A Note on the Propagation Speed of Travelling Waves for a Lotka–Volterra Competition Model with Diffusion

A Note on the Propagation Speed of Travelling Waves for a Lotka–Volterra Competition Model with Diffusion

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO. 217, 693]700 Ž1998. AY975723 NOTE A Note on the Propagation Speed of Travelling Waves...

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

217, 693]700 Ž1998.

AY975723

NOTE A Note on the Propagation Speed of Travelling Waves for a Lotka]Volterra Competition Model with Diffusion Yukio Kan-on Department of Mathematics, Faculty of Education, Ehime Uni¨ ersity, Matsuyama 790, Japan Submitted by G. F. Webb Received April 30, 1997

This paper is concerned with the propagation speed of positive travelling waves for a Lotka]Volterra competition model with diffusion. We show that under a certain boundary condition, the propagation speed of the travelling wave is equal to 0. To do this, we employ the method of moving planes proposed by Gidas et al. Q 1998 Academic Press

Key Words: Lotka]Volterra model; propagation speed; method of moving planes.

1. INTRODUCTION There have been many studies of reaction]diffusion equations wt s D w x x q f Ž w . ,

x g R,

t)0

Ž 1.1.

to explain phenomena which appear in various fields, where w and f are n-dimensional vectors, and D is a diagonal matrix whose elements are positive. One interesting phenomenon is the appearance of travelling waves which are represented as wŽ t, x . s uŽ j ., j s x y st, where uŽ j . is a C 2-class function, and s is a constant to be determined. Such waves necessarily satisfy the system of ODEs 0 s Du jj q su j q f Ž u . ,

j g R.

Ž 1.2a.

693 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

694

NOTE

In order to determine the propagation speed s, we lay the boundary condition u Ž y` . s u Ž q` . s u 0 Ž g R n .

Ž 1.2b.

on the wave. Since as < j < ª q`

u j Ž j . rj s yDy1 f Ž u 0 . q o Ž 1 .

is satisfied, we find that u 0 must be a solution of fŽu. s 0. Here we assume that ŽH.1. the real part of every eigenvalue for the matrix f u Žu 0 . is negative, which means that u 0 is an exponentially stable equilibrium point of u t s fŽu.. Let us consider the case n s 1. Since f uŽ u 0 . - 0 holds because of ŽH.1., we can easily check that every solution of Ž1.2. exponentially decays to u 0 as < j < ª q`. Hence we have uj Ž"`. s 0, and then obtain s

HRu

j

2 Ž j . d j s yH Ž Dujj Ž j . q f Ž u Ž j . . . uj Ž j . d j s 0,

R

which implies that the following property holds for Ž1.2.: ŽP. If there exists a nonconstant solution for s s s0 g R, then s0 s 0 must be satisfied. As the property ŽP. does not always hold for n G 2, one important problem for the travelling wave is the qualitative study on the propagation speed s. In this paper, to approach the problem, we consider the propagation speed of travelling waves for the Lotka]Volterra competition model with diffusion u t s u x x q uf 0 Ž u . ,

¨ t s d¨ x x q ¨ g 0 Ž u . ,

x g R,

t ) 0,

which describes the dynamics of the population u s Ž u, ¨ . g R 2 of two competing species, where d is a positive constant, and f 0 Žu. s Ž f 0 , g 0 .Žu. is a C 1-class function in u. Since u and ¨ are variables which indicate the population density, we restrict our discussion to nonconstant solutions which satisfy 0 s Du jj q su j q f Ž u . , u Ž j . G 0,

¨Ž j

j g R,

. G 0,

u Ž y` . s u Ž q` . s u 0

j g R,

Ž 1.3a. Ž 1.3b. Ž 1.3c.

NOTE

695

for some s g R, where D s diagŽ1, d ., f Žu. s uf 0 Žu., g Žu. s ¨ g 0 Žu., fŽu. s Ž f, g .Žu., and u 0 is a solution of fŽu. s 0 in the first quadrant. From the competitive interaction, we assume the following: ŽH.2. There exist a u ) 0 and a ¨ ) 0 such that f 0 Ž u, 0 . Ž u y a u . - 0 for any u / a u , g 0 Ž 0, ¨ . Ž ¨ y a ¨ . - 0 for any ¨ / a ¨ . ŽH.3.

f 0 ¨ Žu. - 0 and g 0 uŽu. - 0 are satisfied for any u.

