the right hand side of (2) and clearly demonstrated by numerical methods segregation processes under the condition al/b2 < R1/R2 < b~/a2. This work deals with the case where no coexistence is possible in the most simple system (1). Our concern is the case where coexistence is possible and we shall consider segregation processes for (2) and (3) in this situation. In this paper, we consider a slightly simplified version of (2), i.e. 0 82 a~ ul = ~-5 [(~ + fllu2)ul] + (R1 - aul - bu2)ul, (4a)

L ua ~t

((t, x ) 8 8x

~2

= ~

[(a + fl2ul)u2] + (R2 - bul - au2)u2,

(0, +oo) x (0, r)), subject to the boundary and initial conditions u~(t, o ) =

O u~(t,r) = 0 -~

(t > 0 ) ( i = 1,2)

(4b)

and u~(O, x ) = U~o(X) (x e [0, r])(i = 1, 2),

(4c)

respectively. For simplicity only, we restrict the values of the parameters to al -= a2, bl = b 2 , ~ 1 = a2and71 = 7 2 = 0 In Section 2, it is shown that the stationary problem of (4) has spatial patterns exhibiting segregation under appropriate diffusion coefficients ~, fll and f12 when a, b, R1 and R2 are fixed to be a/b > R~/R2 > b/a, implying that (~, u2) is stable with respect to spatially homogeneous perturbations. This condition is different from the one imposed by Shigesada et al. (1979). We also show numerically that the evolution system (4) produces segregation effects for large time, starting from heterogeneous distributions initially. In Section 3, we consider Levin's model with inclusion of the effect of cross-population pressure under a/b < R1/R2 < b/a, which is different from the assumption a/b > R~/R2 > b/a in Section 2. The result

52

M. Mimura and K. Kawasaki

is that cross-diffusion effect has a tendency to enhance the stability assuring coexistence of two species, in comparison with Levin's model. In Section 4, we conclude our paper with a discussion.

2. Spatial Segregation In this section, we suppose a/b > R1/R2 > b/a which indicates that the constant stationary point 071, ~2)(>0) is stable with respect to spatially homogeneous perturbations. We study the problem whether the asymptotic solution (u~(t, x), u2(t, x)) of (4) tends to (~1, ~2) or to heterogeneous steady states exhibiting spatial segregation, starting from heterogeneous distributions (ulo(x), U2o(X)). Using the bifurcation technique, we will demonstrate that the stationary problem of (4) has spatially inhomogeneous solutions exhibiting segregation. The adjustable parameters used here are c~, ~1 and 82. We conveniently introduce perturbation variables vi = u~ - fit (i = I, 2) into (4). The resulting problem is 02 02 02 0 = (~ + #1u2)b-~ b-~ Tt v~ v~ + f~u~ v~ + [~ - ~ (vlv~.) - afllvl - b~lvz - av~ - bvlv2, a

02 02 02 a-i v~ = [3~u~ - ~ v~ + (,~ + [ 3 ~ ) - ~ v~ + ~. - ~ (v~v~) -

(Sa)

b~zvl - a~2v2 - bv~vz - av~

((t, x) ~ (0, +oo) • (0, r)), v,(t, O) = - ~ v,(t, r) = 0

(t > 0),

(5b)

vi(0, x) = U,o(X) - if, (x e [0, r ]),

(5c)

for i = 1, 2. The stationary problem for (5) describes, in the vector form, 0 = D-~ dW

O = - d - ~-

W+

MW+

N

(

W, d W d~W'~ d x ' dx 2 ]

(6a)

(xe(O,r)),

(6b)

(x = O, r),

where W = t(wl, w2), D=[Cz+/31ff2

/ ~2a~

/~1ul ]

M=[-a~l-b~l]

~ +/~a~]'

[-ba~

-aa~a

and N is the nonlinear term N =

t( - a w l ~ -

d2

bwlwg. + ~1 ~

d2

(wlw2), - b w l w 2 - aw~ + ~ ~

)

(wlwz) 9

We now consider the bifurcation problem from the trivial branch W = 0 when ~,/31 and/~ are varying as bifurcation parameters. Let us study the linear eigenValue

Competitive Interaction-Diffusion Equations

53

problem in order to find possible bifurcation points of (%/31,/32), d2 D ~-~ gP + Mdp = MP (x e (0, r)),

(7a)

0 = - dd- ey

(75)

(x = 0, r).

