Existence of travelling waves with their minimal speed for a diffusing Lotka–Volterra system

Existence of travelling waves with their minimal speed for a diffusing Lotka–Volterra system

Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524 www.elsevier.com/locate/na Existence of travelling waves with their minimal speed for ...

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Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524 www.elsevier.com/locate/na

Existence of travelling waves with their minimal speed for a di%using Lotka–Volterra system Ning Fei∗ , Jack Carr Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK Received 13 March 2002; accepted 22 October 2002

Abstract Travelling waves are natural phenomena ubiquitously for reaction–di%usion systems in many scienti5c areas, such as in biophysics, population genetics, mathematical ecology, chemistry, chemical physics, and so on. It is pretty well understood for a di%using Lotka–Volterra system that there exist travelling wave solutions which propagate from an equilibrium point to another one. In this paper, we prove there exists, at least, a wave front—the monotone travelling wave— with its minimal speed. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Travelling waves; Minimal speed; Shooting argument

1. Introduction Consider the di%using Lotka–Volterra system ut = uxx + (1 − u − a1 v)u; vt = d vxx + r(1 − a2 u − v)v;

(1.1)

where u(x; t) ¿ 0, v(x; t) ¿ 0, (x; t) ∈ R×R+ and a1 ; a2 ; d; r are all positive constants. We look for a monotone travelling wave solution of (1.1) u(x; t) = u( );

v(x; t) = v( );

= x + ct ∈ R;

 This work was supported by a scholarship from the Department of Mathematics at Heriot-Watt University, United Kingdom. ∗

Corresponding author. Tel.: +44-131-451-3225; fax: +44-131-451-3249. E-mail addresses: [email protected] (N. Fei), [email protected] (J. Carr).

1468-1218/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1468-1218(02)00077-9

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with minimal speed c = cmin ¿ 0 under the boundary value conditions (u− ; v− ) = (u(−∞); v(−∞))

(u+ ; v+ ) = (u(+∞); v(+∞))

and

which are equilibrium points of (1.1). Translating by travelling wave solution, we get two second order semi-linear ordinary di%erential equations u

− c u + (1 − u − a1 v)u = 0; d v

− c v + r(1 − a2 u − v)v = 0:

(1.2)

There are four equilibrium points of (1.2) in the form of (u; v; u ; v ) as Y1 = (0; 0; 0; 0); Y2 = (1; 0; 0; 0);

and





Y4 = (u ; v ; 0; 0) =

Y3 = (0; 1; 0; 0);



 a1 − 1 a2 − 1 ; ; 0; 0 : (1.3) a1 a2 − 1 a1 a2 − 1

In [13], Tang and Fife, by constructing an absorbing area, treated (1.2) under the boundary value condition (u− ; v− ) = (0; 0);

(u+ ; v+ ) = (u∗ ; v∗ ):

In [2,4], Gardner and Conley, by Leray–Schauder degree, investigated (1.2) under the boundary value condition (u∗ ; v∗ ) to be one of (u− ; v− ), (u+ ; v+ ). In [6], Kanel and Zhou, by shooting argument, studied (1.2) for the boundary value condition (u− ; v− ) = (u∗ ; v∗ );

(u+ ; v+ ) = (1; 0):

In [5], Hosono assumed the natural condition 0 ¡ a1 ¡ 1 ¡ a2

(1.4)

which are di%erent conditions from those in paper [2,4,6,13]. By a1 − 1 a2 − 1 u∗ × v ∗ = × ¡ 0; a1 a2 − 1 a1 a2 − 1

(1.5)

we do not discuss (u∗ ; v∗ ) to be one of boundary value conditions as (u− ; v− ) or (u+ ; v+ ) since both u and v must be nonnegative in Lotka–Volterra model by practical meaning. As for the boundary value conditions of (u− ; v− ) = (0; 1);

(u+ ; v+ ) = (1; 0);

Murray [10] predicted the minimal wave speed c given by  c∗ := 2 1 − a1

(1.6) (1.7)

provided there exists a travelling wave front solution for (1.1), (1.4), (1.6). Kan-on [7], by the method of continuation, proved that there exists a constant c0 ¿ 0 such that for each c ¿ c0 , the boundary value problem of (1.1), (1.4), (1.6) has a travelling wave front. He gave his estimation of the speed c = c0 ¿ c∗ .

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505

Hosono [5], by the method of numerical computation, gave the speed error estimation between the computational speed ccomp (¿ cmin ) and c∗ , where cmin := inf {c: there exists a traveling wave front for c}: We know that (see Section 2) the signs of eigenvalues at equilibrium point Y1 are 11 ¿ 0;

12 ¡ 0;

13 ¿ 0;

14 ¡ 0;

and the second components of corresponding eigenvectors are    1 1 r 1j = − 1 c1j + (a2 − 1) − 1 ; (j = 3; 4): a1 d d In paper [5,11], a same example was mentioned for a special case: d = 1, r = 1, a1 + a2 = 2 for travelling wave front of (1.1) where cmin = c∗ is valid. In their case, we 5nd 13 = 14 = −1 and guess our limit conditions should be guided by 12 ¡ 14 ¡ 0 ¡ 13 ¡ 11 ;

−1 6 14 ¡ 0;

d = 1:

We assume our limited condition 1 − a1 6 r(a2 − 1) ¡ 1;

d = 1:

(1.8)