We should note that Ž0, a ¨ . and Ž a u , 0. are solutions of fŽu. s 0 in the first quadrant. Many authors have studied the existence of nonconstant solutions of the problem Ž1.3a., Ž1.3b. with a variety of boundary conditions Žfor instance, see Gardner w1x, Hosono and Mimura w3x, and Kan-on w4x.. However, so far, we do not yet have enough results on the propagation speed of the travelling wave. The following is the main result in this paper. THEOREM 1.1. Let u 0 be either u 0 s Ž0, a ¨ . or u 0 s Ž a u , 0.. If the assumptions ŽH.1., ŽH.2., and ŽH.3. are satisfied, then the property ŽP. holds for Ž1.3.. In Section 3, we shall prove the above theorem by using the method of moving planes proposed by Gidas et al. w2x.

2. APPLICATION In this section, we consider the case where f 0 Žu. is linear, that is, f 0 Ž u . s 1 y u y c¨ ,

g 0 Ž u . s a y bu y ¨ ,

where a, b, and c satisfy 0 - 1rc - a - b. We can easily check that Ž0, a. and Ž1, 0. are exponentially stable equilibrium points of u t s fŽu.. LEMMA 2.1 ŽTheorem 2.2 and Corollary 2.3 in w4x.. There exist a constant a0 g Ž1rc, b . and a C 1-class function ˆ uŽ?, a. defined on Ž1rc, a0 . Ž resp. Ž a0 , b .. such that for each a g Ž1rc, a0 . Ž resp. a g Ž a0 , b .., Ž1.3. with s s 0 and u 0 s Ž0, a. Ž resp. u 0 s Ž1, 0.. has a unique nonconstant solution ˆ uŽ j , a. up to the translation. We find from w5x that for each a, ˆ uŽ j , a. is an unstable travelling wave of Ž1.1. relative to the space of uniformly continuous functions from R to R 2 with the supremum norm. Combining the above lemma with Theorem

696

NOTE

1.1, we obtain the following: THEOREM 2.2. Let a g Ž1rc, a0 . Ž resp. a g Ž a0 , b .., and let uŽ j . be an arbitrary nonconstant solution of Ž1.3. with u 0 s Ž0, a. Ž resp. u 0 s Ž1, 0.. for s s s0 g R. Then s0 s 0 holds and there exists t g R such that uŽ j . s ˆ uŽ j q t , a. for any j g R.

3. PROOF OF THEOREM 1.1 We only show the proof for the case u 0 s Ž0, a ¨ ., because the other case can be proved in a similar manner. Contrary to the conclusion, we assume that Ž1.3. has a nonconstant solution uŽ j . s Ž u, ¨ .Ž j . for s s s0 Ž/ 0.. Since uŽ2 l y j . is a nonconstant solution of Ž1.3. with s s ys0 for each l g R, we may assume s0 ) 0 without loss of generality. Furthermore by ŽH.2., ŽH.3., and the comparison principle, we have 0 - uŽ j . - a u and 0 - ¨ Ž j . - a ¨ for any j g R. The linearized operator of Ž1.3a. around u s u 0 can be represented as



pu

d

ž /

0

dj

gu Ž u 0 .



d

ž / dj

0

,

where puŽg . s g 2 q s0 g q f uŽu 0 . and p¨ Žg . s dg 2 q s0 g q g ¨ Žu 0 .. Since f u Ž u 0 . - 0,

g u Ž u 0 . - 0,

g¨ Ž u 0 . - 0

are satisfied because of ŽH.1. and ŽH.3., we can define gn" Žgny- 0 - gnq . by the solutions of pn Žg . s 0 Ž n s u, ¨ .. We set G1"s min  gu", g¨" 4 ,

G2"s max  gu", g¨" 4 ,

m "s 2 ya gu", g¨" 4 ,

where aA is the number of elements of the set A. By definition, we have

guyq guqs ys0 Ž - 0 . ,

g¨yq g¨qs ys0rd Ž - 0 . ,

Ž 3.1a.

and then obtain G1qq G2yF y

s0 d

min Ž 1, d . Ž - 0 . .

Ž 3.1b.

697

NOTE

Let us obtain the estimate for uŽ j . in a neighborhood of j s "`. Since uŽ j . satisfies Ž puŽ drd j . q oŽ1.. uŽ j . s 0 as j ª y`, we have q

gu j u Ž j . s Cy Ž 1 q o Ž 1. . u e

as j ª y`,

Ž 3.2a.

where Cy u is a positive constant. From the above estimate, we have 0 s d¨jj Ž j . q s0¨j Ž j . q g Ž u Ž j . . s

d

ž ž / p¨

dj

q o Ž 1.