The Fourier series expansion qb = ~,%o ~ , cos m r x / r yields that the problem (7) is equivalent to M~qb~= Aqb, (n = 0 , 1 , 2 , . . . ) ,

(8)

where M , = M - oJZn2D for ~o = rr/r. The resulting characteristic equation is a 2 + [a(al + a2) + {2~ + (/31U2 +/3~al))~o2n2]a + {a~l + (~ +/31a2)o~2n~} x {aft2 + (a +/32ffl)o~2n

2} - {bal +/31~loJ2n2}{b~2 + t32ff2r

2} = 0.

(9)

It is obvious that (9) has a zero root if and only if a, fil and/32 satisfy {~2 + (/3~22 +/322~)~}o~'n' + {a(al + ff2)~ + ~1~2(a~2 - bnl) +/32~1(a~1 -

b~2)}oJ2n 2 + (a 2 -

b 2 ) ~ f f 2 = O.

1. Suppose that a >>/31, f12- Then Re (An) < 0 for n = 0, 1. . . . . thus W = 0 is linearly stable. Furthermore, for the special case when/31 =/32 or R~ = R2, W = 0 is also linearly stable.

Remark

From this remark, we find that, if the cross-population pressure is not introduced, diffusion-induced instability does not occur (see Levin, 1974; Rosen, 1977). For the specific case when /31 =/3 and /32 = 0, let us investigate the set of bifurcation points with respect to (a,/3). The bifurcation curve is represented by the following curves {C~} in the positive quadrant of (a,/3)-space: C,:fl = -

c~

+ a ( a l + 22)~o2n%~ + ( a 2 - b2)21~2 ~2oJ2n2{oJ2n2~ + (a~2 - bffl))

(see Fig. 1). 2. Let ~ be a~ = - (aft2 - b ~ ) / 2 . If a~ ~< a, W = 0 is linearly stable. If 0 < ~ < ~ , W = 0 is unstable for appropriate values/3 > 0. Remark

Our system does not fall within the category of activator-inhibitor relationship. However, one knows that if aa~ - b~1 < 0, the cross-population pressure leads to cross-diffusion-induced instability. We consider the bifurcation problem when (a,/3) varies along any fixed path (a(a),/3(~)) in such a manner that it starts from the stable region and goes into the unstable region. Here (~(~),/3(a)) is a smooth mapping from an open interval I containing zero to R~+ in a way that (a(0),/3(0)) intersects transversally with the bifurcation curve at a = 0 but that it is not an intersecting point of two curves of {C,}. Thus, the system (62) is reduced to a system with only one parameter ~. In this situation, the problem (6) can be discussed by the use of the theory of bifurcation at simple eigenvalues. Mimura et al. (1979) studied diffusive prey-predator models

54

M. Mimura and K. Kawasaki Cn

unstable regio I

C~,

Ci

,

stable region

II II AI I

B/nt[~

I]1

0

A/rl z

~Ot

Fig. 1. Schematic bifurcation curve in (~,/3) space. A

at22 - bt21 and B = co2

(a2 - b2)t~d22 ~2oJ2(at22 -- bt21)

with/31 =/32 = 0 and showed the existence o f heterogeneous steady state solutions, using this bifurcation theory. The introduction o f cross-diffusion makes the situation in this paper quite different. However, notice that the nonlinear term involves the second derivative d2/dx 2 only linearly. T h e L y a p u n o v - S c h m i d t m e t h o d can be applied to our problem. Then we have Theorem 3. Consider the problem (6) with/31 = [3and~32 = O. There exists a constant eo such that (6) has a unique one parameter family of nontrivial classical solutions (~(r W(e)) for 0 < [el < Co. Here e(e) satisfying ~(0) = 0 and W(e) are

smooth and W(8) = 8 r 0 c o s

n~x

+ 0(8) a s 8

r

-+ 0,

(10)

where ~o is defined in the proof. Sketch of Proofi Consider (6) in (L~((0, r))) 2, which has a real p a r a m e t e r a, L(cr)W + N ( W ) = 0,