Theorem 1.1 (Monotone travelling wave solution). (I) Under the conditions (1.4), (1.8), for the boundary value problem of (1.1), (1.6), there exist positive increasing travelling wave solutions (u(x + ct); v(x + ct)) with speed c satisfying  c ¿ 2 r(a2 − 1): (II) There do not exist travelling wave solution (u(x + ct); v(x + ct)) with speed c if 1=2   r c6 (2 − 2 ) + r ln 2  where =

a2 − 1; |14 |

=

a2 − a2 ; |14 |

|14 | =

1 − r(a2 − 1) : a1

Theorem 1.2 (Non existence of travelling wave solution). There does not exist travelling wave front solution (u; v) of problem (1.1), (1.4) under boundary condition (u− ; v− ) = (1; 0); (u+ ; v+ ) = (0; 0)

or

(u− ; v− ) = (0; 1); (u+ ; v+ ) = (0; 0):

Steven in [12], Kapel in [8], Alan, etc. in [1] dealt with monotone travelling wave solutions of Reaction Di%usion equations which are di%erent from system (1.1). We got useful idea for our question from their papers. In this paper, several mathematical tools have been used, such as Maximum Principle, Inverse Function Theory, Projection of Shooting Argument, Minimum Value Lemma for Fisher scalar equation, Compared Lemma, local and global monotonicity on parameter.

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The rest of this paper is organized as follows. 1. Translating into ordinary di%erential system 2. Analysis of the phase plane 2.1. Constructing a local explicit solution 2.2. Constructing an available area on uv-phase plane for Maximum Principle 2.3. Extending the local explicit solution 3. Two famous lemmas 3.1. Fisher–Kolmogorov–Petrovskii–Piskunov minimum value lemma 3.2. Compared lemma 4. Shooting argument of projection 5. Monotonicity on parameter 5.1 Local monotonicity on parameter 5.2. Global monotonicity on parameter 6. Existence of monotone travelling wave fronts 7. Estimating the minimal speed 8. Discussion 8.1. SuNcient and necessary conditions about travelling wave speed 8.2. An example and an open problem 2. Translating into ordinary dierential system Set Y = (y1 ; y2 ; y3 ; y4 )T , where y1 = u( ) ¿ 0; y2 = v( ) ¿ 0; y3 = u ( ); y4 = v ( ); ( = x + ct ∈ R): Then (1.1) can be translated into an ordinary di%erential system  dy   3 = cy3 + (−1 + y1 + a1 y2 )y1 ;  d

 c r dy   4 = y4 + (−1 + a2 y1 + y2 )y2 : d

d d

(2.1)

There are four equilibrium points for (2.1): Y0 = (0; 0; 0; 0)T ; Y1 = (1; 0; 0; 0)T ;

 and

Y3 =

T

Y2 = (0; 1; 0; 0) ;

a1 − 1 ; a1 a2 − 1

a2 − 1 ; 0; 0 a1 a 2 − 1

T :

(2.2)

Y0 is an unstable point because the real parts of its eigenvalues at Y0 of the linearization of (2.1) are all positive numbers. Therefore, Theorem 1.2 holds. Y1 is a saddle point. Its four real eigenvalues are   1 1 [c + c2 + 4rd(a2 − 1)] ¿ 0; 11 = [c + c2 + 4] ¿ 0; 13 = 2 2d  1 12 = [c − c2 + 4] ¡ 0; 2

14 =

 1 [c − c2 + 4r d(a2 − 1)] ¡ 0: 2d

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Their corresponding eigenvectors are     1 1      0   0      p11 =   ; p12 =  ;  11   12      0

0



1



   13    p13 =  ;  13    13 13



1

507



   14    p14 =  ;  14    14 14 (2.3)

where, by limited condition (1.8), 12 ¡ 14 ¡ 0; 1 ¡  := 12 =14 ¡ + ∞, and    1 1 r 1j := (2.4) − 1 c1j + (a2 − 1) − 1 ; (j = 3; 4): a1 d d By limit condition (1.8), we have −1 = 14 , or −1 ¡ 14 ¡ 0. Y2 is a saddle point. Its four real eigenvalues are     1 1  c + c2 − 4(1 − a1 ) ¿ 0; 23 = c + c2 + 4 dr ¿ 0; 21 = 2 2d      1  1 c − c2 + 4 dr ¡ 0; c − c2 − 4(1 − a1 ) ¿ 0; 24 = 22 = 2d 2 where



c ¿ c∗ := 2

1 − a1 :

(2.5)

(2.6)

We do not consider the boundary condition (u− ; v− ) or (u+ ; v+ ) to be the equilibrium point Y3 because of (1.5). 3. Analysis of the phase plane The phase space of Eqs. (2.1) is a R4 space in the form of (u; v; u ; v ). In order by applying shooting argument of projection to prove the existence of travelling wave solutions of the boundary problem (1.1), (1.4), (1.6), with the minimal speed cmin ¿ 0, we analyze the (u; v)-projection of Phase Space as follows. 3.1. Constructing a local explicit solution On the uv-phase plane, we take the projection point A = (1; 0) of equilibrium point Y1 = (1; 0; 0; 0) to be our starting point (u+ ; v+ ) = (1; 0). As the independent variable

= x + ct ∈ R goes back from +∞ to −∞, we check the properties of trajectories of the initial problem  u

− c u + (1 − u − a1 v)u = 0;    d v

− c v + r(1 − a2 u − v)v = 0; (3.1)    u(+∞) = 1; v(+∞) = 0:

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By Tailor’s expression Y = Y1 + k11 e11 · p11 + k12 e12 · p12 + k13 e13 · p13 + k14 e14 · p14 + O((Y1 − Y )2 ) and 0 ¡ a1 ¡ 1 ¡ a2 d = 1;

(natural condition)

1 − a1 6 r(a2 − 1) ¡ 1

(limited condition)

we express the local explicit solution (3.2) of the initial problem (3.1) in a left neighborhood of → +∞ in the form u = 1 + k14 · e14 + k12 · e12 + O((1 − u)2 ) = 1 − s + k · s + O(s2 ) = u(s; k) → 1−

for → + ∞

v = 0 + k14 · 14 · e14 + O(v2 );

or s := |k14 |e14 → 0+

= |14 |s + O(s2 ) = v(s; k) → 0+ ;