¨Ž j

q

. y a ¨ . y C1 e g u j Ž 1 q o Ž 1 . .

as j ª y` and then obtain ¨Ž j

. y a¨

¡C e 1

s

~

gq uj

p¨ guq

Ž

q

Ž 1 q o Ž 1 . . q C2 eg j Ž 1 y o Ž 1 . . ¨

.

if guq / g¨q ,

q

C1 < j < e g u j y Ž 1 q o Ž 1. . d q p Žg . dg ¨ u

if guqs g¨q

¢

Ž) 0., and C2 is a suitable constant. as j ª y`, where C1 s yg uŽu 0 .Cy u By the definition of guq and g¨q, we have d dg

q

p¨ Ž g¨

p¨ guq

. ) 0,

Ž

.

½

- 0 if guq- g¨q, ) 0 if guq) g¨q.

Since 0 - ¨ Ž j . - a ¨ holds for any j g R, we see that C2 - 0 must be satisfied when guq) g¨q holds. Hence we have ¨Ž j

q

q

< < m e G1 j Ž 1 q o Ž 1 . . . s a ¨ y Cy ¨ j

as j ª y`,

Ž 3.2b.

where Cy ¨ is a positive constant. In a similar manner, we can obtain y

gu j u Ž j . s Cq Ž 1 q o Ž 1. . , u e

¨Ž j

y

y

< < m e G2 j Ž 1 q o Ž 1 . . . s a ¨ y Cq ¨ j

Ž 3.2c.

698

NOTE

q Ž . Ž . as j ª q`, where Cq u and C¨ are positive constants. By 3.1 and 3.2 , we obtain

uŽ 2 l y j . uŽ j . ¨ Ž2 l y j ¨Ž j

. y a¨

. y a¨

q

s

2g u l Cq u e

Cy u y

s

e s 0 j Ž 1 q o Ž 1 . . - 1, y

< < m e 2 G2 l Cq ¨ 2l y j y<

C¨ j < m

q

y

q

eyŽ G2 qG1

.j

Ž 1 q o Ž 1. . - 1

as j ª y` for any fixed l g R, which implies that uŽ 2 l y j . - uŽ j . , ¨ Ž2 l y j

. ) ¨ Ž j . in a neighborhood of j s y`

Ž 3.3.

hold for any fixed l g R. We set UŽ j , l. Žs ŽU, V .Ž j , l.. ' uŽ j . y uŽ2 l y j .. Let us show that there exists l0 g R such that V Ž j , l. F 0 F UŽ j , l. holds on Žy`, lx for any l G l0 . Since uŽ j . is not oscillatory near j s "`, we obtain y` jyF jq- q`, where jys minŽ jyu , jy¨ ., jqs maxŽ jqu , jq¨ .,

jyu s sup  t < uj Ž j . G 0 on Ž y`, t 4 , jqu s inf  t ¬ uj Ž j . F 0 on w t , q` . 4 , jy¨ s sup  t ¬ ¨j Ž j . F 0 on Ž y`, t 4 , jq¨ s inf  t ¬ ¨j Ž j . G 0 on w t , q` . 4 . We assume that there exists t Ž) jq . such that uj Žt . ¨j Žt . s 0 holds. By ŽH.3., we have ujj Ž t . s 0,

0 G ujjj Ž t . s yf¨ Ž u Ž t . . ¨j Ž t . G 0 when uj Ž t . s 0,

¨jj Ž t . s 0,

0 F d¨jjj Ž t . s yg u Ž u Ž t . . uj Ž t . F 0 when ¨j Ž t . s 0,

which implies that both of u j Žt . s 0 and u jj Žt . s 0 hold. Since u j Ž j . is a solution of 0 s D wjj q s0 wj q f u Ž u Ž j . . w,

j g R,

Ž 3.4.

699

NOTE

we find that uŽ j . must be a constant function. This is a contradiction. Hence we obtain uj Ž j . - 0 - ¨j Ž j .

for any j ) jq .

Ž 3.5.

Since Uj Ž j , l . s uj Ž j . q uj Ž 2 l y j . - 0, Vj Ž j , l . s ¨j Ž j . q ¨j Ž 2 l y j . ) 0 holds on w jq, l. for any l G jq, we have V Ž j , l. - 0 - U Ž j , l.

on w jq , l .