~ e I,

(11)

where L(e) = D(o) d2/dx 2 + M. T h e domain of L(g) is O(L(cr)) = D(L(0)) = (H~((0, r))) z where H~((0, r)) is the closure of {cos (mrx/r)}~= o in the usual Sobolev space H2((0, r)). Let P be the orthogonal projection operator of (L2) 2 onto N(L(O)) where N(L(O)) is the null space of L(0). It is easily seen that N(L(O)) is spanned by the vector ~o cos (mrx/r) with

r

= kt(b~l +/3~loJ2n 2, - a ~ l - (a +/3~2)co2n~),

Competitive Interaction-Diffusion Equations

55

where k, is a normalized constant such that 10~ = V'2-'~/r. A solution W will be obtained in the form of

W = P W + ( I - P)IfV --- 80 ~ cos mrx + IV, r

(12)

where e is a real parameter. Hence, the Lyapunov-Schmidt method can be applied to (11) together with (12). Following the method of Proof of Theorem 3-1 in Mimura et al. (1977), we can show directly that W(e) ~ (H~) 2 uniquely exists for sufficiently small e and ~ such that if(e) = o(e) as e --~ 0, together with the relation between 8 and or. Here we used the fact that W e (H2) 2 leads to N(W) ~ (L2) 2, since N(W) is linear with respect to d2W/dx 2. Since (6a) can be rewritten as dz

1

dx---~wl = o~+ fll(fi2 + w~) x

[31(~1+ w l ) ~

w2 + a~lw~ + b~lw2 + aw~ + bw~w2

d2 ~ - ~ w2 --- b~2wl + a~2w2 + bwlw2 + aw~, we see that w2 e H ~ and then wl e H 4, since c~ + ~ ( ~ + wa) > 0 for a sufficiently small e > 0. Thus, we obtain a classical solution (c~(e(e)), fl(e(e)), W(e)), which completes the proof. If e is sufficiently small, Theorem 3 indicates that spatial patterns of IV(e) are determined by the first term of the right hand side of (10). Thus we find that W(e) exhibits segregation phenomena in the sense that the sign of two components of O~ are different (see Fig. 2).

Remark 4. From the relation between e and e, we know that the bifurcation is onesided.

Remark 5. For more general nonlinear terms N(W, dW/dx, d2W/dx2), Theorem 3 remains valid only if N involves the operator of second derivative d~/dx ~ linearly. Another model was proposed by Rosen (1977) 0 = ~d

[(

dl~ d (log u ~ ) + d~z ~x (log uz)}u~]+ f(u~, us), (13)

d [( d21 ~-~ d(logu~)+d22 ~-~ d (log u2)} u2] + g(ul, u2), 0 = ~-~

oI

Ir

w~lX;~)

/

Fig. 2. A bifurcating solution with the mode number 2, W(x; e) = ~(Wl(X;e), w2(x; e)), of (6) for a sufficiently small e>0

56

M. Mimura and K. Kawasaki

w h e r e a~s (i, j = 1, 2) a r e n o n n e g a t i v e c o n s t a n t s a n d f a n d g a r e a p p r o p r i a t e e c o l o g i c a l i n t e r a c t i o n s . B y i n t r o d u c i n g w~ = log(u~/~0 (i = 1, 2) i n t o (13), it f o l l o w s , in v e c t o r f o r m ,

0 = D - ~ W + M W + N W,--~ . H e r e (ul, ~2) is a p o s i t i v e s o l u t i o n o f f = g = 0, D is a full m a t r i x w i t h e l e m e n t s {d~s} (i,j = 1, 2), M is t h e J a c o b i a n o f ( f , g) a t u~ = ~, (i = 1, 2) a n d N i s a c e r t a i n nonlinear term.