(3.2)

where k11 = k13 = 0 ¿ k14 , k := k12 · |k14 |− , −1 6 14 ¡ 0, and 12 ¡ 14 ¡ 0 ¡ 13 ¡ 11 ; We have two equivalent forms  u ( ; k ) → 0+ ;    v ( ; k ) → 0− ; ⇔    for → +∞;

1 ¡  :=

12 ¡ + ∞: 14

in derivative  us ( s; k ) → 0− ;    vs ( s; k ) → 0+ ;    for s → 0+ :

(3.3)

(3.4)

By inverse function theorem, we have an inverse function s = s(u; k) from u = u(s; k) since us (s; k) = 0, So, we have the inverse function = (u; k) from u=u( ; k) in a left positive in5nity neighborhood → +∞ by translation s := −k14 e14 where k14 ¡ 0. We regard k := k12 · |k14 |− as a parameter and translate the explicit solution (3.2) into v = v(u; k) in a local neighborhood u ∈ [1 − ; 1) for suNcient small  ¿ 0,  

= (u; k); u = u( ; k );        v = v(u; k); v = v( ; k); ⇔ v = v( ; k ); ⇔ (3.5)   for u → 1− :     −

→ +∞; u→1 ;

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509

v B C

u O

D

A

Fig. 1. Available area for maximum principle.

3.2. Constructing an available area on uv-phase plane for Maximum Principle On (u; v) phase plane, we set quadrilateral ABCD by its four vertexes (Fig. 1) A = (1; 0) 

1 B = 0; a1

the projection of Y1 = (1; 0; 0; 0); 

C = (0; 1)

the intersection of 1 − u − a1 v = 0 with u = 0;

 D=

the projection of Y2 = (0; 1; 0; 0); 

1 ;0 a2

the intersection of 1 − a2 u − v = 0 with v = 0:

The quadrilateral ABCD is called an Available Area where we can apply Maximum Principle for (u( ; k); v( ; k)) and investigate monotonicity of (u( ; k); v( ; k)) on its parameter k. Lemma 3.1. There exists (u( ; k); v( ; k)) to be a solution of initial problem of (3.1), (1.4), (1.6), such that u ¿ 0;

v ¿ 0;

1 − u − a1 v ¿ 0;

1 − a2 u − v 6 0

for → +∞:

That is, orbit (u( ; k); v( ; k)) stays in the quadrilateral ABCD for → +∞. Proof. By explicit solution form (3.2), (3.3), we have u = 1 − s + ks + O(s2 ) ¿ 0 2

v = |14 |s + O(s ) ¿ 0

for s → 0+ ;

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and then 1 − u − a1 v = (1 − a1 |14 |)s − ks + O(s2 ) = r(a2 − 1)s − ks + O(s2 ) ¿ 0; 1 − a2 u − v = −(a2 − 1) + (a2 − |14 |)s − a2 ks + O(s2 ) ¡ 0: By local translation s = |k14 | e14 , (14 ¡ 0), for → +∞, (u( ; k); v( ; k)) stays in the quadrilateral ABCD and u( ; k) ↑ is a strictly increasing function of variable in its positive in5nite neighborhood of → + ∞ with its 5xed parameter k while v( ; k) ↓ is a strictly decreasing function of variable . For example, if = M , a suNcient large positive constant, then u (M; k) ¿ 0

and

v (M; k) ¡ 0:

On the uv-phase plane, orbit v(u; k) stays in the quadrilateral ABCD, provided u ∈ [1 − ; 1) where u(M; k) = 1 − . 3.3. Extending the local explicit solution Set ! = − , then, for (u; v) in the quadrilateral ABCD, we have  u!! + cu! = (−1 + u + a1 v)u 6 0 as ! ↑−∞ : dv!! + cv! = r(−1 + a2 u + v)v ¿ 0 By Maximum Principle, u(!; k) cannot arrive its local minimum at any 5xed value of ! while v(!; k) cannot arrive its local maximum at any 5xed value of !. Then by u = −u! ¿ 0, v = −v! ¡ 0, we have  u( ; k) has no local maximum at if (u; v) ∈ ABCD as ↓+∞ : v( ; k) has no local minimum at if (u; v) ∈ ABCD Now, we extend to left side of its positive in5nite neighborhood → +∞. That is,

∈ [m; M ], where m could be a negative constant. Then, u( ; k) is a strictly increasing function for ∈ [m; M ] and v( ; k) is a strictly decreasing function since at = M , u (M; k) ¿ 0 and v (M; k) ¡ 0. (u( ; k); v( ; k)) can not stop at any point in the quadrilateral ABCD if → −∞, since there is no equilibrium point there by (3.2). So, we have u ( ; k) ¿ 0;

v ( ; k) ¡ 0

for ∀[m; +∞):

Since u ( ; k) ¿ 0 for ∀ ∈ [m; +∞), we have inverse function = (u; k) from u = u( ; k). Staring from point A = (1; 0) for trajectory v(u; k) in the quadrilateral ABCD, as ↓+∞ , v(u; k) does not stay at any point in the quadrilateral ABCD. v(u; k) could intersect four sides AB, BC, CD, DA.