Ž 3.6a.

for any l G jq because of UŽ l, l. s 0. It follows from Ž3.3. that there exists j 1 ŽF jy . such that V Ž j , jq . - 0 - UŽ j , jq . on Žy`, j 1 x. Since Ul Ž j , l . s y2 uj Ž 2 l y j . ) 0,

Vl Ž j , l . s y2 ¨j Ž 2 l y j . - 0

are satisfied on Žy`, l. for any l G jq because of Ž3.5., we have V Ž j , l. - 0 - U Ž j , l.

on Ž y`, j 1

Ž 3.6b.

for any l G jq. By Ž3.5., we can take j 2 ŽG jq . as satisfying uŽ j 2 . F

min

jg w j 1 , jq x

uŽ j . ,

¨ Ž j2. G

max

jg w j 1 , jq x

¨Ž j .,

and then obtain V Ž j , l. F ¨ Ž j . y ¨ Ž j 2 . F 0 F u Ž j . y u Ž j 2 . F U Ž j , l.

on w j 1 , jq x

Ž 3.6c.

for any l G j 2 . Hence we arrive at V Ž j , l. F 0 F UŽ j , l. on Žy`, lx for any l G j 2 . Setting

l0 s inf  l ¬ V Ž j , t . F 0 F U Ž j , t . on Ž y`, t for any t G l4 , we have l0 F j 2 , and V Ž j , l0 . F 0 F UŽ j , l0 . on Žy`, l0 x. By the definition of jqu , we find that there exists j 3 Ž- jqu . such that uj Ž j . - 0 for any j g Ž j 3 , jqu .. Since UŽ l, l. s 0 and Uj Ž l, l. s 2 uj Ž l. - 0 hold for any l g Ž j 3 , jqu ., we see that for each l g Ž j 3 , jqu ., UŽ j , l. - 0 is satisfied for some j - l. By the definition of l 0 , we obtain jqu F l0 . Similarly we can prove jq¨ F l 0 . Hence we have jqF l0 .

700

NOTE

We consider the case l 0 s jqu . By definition, we have uj Ž l0 . s 0,

ujj Ž l0 . F 0,

U Ž l0 , l 0 . s 0,

Uj Ž l0 , l0 . s 0,

¨j Ž l 0 . G 0,

Ujj Ž l 0 , l0 . s 0.

By s0 ) 0 and ŽH.3., we obtain 0 G Ujjj Ž l 0 , l 0 . r2 s ujjj Ž l0 . s ys0 ujj Ž l0 . y f¨ Ž u Ž l 0 . . ¨j Ž l0 . G 0 and then find that ujj Ž l0 . s 0 and ¨j Ž l0 . s 0 are satisfied. Similarly we can prove ¨jj Ž l0 . s 0 by the change of the role of u and ¨ in the above argument. Hence we have u j Ž l0 . s 0 and u jj Ž l0 . s 0. Since u j Ž j . is a solution of Ž3.4., we see from uniqueness that uŽ j . must be a constant function. This contradiction implies that l0 ) jqu holds. Analogously we can prove l 0 ) jq¨ . Therefore we obtain l0 ) jq. We see from Ž3.6. that there exists j 4 g w j 1 , jq x such that UŽ j 4 , l0 .V Ž j 4 , l0 . s 0 holds. If UŽ j 4 , l0 . s 0 is satisfied, then we have Uj Ž j 4 , l0 . s 0 and 0 F Ujj Ž j 4 , l0 . s s0 uj Ž 2 l0 y j 4 . y s0 uj Ž j 4 . q f Ž u Ž 2 l0 y j 4 . . y f Ž u Ž j 4 . . s 2 s0 uj Ž 2 l 0 y j4 . y C3V Ž j 4 , l0 . - 0 because of s0 ) 0, ŽH.3., and Ž3.5., where C3 s

1

H0 f Ž u u Ž j . q Ž 1 y u . u Ž 2 l ¨

4

0

y j 4 . . du Ž - 0 . .

Similarly we can derive a contradiction when V Ž j 4 , l0 . s 0 is satisfied. These contradictions imply that s0 s 0 holds. Thus the proof of Theorem 1.1 is completed.

REFERENCES 1. R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations 44 Ž1982., 343]364. 2. B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 Ž1979., 209]243. 3. Y. Hosono and M. Mimura, Singular perturbation approach to traveling waves in competing and diffusing species models, J. Math. Kyoto Uni¨ . 22 Ž1982r1983., 435]461. 4. Y. Kan-on, Existence of standing waves for competition]diffusion equations, Japan J. Indust. Appl. Math. 13 Ž1996., 117]133. 5. Y. Kan-on, Instability of stationary solutions for a Lotka]Volterra competition model with diffusion, J. Math. Appl. Anal. 208 Ž1997., 158]170.