)X

u{t.x)

v(t,x)

Fig. 3. Segregation in the evolution system (4) R1 = 5.0, R2 = 4.5, a = 1.2, b = 1.0, a = 0.5, ~x = 7.5 a n d / ~ 2 = 0

>X

u(t.x)

v[t,x)

Fig. 4. Segregation in the evolution system (4) Rx, R~, a, b,/~2 are the same values as in Fig. 3 except ~ = 0.2 and/]1 = 0.5

Competitive Interaction-Diffusion Equations

57

In Figs. 3 and 4, it is numerically shown for appropriate r and/3, that a solution V(t, x) of the evolution equation (4a) tends, as t - + ~ , to the steady state W ( x ) stated in Theorem 3. Thus, we find that the competitive system exhibits segregation process due to the effect of cross-population pressure.

3. A Two-Patch Model with Cross-Population Pressures Levin (1974) considered general patch models o f competing species with small passive migration and showed the possibility of coexistence of the species in individual patches, even in the situation in which the species could not coexist in a single patch. For a simple model, he proposed a completely symmetric system in two patches (see (3)) and analyzed this model more precisely. We introduce the cross-population pressures proposed in Section 2 into his model. The resulting system is written as f~, = (~ +/31vj)u~ - (~ + [31v,)u, + ( R - au, - bvOu,, ( i , j = 1, 2, i # j ) ( l l ) = b, = (~ +/32uj)vj - (~ +/32ui)v, + ( R - bu, - av~)v~,

where the parameters a and b are chosen such that 0 < a < b. This choice of the parameters differs from that in Section 2 and implies the survival of only one species, that is, no coexistence in a single patch situation. (11)~ is considered to be a spatially discrete model of (4a) and (4b). We show that the effect of population pressures enhances the stability assuring coexistence of segregation phenomenon in comparison with Levin's one. We note that (11)~ has at most 16 stationary points, since the nonlinear term of each equation in (11)~ is a polynomial of the second order. As the simplest case, we first consider the system (11)o with fll =/32 = 0. Then the stationary points, say (~, V), in each patch are given by

Io:~=0, O=0, II0: ~ = R/a, ~ = 0, IIIo: ~ = 0, ~ = R/a, IVo: ~ = ~ = R / ( a + b), and then the stationary points of (11)o with fll = f12 = 0, (ul, v~; u2, v2), are obtained from the combinations of the above four cases. These are denoted by (I, I)o . . . . , (IV, IV)o. From the condition a < b, it is found that the 4 points (II, II)o, (III, III)o, (II, III)o and (III, II)o are stable. For the system (11)= with c~ > 0 and fl~ = f12 = 0, that is, Levin's model, all 16 stationary points are obtained explicitly. In this ease, 10 points are in our study and other 6 points are discarded since these are not in the positive quadrant of R 4 (see Appendix). It is not difficult to see that (I, I)~ is unstable, (II, II)= and (III, III)~ are stable for any ~ > 0, that is, there is no bifurcation point along these paths. On the other hand, along the paths (II, III)~, (III, II)~ and (IV, IV)=, the global bifurcation diagrams can be studied (see Fig. 5).

58

M. Mimura and K. Kawasaki

(ll,lV)e (11.111) 0

(IV, Ill) 0

I

\

(IV, IV )o (lU, lY)O

"~N[

i./

J

(III,II| o

(IV,Ir)o

I

i

a*(O)

~.ot

B

Fig. 5. Global bifurcation diagram along the paths (It, III)., (Ill, II)~ and (IV, IV). with fll = flz = 0. Solid lines are stable paths and dashed lines are unstable ones. ~ = R ( b - a ) / (2(b + a))

Now our interest is in the study of (11). with fil > 0 and/or flz > 0. We first note that the points (I, I)., (1I, II)., (III, III)., (IV, IV)., (II, III). and (III, II). are still stationary points of (11). in spite of ill > 0 and/or f12 > 0, and that the remaining 4 points (II, IV)~, (IV, II)., (III, IV). and (IV, III). are not. It should be noted that the characteristics of stability on (I, I)., (II, II)., (III, III). and (IV, IV). does not change in spite of fil > 0 and f12 > 0. Thus, it suffices to study the question, ' How does the bifurcation diagram along (II, III). and (III, II). change as a result of fll > 0 and/or f12 > 0 ?'. We are interested in studying this problem because each path represents spatial segregation. Theorem 6. Consider the system (11)~. Then the stationary point (II, III). is locally stable for R(b - a) b - a + 0 ~< ~ < ~*(/3) = 2(a + 2b) b _ a + 2b__._.~_~ a + 2b and unstable for R(b - a) =*(/3) < ~ ~ 2(b T a)" Proof. As (II, III)~ is obtained explicitly, a linear analysis can show the proof.