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511

4. Two famous lemmas Considering Fisher equation dp=du = c − f(u)=p, f(u) ∈ C 1 [0; 1] with speed c ¿ 0, we have Lemma 4.1 (Fisher–Kolmogorov–Petrovskii–Piskunov minimum value lemma). Let f(0) = f(1) = 0, f (0) ¿ 0, f (1) ¡ 0, f(u) ¿ 0, and f (u) ¡ f (0) for u ∈ (0; 1). There exists a unique solution p = p(u) such that p(1) = 0, p(u) ¿ 0 for u ∈ (0; 1). Also,   p(0) ¿ 0 if c ¡ 2 f (0): (4.1) p(0) = 0 if c ¿ 2 f (0); This is Minimum Value Lemma. Its proof can be found in [3,9]. Lemma 4.2 (Compared Lemma). Let pi = pi (u), (i = 1; 2), be solutions of Fisher equations dpi =du=c−(fi (u)=pi ) in some interval (u0 ; 1) ⊂ (0; 1); pi (1)=0, pi (u) ¿ 0, fi (u) ∈ C(u0 ; 1), f1 (u) ¡ f2 (u) for u ∈ (u0 ; 1), and 0 6 f1 (u) on [u1 ; 1) ⊂ (u0 ; 1), then 0 ¡ p1 (u) ¡ p2 (u)

for u ∈ (u0 ; 1):

(4.2)

This Compared Lemma is from [6]. 5. Shooting argument of projection Lemma 5.1. The projection (u( ; k); v( ; k)) cannot stay in the interior of the quadrilateral ABCD if → −∞ (Fig. 2). Proof. Since there is no equilibrium point of system (3.1) in the interior of the quadrilateral ABCD and by maximum principle, u ¿ 0 and v ¡ 0 if (u; v) is in the quadrilateral ABCD, then the projection (u( ; k); v( ; k)) cannot stay in the interior of the quadrilateral ABCD if → −∞.

v

v B

B 1 - u - a1 v = 0 C

1 - a2 u - v = 0

v = σ (u) (1- u)

C

Limit condition

Limit condition L

ρ14

k2<0 O

k1>0

L

k1>0

k2<0 D A impossible

u

O

D A impossible

Fig. 2. Shooting argument of projection.

u

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Proof. By inverse function theorem, it is enough for us to get inverse function =

(u; k) from u = u( ; k) by proving u ( ; k) ¿ 0; ∀ ∈ [m; M ]. Set translation = −!, we have  u = u( ; k) = u(!; k); u = −u! ; (5.1) v = v( ; k) = v(!; k); v = −v! : If (u; v) is in the quadrilateral ABCD, from Eqs. (1.8), we have  u!! + cu! = −(1 − u − a1 v)u 6 0; dv!! + cv! = −r(1 − a2 u − v)v ¿ 0:

(5.2)

By Maximum Principle on elliptic di%erential equations (5.2), respectively, u(!; k) cannot arrive its positive minimum value and v(!; k) cannot arrive its positive maximum value. This fact tells us that u! will keep its negative sign while v! will keep its positive sign when ! increases from −∞ to any bounded value. So, we have u ( ; k) ¿ 0;

v ( ; k) ¡ 0;

∀ ∈ [m; +∞):

(5.3)

Lemma 5.3. If (u( ; k); v( ; k)) is the solution of initial problem of (3.1), (1.4), (1.8), then (u( ; k); v( ; k)) cannot intersect the side AD of the quadrilateral ABCD for any real parameter k ∈ R where ∈ [m; +∞). Proof. Since u ( ; k) ¿ 0; ∀ [m; +∞), by the inverse function theorem, we obtain the inverse function = (u; k) from u = u( ; k). There are exact de5nitions for v = v(u; k) := v( ; k); p = p(u; k) = p( ; k) := u (¿ 0); q = q(u; k) = q( ; k) := v (¡ 0): From (1.8), we have  dp (−1 + u + a1 v)u  ;   du = c + p  dq q r(−1 + a2 u + v)v  d =c + ; du p p and dv q = ; du p

  dq 1 d2 v dp : p = − q du2 p2 du du

(5.4)

(5.5)

(5.6)

By limit condition (1.8), (where d = 1), we have p2

 d2 v dv r(1 − a2 u − v)v  c + (−1 + u + a v)u + = − c q = 0: 1 du2 du d d

(5.7)

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Since r(1 − a2 u − v)v ¡ 0, by Maximum Principle, v = v(u; k) cannot achieve its interior positive maximum in the quadrilateral ABCD. Otherwise, if v intersected the straight segment AD at an another point except the point A = (1; 0), then v(u; k) would achieve its maximum point. This is a contradiction. Lemma 5.4. If v=v(u; k) is a solution of initial problem of (3.1) with natural condition (1.4) and limited condition (1.8), then v = v(u; k) can intersect the side AB from the inside of quadrilateral ABCD for some positive parameter k = k1 ¿ 0. Proof. For any small s1 in local neighborhood of s → 0+ , we set k = k1 = s11− ¿ 0 and have 1 − u(s1 ; k1 ) − a1 v(s1 ; k1 ) = (1 − a1 |14 |)s1 − k1 s1 + O(s12 ) = −a1 |14 |s1 + O(s12 ) ¡ 0: For this 5xed k1 ¿ 0, if we take s ¿ 0 suNciently small, then 1 − u(s; k1 ) − a1 v(s; k1 ) = (1 − a1 |14 |)s − k1 s + O(s2 ) ¿ 0: The projection v = v(s; k1 ) can intersect the line 1 − u − a1 v = 0 by changing small positive variable s, which means that v = v(u; k1 ) can intersect the side AB from the inside of quadrilateral ABCD in the neighborhood u → 1− . Lemma 5.5. If v = v(u; k) is the solution of initial problem of (3.1), (1.4), (1.8), then v = v(u; k) can intersect the side CD from the inside of the quadrilateral ABCD for some negative parameter k = k2 ¡ 0. Proof. Set translation w = w(u; k) :=

v 1−u

(5.8)

we have

    1 1 q dv dw = +w (1 − u) + v = du (1 − u)2 du 1−u p

and d2 w dw − [2p2 − u(1 − u)(−1 + u + a1 v)] + H (u; v)w = 0 du2 du where H (u; v) := r(1 − a2 u − v)(1 − u) + (1 − u − a1 v)u. It is equivalent between H (u; v) = 0 and v = )(u)(1 − u), where (1 − ra2 )u + r )(u) := ; ∀u ∈ [0; 1]: (a1 − r)u + r p2 (1 − u)

Set limit line (corresponding limited condition (1.8)) 1 v = 0; (−1 6 14 ¡ 0); L: 1−u− |14 |

(5.9)

(5.10)

(5.11)

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we have following two results: 1. The curve v = )(u)(1 − u) is through the two points A = (1; 0); C = (0; 1). 2. The curve v = )(u)(1 − u) is above the straight line L for u ∈ (0; 1). In fact, we can extend v = h(u) = )(u)(1 − u) at u = 1 to get Tailor’s Extension v = h(u) = |14 |(1 − u) + 12 h (1)(1 − u)2 + · · · ;

for u → 1− ;

where we used limit condition (1.8) for −1 6 14 =(r(a2 −1)−1)=a1 ¡ 0 and ∀u ∈ [0; 1], h (u) = ) (u)(1 − u) − 2) (u) =

2r[r(a2 − 1) − (1 − a1 )] (a1 − r)(1 − u) [(a1 − r)u + r]3 +

2r[r(a2 − 1) − (1 − a1 )] ¿ 0: [(a1 − r)u + r]2

By Maximum Principle on elliptic equation (5.9), w(u; k) cannot achieve its positive interior minimum value if H (u; v) ¿ 0. In following analysis, we look for a parameter k2 ¡ 0 and apply Maximum Principle to 5nd a contradiction if (u( ; k2 ); v( ; k2 )) would not intersect the side CD of the quadrilateral ABCD by decreasing from +∞ to −∞. The explicit solution (1.8) satis5es 1 v(s; k) = −ks + O(s2 ) for  ∈ (1; +∞): 1 − u(s; k) − (5.12) |14 | For small s2 ¿ 0, we choice k = k2 = −s2− ¡ 0, then 1 1 − u(s2 ; k2 ) − v(s2 ; k2 ) ¿ 0; |14 |

(5.13)

which means (u(s2 ; k2 ); v(s2 ; k2 )) is below the straight line L. We assert the solution v=v(u; k2 ) is below L for u ∈ (0; 1]. In fact, if v=v(u; k2 ) intersects with L at (u; ˆ v(u; ˆ k2 )) and (u; v(u; k2 )) is below L for uˆ ¡ u ¡ u(s2 ; k2 ) ¡ 1, then from the two de5nitions of w(u; k) in (5.8) and L in (5.11), we have (Fig. 3) w(u; ˆ k2 ) = |14 |; w(u; k2 ) ¡ |14 |

for uˆ ¡ u ¡ u(s2 ; k2 ) ¡ 1;

v = |14 |: 1−u w(u; k2 ) achieves its positive interior minimum at some u∗ ∈ (u; ˆ 1). (u∗ ; v(u∗ ; k2 )) is ∗ ∗ below the line L so that H (u ; v(u ; k2 )) ¿ 0. By Maximum Principle, w cannot achieve its positive interior minimum. This is a contradiction. Since w(1; k2 ) = |14 |

since lim+ w(u(s; k2 ); k2 ) = lim s→0

u→1−

v = v(u; k2 )

cannot stay in the quadrilateral ABCD;

v = v(u; k2 )

cannot intersect the side DA;

v = v(u; k2 )

cannot intersect straight line L;

(5.14)

N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

v

515

v

B

B

C

C v = σ (u) (1-u)

ρ14

L

L O

D

u

A

O

D

A

u

Fig. 3. Limit condition: 14 ∈ [ − 1; 0).

the projection v = v(u; k2 ) must intersect the side CD if u goes out of the left side of its local neighborhood [1 − ; 1]. 6. Monotonicity on parameter We want to prove the solution v = v(u; k) of (4.1) is global monotone increase depending on its parameter k. That is, set vi = v(u; ki );

pi = p(u; ki );

qi = q(u; ki );

we take two subsections to prove that  Qv = v1 − v2 ¿ 0 k1 ¿ k2 ⇒ Qp = p1 − p2 ¡ 0

(i = 1; 2);

for u ∈ (0; 1):

(6.1)

6.1. Local monotonicity on parameter From (5.10), at neighborhood of u = 1− , v = 0+ , we have I. The local explicit solution (3.2), we have lim (u; v) = (1; 0);

s→0+

lim+

s→0

dv |14 | + O(s) = lim+ = −|14 | 6 0: du s→0 −1 + ks−1 + O(s)

II. By (5.11), the limit line L is expressed as 1 dvL L: 1 − u − v = 0 with its slope = −|14 |: |14 | du III. If k1 ¿ k2 , then dv1 |14 | + O(s) dv2 |14 | + O(s) ¡ 0: ¡ = = −1 −1 du −1 + k1 s −1 + k2 s du + O(s) + O(s)

(6.2)

(6.3)

(6.4)

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N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

We have dv2 dQv dv1 = − ¿ 0 ⇒ Qv ↑ du du du

for u ∈ [1 − ; 1):

(6.5)

Also, Qv|u=1 = v(1; k1 ) − v(1; k2 ) = 0 + 0 = 0. Therefore, Qv ¡ 0; u ∈ [1 − ; 1). IV. If parameter k1 ¿ k2 , then in the neighborhood of s → 0+ , we have p2 =

du(s; k2 ) ds · = |14 |s − |14 |k2  · s + O(s2 ) ds d

¿ |14 |s − |14 |k1  · s + O(s2 ) = p1 ¿ 0:

Therefore, we have local monotonicity:  Qv = v1 − v2 ¿ 0 If k1 ¿ k2 ⇒ Qp = p1 − p2 ¡ 0

for u ∈ [1 − ; 1):

(6.6)

(6.7)

6.2. Global monotonicity on parameter From (5.5), (5.6), we have  dp (1 − u − a1 v)u   =c−   du p     dq q r(1 − a2 u − v)v =c −  du p p      dv q   = du p

 du    p = d ¿ 0; where  dv  q= ¡ 0: d

Set constant K ¿ 0 to be determined late, we have   1 dv d − [K(1 − u − a1 v)u − r(1 − a2 u − v)v] [Kp − q] = c K − du p du and then  1 u

d [Kp(!) − q(!)] d! d!   1  1  K(1 − ! − a1 v)! − r(1 − a2 ! − v)v dv d! − c K− = d!: d! p u u

By v(1; k) = p(1; k) = q(1; k) = 0, we have Kp(u) − q(u) + cK(1 − u) + cv(u)  1 K(1 − ! − a1 v)! − r(1 − a2 ! − v)v = d!; p u

(6.8)

N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

that is, q(u) 1 1 =K + [cK(1 − u) + cv(u)] − p(u) p(u) p(u) =K +



1

u

517

Q(!) d! p(!)

1 J p(u)

(6.9)

where Q(u) = K(1 − u − a1 v)u − r(1 − a2 u − v)v;  1 1 Q(!) J = cK(1 − u) + cv(u) − d!: p(u) u p(!)

(6.10)

By (5.5), we have (1 − u − a1 v2 )u a1 u d Qp = Qp + Qv du p1 p2 p1

(6.11)

d c Qv = Qv + G(u); du p2

(6.12)

and

where

   1 Qp Q2 (!) 1 dv1 + Qp(!) d! G(u) = K − du p1 p2 p 2 u p 1 p2  1 1 R(!) + Qv(!) d!: p2 u p1

(6.13)

In the quadrilateral ABCD, we have two inequalities: Q2 (u) = K(1 − u − a1 v2 )u − v2 (1 − a2 u − v2 ) ¿ 0; R(u) = 1 − a2 u − v1 − v2 + Ka1 u ¡ 0;

∀K ∈ R+

∃K ¿ 0:

(6.14)

In fact, we take K = min{K1 ; K2 } ¿ 0 such that (6.14) is hold and then K satis5es all conditions needed above, where   a2 (1 − ) − 1 1 min v(u; k) 0 ¡ K1 ¡ ; 0 ¡ K2 ¡ a1 a1 (1 − ) u∈[0;1−] and



R(u) = [(1 − a2 u − v1 ) + K1 a1 u] − v2 ¡ 0

if u ∈ (1 − ; 1);

R(u) = (1 − a2 u − v1 ) − (v2 − K2 a1 u) ¡ 0

if u ∈ (0; 1 − ]:

So, G(u) ¡ 0

if Qp ¡ 0; Qv ¿ 0; ∀(u; v) ∈ ABCD:

(6.15)

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N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

By solving (6.11), (6.12), we have  u [1 − ! − a1 v2 (!)]! Qp = Qp(1 − ) exp d! p1 (!)p2 (!) 1−  u  u a1 ! [1 − 1 − a1 v2 (1)]1 Qv(!) exp d1 d! + (!) p1 (1)p2 (1) p 1 ! 1− and

 Qv = Qv(1 − ) exp

u

1−

c d! + p2 (!)



u

1−

 G(!) exp

u

!

c d1 d!: p2 (1)

(6.16)

(6.17)

Set u1∗ to be the intersect of v1 (u) with v2 (u) and u2∗ the intersect of p1 (u) with p2 (u), that is, suppose  Qv(u) = v1 (u) − v2 (u) ¿ 0; ∀u ∈ (u1∗ ; 1);       Qp(u) = p1 (u) − p2 (u) ¡ 0; ∀u ∈ (u2∗ ; 1); (6.18) ∗  Qv(u ) = 0;  1     Qp(u2∗ ) = 0; then we have by local monotonicity (6:7) ⇒ Qp(u) ¡ 0 by supposing (6:18) ⇒ Qv(u) ¿ 0 by formula (6:16) ⇒ Qp(u) ¡ 0

for u = 1 − ;

for u ∈ (u1∗ ; 1 − ]; for u ∈ [u1∗ ; 1 − ];

and Qp(u) ¡ 0;

∀u ∈ (u1∗ ; 1 − ) ∪ [1 − ; 1) ⊂ (u2∗ ; 1):

(6.19)

So, u2∗ ¡ u1∗ ¡ 1 − . We check the sign on both sides of (6.18) at u = u1∗ and 5nd a contradiction that the left side of (6.17) vanishes while the right side of (6.17) is positive since Qv(1 − ) ¿ 0

and

G(!) 6 0

for ! ∈ [u1∗ ; 1):

Therefore, the orbits v1 = v(u; k1 ), v2 = v(u; k2 ) cannot intersect each other in the quadrilateral ABCD if k1 ¿ k2 . This is the end of proof of global monotonicity. 7. Existence of monotone travelling wave fronts proof of Theorem 1.1 (part I) By global monotonicity of (u( ; k); v( ; k)) on its parameter k, by classical continuous dependence theory (Gronwall’s inequality) for initial value problem, and by Lemmas 5.1–5.5, there exists only one parameter k0 ∈ (k2 ; k1 ) such that the projection v=v(u; k0 )