In this theorem, we note Proposition 7. ~*(/3) is a monotonically increasing function satisfying c~(O) = R ( b - a)

2(a + 2b)

and ~-.~ lim c~*(/3) =

R ( b - a)

4b

The other point (III, II)~ has the same property as the above. This proposition leads to a conclusion that as an effect of cross-population pressures the interval assuring the stable coexistence tends to be longer. We next consider the bifurcation from the

Competitive Interaction-Diffusion Equations

59

points (II, III)~.(~). It is convenient to rewrite the right hand side of (11). such that (II, I I I ) . = (~, ~ ; fig, ~g) is transformed into the origin (0, 0; 0, 0). Then it follows that

= *(G G G ~), {g~ + fi~M~ + fl~Mg}E + Qo(E, E) + fl~Q~(E, E) + [3gQ~(E, E) = 0,

(12)

where RM~ =

O~ - b ~ 2affi - b~i - a 0 R - 2av~ - bui - ~z - - bgi (g 0 R - 2a~2- b~2- a R - 2a~g - bff2- a ] -- bog 0

0

0

d], Ng=

0

N i =

Q0

=

[

0

-a~ -a~ -a~ -a~

0

0 -- vl

0 -- fii

0

0

- b~:lG] b~i~g[, b~[ b~.]

Qi

0

~1761 vg

02

0 --,Tg -ug_l

=

and o

Q~ =

-Kfd uKfd~-

, fd,)_l

Here we note that M~,<0> = Me, is singular. Using the Lyapnov-Schmidt method in the finite dimensional case (see Stakgold, 1971), we obtain the bifurcation equation for a, fll and fi2. Let us seek a solution E in the form E = e@ + tF,

(13)

where (I) = *(41, 4g, 43, 44) is given from Mr = 0, that is, @ satisfies 4~ + 44 = 4g + 48 = 0 and (R - 2a~1 - b ~ - a)41 = (bff~ + a)42, and tF is the remainder. After substituting (13) into (12), we obtain

(4M~ - M~.)~ + / ~ g & ~ + ,r) + / ~ M & ~ + T) + Qo(~, + T, ~@ + ~ ) + fi~Q~(ea) + qp, ca) + ~) + flgQ2(ea) + w, ~@ + xF), @*) = 0. (14) Here @* is given from M** ~* = 0 where M** is the transposed matrix of M~.. Thus, (I4) leads to the relation

,(~ - ~*)A + ,(~i + &)B + , ~ ( ~ - & ) c + ~ D + - . . where A, B, C and D are described by

a = -2(4~ + 4~.)(4" + 4"), B = -2(a=4~ + a~4=)(4" + 4 % C= and D=

(Qo(@, tF~2) + Qo(tF~2, @), @*)

(15)

M. Mimura and K. K a w a s a k i

60

and ~F~2 = (ffl,2, ff2~2, ~ba,2, ~b4~@ is the coefficient vector of the order 82 in W. Here we used the relation that (Qo(~, ~), o*) = 0, Q~(qb, ~) = 0 (i = 1, 2) and if* + if* = if2* + if* = 0. The bifurcation diagram derived from (15) is drawn in Fig. 6. The global bifurcation diagram from numerical computations, is shown in Fig. 7.

,,':r a*(~) (b) (,e~-,S,) C > 0

~(0)

(a) ~l = .Sz = 0

.,~a*(B)

(c) (~l-.821 C <

0

Fig. 6. Bifurcation diagram along the p a t h (II, III)~ for a sufficiently small 9 > 0 where #1 + #2 = # > 0

m.lv)o ,,

J,

(litlll) o |IV.NI) o

r

J

\

(Iv.IV)o

I

k.