N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

519

arrives the vertex point C = (0; 1) from the vertex point A = (1; 0) in the quadrilateral ABCD on uv phase plane as variable goes back from +∞. Set the 5xed value = 0 such that u( 0 ; k0 ) = 0, v( 0 ; k0 ) = 1 and (u( ; k0 ); v( ; k0 )) in the quadrilateral ABCD for ∈ ( 0 ; +∞), then p = u ( ; k0 ) ¿ 0;

q = v ( ; k0 ) ¡ 0;

∀ ∈ ( 0 ; +∞):

 Lemma 7.1. If speed constant c ¿ 2 r(a2 − 1), then u ( 0 ; k0 ) = lim+ p( ; k0 ) = lim+ p(u; k0 ) = 0; u→0

→ 0

v ( 0 ; k0 ) = lim+ q( ; k0 ) = lim+ q(u; k0 ) = 0; u→0

→ 0

Proof. By Lemma 5.5, the projection (u( ; k0 ); v( ; k0 )) is above the limit line L which means 1 − u( ; k0 ) −

1 v( ; k0 ) 6 0; |14 |

∀ ∈ R:

We have [1 − u − a1 v]u 6 (1 − a1 |14 |)(1 − u)u = r(a2 − 1)(1 − u)u:

(7.1)

By Compared Lemma 4.2, 0 ¡ p(u; k0 ) ¡ P(u; k0 )

for 0 ¡ u ¡ 1

(7.2)

where (1 − u − a1 v)u dp =c− p du

dP r(a2 − 1)(1 − u)u =c− : du P  By Fisher Lemma 4.1, we know P(0; k0 ) = 0, if c ¿ 2 r(a2 − 1). We take the limit u → 0+ on both sides of inequality (7.2) to get  p(0; k0 ) = 0; if c ¿ 2 r(a2 − 1): and

Also, the slope dv dv d

q(u; k0 ) = · = du d du p(u; k0 )

(7.3)

is bounded because v = v(u; k0 ) is located in the area by the four sides lines 1 − u − a1 v = 0;

1 − a2 u − v = 0;

1−u−

1 v = 0; 14

Let u → 0+ on the both sides of the slope (7.3), then p(0; k0 ) = 0 ⇒ q(0; k0 ) = 0:

v = 1:

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N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

Therefore, du( 0 ; k0 ) = p( 0 ; k0 ) = 0; d

dv( 0 ; k0 ) = q( 0 ; k0 ) = 0: d

(7.4)

Lemma 7.2. 0 = −∞. Proof. We have three results for orbit (u( ; k0 ); v( ; k0 )): 1. (u( ; k0 ); v( ; k0 )) is a solution of initial problem  u

− cu + (1 − u − a1 v)u = 0;    dv

− cv + r(1 − a2 u − v)v = 0;    u(+∞; k0 ) = 1; v(+∞; k0 ) = 0:

(7.5)

2. (u( ; k0 ); v( ; k0 )) arrives the point C = (0; 1) at = 0 , that is u( 0 ; k0 ) = 0;

v( 0 ; k0 ) = 1:

3. (u( ; k0 ); v( ; k0 )) stays in the quadrilateral ABCD for ∈ [m; M ] ∪ (M; + ∞) where we can use inverse function theory to get = (u; k0 ) from u = u( ; k0 ). We have u = u( ; k0 ) ¿ 0; v = v( ; k0 ) ¿ 0; u = p( ; k0 ) = p(u; k0 ) ¿ 0;

∀ ∈ [m; M ] ∪ (M; +∞);

v = q( ; k0 ) = q(u; k0 ) ¡ 0; where M ¿ 0 is a suNcient large positive number. Especially, u( ; k0 ) is a solution of the boundary problem of scalar equation  u

− cu + (1 − u − a1 v( ; k0 ))u = 0; ∀ ∈ ( 0 ; +∞); u(+∞; k0 ) = 1; u( 0 ; k0 ) = 0; u ( 0 ; k0 ) = 0;

(7.6)

where we regard v( ; k0 ) as a given function. By ODEs theory, we know that u = 0 is an equilibrium point of (7.6). By uniqueness of solution u( ; k0 ) of (7.6), u( ; k0 ) cannot arrives its equilibrium point at a limited value for its independent variable . So, 0 = ±∞. By u(+∞; k0 ) = 1 and u ( ; k0 ) ¿ 0; ∀ ∈ [m; M ] ∪ (M; +∞), we have 0 = +∞. Therefore, 0 = −∞. From the proof above, the solution (u( ; k0 ); v( ; k0 )) of initial problem (3.1) does arrive the equilibrium point Y2 = (0; 1; 0; 0) under the natural condition (1.4) and limit condition (1.8). (u(x + ct; k0 ); v(x + ct; k0 )) is the solution of boundary problem (1.1), (1.4), (1.8), if its speed constant c satis5es  c ¿ 2 r(a2 − 1): This is the end of proof of Theorem 1.1 in Part I.

N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

521

8. Estimating the minimum speed proof of Theorem 1.1 (part II ) By Theorem 1.1 (part I), there exits, at least, a monotone travelling wave  solution of problem (1.1), (1.4), (1.6), (1.8) if the speed parameter satis5es c ¿ 2 r(a2 − 1). If we choose r(a2 − 1) = 1 − a1 , one of limited√condition of (1.8), then the minimal speed can be expressed by√a formula: cmin = 2 1 − a1 . This thing can be checked by (2.5), (2.6). If cmin ¡ 2 1 − a1 , then the eigenvalues 21 , 22 would be complex numbers and the equilibrium point Y2 would be a center or a spiral point. This is a contradiction with monotone travelling wave solution. For r(a2 − 1) = 1 − a1 , we estimate the speed c of problem (1.1), (1.4), (1.6) by compared Lemma 4.2 as follows. Proof of Theorem 1.1 (part II). Set translation u( ; k0 ) = a1 U;

v(u; k0 ) = (1 − V );