(lll,lV)o

/

J

/

/

(111,|1)o

(iV,ll)o

........ 0

/I

'

i

1

a~'(OI a'(~)

)

,

ao(~)

Fig. 7. Global bifurcation diagram along the path (II, III)., (III, II)~ a n d (IV, IV)~ when flz = 0 and fll = 13 > 0 is fixed

Competitive Interaction-Diffusion Equations

61

P r o p o s i t i o n 8. For a sufficiently small fl, there exists a constant ~o(~) such that a new stable branch bifurcating f r o m (II, III),:(~) or ( I I I , II)~.(a) exists f o r cc*(~) < c, <

0~0(~).

O n the b a s i s o f (11)~, we n e x t c o n s i d e r the initial v a l u e p r o b l e m w h e r e initial values a r e c h o s e n as u~(0) = u ~

vl(0) = v ~

u2(0) = v2(0) = 0,

(16)

t h a t is, t w o species live in o n l y o n e p a t c h a t the initial stage. W e are i n t e r e s t e d in the q u e s t i o n w h e t h e r c o e x i s t e n c e o f t w o species is r e a l i z e d or not. W h e n coexistence is realized, we say t h a t s e g r e g a t i o n is realized, b e c a u s e a t coexistence e q u i l i b r i a (II, I I I ) , , o r ( I I I , I I ) , , e a c h p a t c h r e p r e s e n t s s p a t i a l s e g r e g a t i o n . W e s t u d y n u m e r i c ally t h e initial v a l u e p r o b l e m (11), a n d (16). T h e results are t h e f o l l o w i n g : W h e n fil = fi2 = 0 ( n o c r o s s - p o p u l a t i o n p r e s s u r e , L e v i n ' s case), s p a t i a l s e g r e g a t i o n o r

parch I 0

I 5.

I I0.

time

patch

0

5.

I0.

I 15.

2 I

time

15.

Fig. 8. Simulation of segregation process which the patch model (11)~ exhibits. R = 2.0, a = 0.8, b = 1.2, = = 0.1,/3, = 4.0,/32 = 0, u ~ = 1.0 and v~ = 0.8. Solid lines denote u~(t) and dashed lines denote v~(t)(i = 1, 2)

B 1.0

s e g ~

.5

0

uniformity ~ unof~irm.ti . 0.5 i.o

y i.~

Uo

Fig. 9. The regions of segregation and uniformity in the (u ~ fl) plane for fixed v~ = 0.8. fll = fl, f12 = 0

62

M. Mimura and K. Kawasaki ~8.1.6 0.8

0.4

0.2

0.2 0.4

Vo

1.0 0.8

1.6

0.5

0

0.5

1.0

1.5

Uo

Fig. 10. The effect of t h e cross-population pressure (131 = /3,/32 = 0) o n segregation. Curves

with respect to/3 in the (u~ v~ plane are contour lines determining either segregation or uniformity

coexistence does not arise for the initial value problem. Only one species survives in both patches and the other species becomes extinct in both patches. On the other hand, when/31 > 0 and/or/32 > 0, segregation can occur for appropriate values of /31 and/32. Here we study only a specific case when/31 =/3 > 0 and/32 = 0. The result implies that for any fixed (u ~ v~ there exists a positive constant/3* such that spatial segregation arises for/3* < /3 and does not arise for 0 ~
4. Concluding Remark We have considered a two-component competitive system described by a reactiondiffusion equation. Section 2 stated that the cross-population pressure or crossdiffusion is an important mechanism to produce heterogeneous steady states which exhibit segregation phenomena. We also showed numerically that the evolution system describes segregation processes for large time. Section 3 dealt with a model of two competing species in two patches where mutual migration is allowed. This model is a spatially discrete version of that in Section 2. Two results were shown. One is that the stability condition assuring the coexistence of two species is weaker than that for the model without cross-population pressure. The other is the realization of segregation in the sense that although two species live initially in only one patch, for large time they can coexist separately in two patches where one species

Competitive Interaction-Diffusion Equations

63

is dominant in one patch and rare in the other. Thus, we find that the crosspopulation pressure is essential for the realization of spatial segregation, because if such a pressure is not introduced, segregation can not take place and only one species survives and the other becomes extinct in both patches. The idea in Section 3 can be extended to continuous models in higher space dimensions. In this case, the model is described by a ~ ul = A[(a +/31u~)ua] + (R1 - au~ - bu2)ul, a-~ u2 = A[(~ +/32u~)u=] + (R= - bu~ - au~)u~,

where A is the Laplacian in R" (n >1 2). If the habitat f~ is, for example, of dumbbell shape in R 2, segregation process in the above sense seems to be realized. This study will be reported in a forthcoming paper.