P=

then p=

du = a1 P ¿ 0; d

q=

dv = −Q ¡ 0 d

dU ; d

dV ; d

(8.1)

  1 for U ∈ 0; : a1

(8.2)

Q=

We rewrite the second equation of (5.5) by translation (8.1) to be dq r(1 − a2 u − v)v dQ r(1 − V )(a1 a2 U − V ) =c− ⇔ =c− : dv q dV Q

(8.3)

By Lemma 5.5, the trajectory v = v(u; k0 ) is above the line L in (5.11), which means   1 1 1 + 1−u− v(u; k0 ) ¡ 0 ⇔ a1 U ¿ 1 − V; |14 | |14 | |14 | we have r(1 − V )(a1 a2 U − V ) ¿ f(V ); where

f(V ) :=

    0

(8.4)   ;     for V ∈ ;1 ;  

for V ∈

    r(1 − V )(V − )

and  :=

a2 − 1; |14 |

 :=

a2 − a2 : |14 |

0;

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N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

By Compared Lemma 4.2, we have Q(V ) ¿ p∗ (V ); ∀V ∈ (=; 1), where p∗ (V ) is a solution of  dp∗ f(V )    dV = c − p∗ ; (8.5)      p∗  = 0; p∗ (1) = 0; p∗ (V ) ¿ 0;  Integrating (8.5) from = to 1 and from = to V , respectively, we have two forms    1  f(V ) dV = c 1 − ¡ c and p∗ (V ) ¡ cV: ∗ (V ) p  = From



1

=

f(V ) dV ¡ cV



1

=

f(V ) dV ¡ c; p∗ (V )

we have the estimation for the speed c as 1=2   1=2 1 r(1 − V )(V − ) r 2  2 dV ( −  ) + r ln = : c¿  V 2 = This is the end of proof of Theorem 1.1 (part II). 9. Discussion 9.1. Su
 1+

 

 + r ln

 ¡ r(a2 − 1) ¡ 4r(a2 − 1); 

Part I is a suNcient condition of speed c for the existence of travelling wave and Part II is a necessary condition of speed c for the existence of travelling wave.

N. Fei, J. Carr / Nonlinear Analysis: Real World Applications 4 (2003) 503 – 524

523

9.2. An example and an open problem If 14 = −1 such as d = 1; r = 1; a1 + a2 = 2 or r(a2 − 1) = 1 − a1 , then, by   2 r(a2 − 1) = 2 1 − a1 = c∗  and

r 2  ( − 2 ) + r ln 2 



1 2

=

c∗ 1 − a1 = √ ; 2 2 2

we have √ • If c ¿ 2 1 − a1 , there exist positive increasing travelling wave solution (u(x + ct); v(x + ct)) for boundary √ problem (1.1), (1.4), (1.6), (1.8). • If (1 − a1 )=2 ¡ c ¡ 2 1 − a1 , there exists no positive increasing travelling wave solution (u(x + ct); v(x + ct)) for boundary problem (1.1), (1.4), (1.6), (1.8). Since from (2.5), we know that 21−2 are complex numbers and there are no monotonic travelling  wave solution for boundary problem (1.1), (1.6). • If c 6 (1 − a1 )=2, there exist no travelling wave solution (u(x + ct); v(x + ct)) for boundary problem (1.1), (1.4), (1.6), (1.8). Open Problem: When d = 1, the question of minimum speed of problem (1.1), (1.4), (1.6) has not been answered yet. Acknowledgements This work was supported by a scholarship from the Department of Mathematics at Heriot-Watt University, Great Britain. We would like to thank Li Zhou for a useful conversation about the global monotonicity on parameter. References [1] A. Champneys, S. Harris, J. Toland, J. Warren, D. Williams, Algebra, analysis and probability for a coupled system of reaction-di%usion equations, Phil. Trans. R. Soc. Lond. A 350 (1995) 69–112. [2] C. Conley, R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction di%usion model, Indiana Univ. math. J. 33 (1984) 319–343. [3] R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 335–369. [4] R.A. Gardner, Existence and stability of traveling wave solutions of competition models: a degree theoretic, J. Di%erential Equations 44 (1982) 343–363. [5] Hosono, The Minimal Speed of Traveling Fronts for a Di%usive Lotka–Volterra Competition Model, Bull. Math. Bio. 60 (1998) 435 – 448. [6] J.I. Kanel, L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition– di%usion system, Nonlinear Anal. TMA 27 (1996) 579–587. [7] Kan-on, Fisher wave fronts for the Lotka–Volterra competition model with di%usion, Nonlinear Anal. TMA 28 (1997) 145 –164. [8] A.Ya. Kapel, The existence of travelling wave type solutions for the Belousov-Zhabotinskii system of equations, Sib. Math. J. 32 (3) (1991) 47–59. [9] Kolmogorov, Petrovskii, Piskunov, A study of the equation of di%usion with increase in the quantity of matter, and its application to a biological problem, Bull. Moscow State Univ. 17 (1937) 1–72.

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[10] J.D. Murray, Mathematical Biology, in: Biomathematics Text, Vol. 19, Springer-Verlag, Heidelberg and New York, 1989. [11] A. Okubo, P.K. Maini, M.H. Williamson, J.D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B 238 (1989) 113–125. [12] S.R. Dunbar, Travelling wave solutions of di%usive Lotka–Volterra equations: A heteroclinc connection in R4 , Trans. Amer. Math. Soc. 286 (2) (1983) 557–588. [13] M.M. Tang, P.C. Fife, Propagating fronts for competing species equations with di%usion, J. Math. Anal. Appl. 73 (1980) 69–77.