Acknowledgement. The authors wish to express their gratitude to Professor Ei Teramoto and Dr. Nanako Shigesada for their stimulating discussions, and to Professor Akira Okubo for reading the manuscript and offering many suggestions. They also wish to thank Professors Hiroshi Fujii and Yuzo Hosono for giving kindly an opportunity of using their programs to obtain the numerical data in Section 2.

Appendix For ~ > 0 and fll =/32 = 0, all stationary points (~1, vl; u2, i2) of the system (11)~ are given in the following. In the positive quadrant, there are 10 stationary points (I, I),~:(O, 0;0, 0), (II, II)~: (R ' 0 ; Ra ' 0)

(III, III)~: (0, R. 0, R) a'

'

'

(IV, IV)~: a + b' a + b' a + b' a +-b ' (II, III)~: (p+, p - ; p - , p+),

p.

R-2o~

1J

= 2-------a--- -+ 2-a

(III, II)~: (p-, 1)+ ;p+,p-),

(

_ b+a,

(R - 2~) a - 2a b - - a ) '

(II, IV),: (q+, q- ; r -, r +),

(IV, III)~: (r +, r - ; q-, q+),

(III, IV)~:(q-,q+;r+,r-),

(IV, II),:(r-,r+;q+,q-),

1 (

q*=Ta

b-as2 b+a

R-~

+ts_) ' bb-as2+ +a

r~=~-~1 ( R - ~ 1(;

s~ = ~ t=

+_ts+),

2a.

R+~_a+

2a~ R + a +"-'--~'

;

2 b + a]

R-2ab_a],

64

M. Mimura and K. Kawasaki

and in the other quadrants, there are an additional six (I, IV)~: ( m - , m - ; m +, m+), (IV, I)~: (m +, m + ; m - , m - ) , (III, I)~: (0, n § ; 0, n - ) , (II, I)~: (n § 0; n - , 0), (I, III)~: (0, n - ; 0, n+), (I, II)~: ( n - , 0; n § 0), rn~ = 2(a'-'--'-~-I b ) ( R - 2~ _+ ~/R2 _ 4~2) ' n~ = a_.__a_ + b m~"

References Gurney, W. S. C., Nisbet, R. M.: The regulation of inhomogeneous populations. J. Theor. Biol., 52, 441-457 (1975) Gurtin, M. E. : Some mathematical models for population dynamics that lead to segregation. Quart. Appl. Math., 32, 1-9 (1972) Gurtin, M. E., MacCamy, R. C.: On the diffusion of biological populations. Math. Biosci., 33, 35-49 (1977) Kerner, E. H.: Further considerations on the statistical mechanics of biological associations. Bull. Math. Biophysics, 21,217-255 (1959) Levin, S. A. : Dispersion and population interactions., Amer. Natur., 108, 207-228 (1974) Mimura, M., Nishiura, Y., Yamaguti, M. : Some diffusive prey and predator systems and their bifurcation problems. Anal. New York Acad. of Sci., 316, 490-510 (1979) Okubo, A. : Ecology and diffusion. Tokyo: Tsukiji Shokan 1975 Rosen, G. : Effects of diffusion on the stability of the equilibrium in multi-species ecological systems. Bull. Math. Biol., 39, 373-383 (1977) Segel, L. A., Levin, S. A.: Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions. AIP Conf. Proc. 27, 123-152 (1976) Shigesada, N, Kawasaki, K, Teramoto, E.: Spatial segregation of interacting species, J. Theor. Biol. 79, 89-99 (1979) Stakgold, I.: Branching of solutions of nonlinear equations. SIAM Review, 13, 289-332 (1971)

Received May 2/Revised July 19, 1979