Existence of travelling wave solutions in delayed reaction diffusion systems with applications to diffusion competition systems

Transcript

1 INSTITUTE OF PHYSICS PUBLISHING Nonlineariy 9 (2006) NONLINEARITY doi:0.088/ /9/6/003 Exisence of ravelling wave soluions in delayed reacion diffusion sysems wih applicaions o diffusion compeiion sysems Wan-Tong Li, Guo Lin and Shigui Ruan 2 School of Mahemaics and Saisics, Lanzhou Universiy, Lanzhou, Gansu , People s Republic of China 2 Deparmen of Mahemaics, Universiy of Miami, PO Box , Coral Gables, FL , USA Received June 2005, in final form 4 April 2006 Published 5 May 2006 Online a sacks.iop.org/non/9/253 Recommended by B Eckhard Absrac This paper is concerned wih he exisence of ravelling wave soluions in a class of delayed reacion diffusion sysems wihou monooniciy, which concludes wo-species diffusion compeiion models wih delays. Previous mehods do no apply in solving hese problems because he reacion erms do no saisfy eiher he so-called quasimonooniciy condiion or non-quasimonooniciy condiion. By using Schauder s fixed poin heorem, a new cross-ieraion scheme is given o esablish he exisence of ravelling wave soluions. More precisely, by using such a new cross-ieraion, we reduce he exisence of ravelling wave soluions o he exisence of an admissible pair of upper and lower soluions which are easy o consruc in pracice. To illusrae our main resuls, we sudy he exisence of ravelling wave soluions in wo delayed wo-species diffusion compeiion sysems. Mahemaics Subjec Classificaion: 35K57, 35R20, 92D25. Inroducion In recen years, grea aenion has been paid o he sudy of he exisence of ravelling waves in reacion diffusion sysems wih delays. In a pioneering work, Schaaf [27 sysemaically sudied wo scalar reacion diffusion equaions wih a single discree delay for he so-called Huxley nonlineariy as well as Fisher nonlineariy by using he phase space analysis, he maximum principle for parabolic funcional differenial equaions and he general heory for ordinary funcional differenial equaions. For reacion diffusion sysems wih quasimonooniciy (QM) and a single discree delay, Zou and Wu [36 esablished he exisence /06/ $ IOP Publishing Ld and London Mahemaical Sociey Prined in he UK 253

2 254 W-T Li e al of ravelling wave frons by firs runcaing he unbounded domain and hen passing o a limi. Wu and Zou [33 furher considered more general reacion diffusion sysems wih a single delay of he form 2 u(x, ) = D x u(x, ) + f(u (x)), (.) 2 where R, x R, D = diag(d,,d n ), d i > 0, i =, n, f C([ τ,0, R n ) is coninuous and saisfies f(ˆ0) = f(ˆk) = 0; here û denoes he consan vecor funcion on [ τ,0 aking he value u, and for any fixed x R, u (x) C([ τ,0, R n ) is defined by u (x) = u( + θ,x),θ [ τ,0. If he reacion erm f saisfies eiher he QM condiion (QM) here exiss a marix β = diag(β,β 2,...,β n ) wih β i 0 such ha f ( ) f( )+ β( (0) (0)) 0 for, C([ τ,0, R n ) wih 0 (s) (s) K,s [ τ,0 or he non-quasimonooniciy (QM ) condiion (QM ) here exiss a marix β = diag(β,β 2,...,β n ) wih β i 0 such ha f ( ) f( )+ β( (0) (0)) 0 for, C([ τ,0, R n ) wih (i) 0 (s) (s) K,s [ τ,0 and (ii) e βs ( (s) (s)) is non-decreasing in s [ τ,0; hen some exisence resuls are esablished for ravelling wave frons connecing he rivial equilibrium 0 and he non-rivial equilibrium K, where he well-known monoone ieraion echniques for ellipic sysems wih advanced argumens are used [20, 23. The resuls are applicable no only o delayed scalar equaions (Lan and Wu [9) bu also o delayed sysems, such as delayed diffusion cooperaion sysems (Huang and Zou [4) and he delayed Belousov Zhaboinskii model (Huang and Zou [5). Following Wu and Zou [33, Ma [22 employed he Schauder s fixed poin heorem o an operaor used in Wu and Zou [33 ina properly chosen subse in he Banach space C(R, R n ) equipped wih he so-called exponenial decay norm. The subse is consruced in erms of a pair of upper lower soluions, which is less resricive han he upper lower soluions required in [33. This makes he search for he pair of upper lower soluions slighly easier. Since Ma [22 only considered delayed sysems wih quasimonoone reacion erms, Huang and Zou [5 exended he resuls of Ma [22 o a class of delayed sysems wih QM reacion erms. For relaed resuls on reacion diffusion equaions wih non-local delays, we refer o Ashwin e al [, Al-Omari and Gourly [2, Billingham [3, Li, Ruan and Wang [2, Wang, Li and Ruan [3 and references cied herein. However, i is quie common ha he reacion erm in a model sysem arising from a pracical problem may no saisfy eiher he QM condiion or he QM condiion. Two ypical and imporan examples are he wo species compeiion sysems [26, 32: u 2 (x, ) = d x u (x, ) + r 2 u (x, )[ a u (x, ) b u 2 (x, τ ), u 2 (.2) 2(x, ) = d 2 x u 2(x, ) + r 2 2 u 2 (x, )[ b 2 u (x, τ 2 ) a 2 u 2 (x, ) and u 2 (x, ) = d x u (x, ) + r 2 u (x, )[ a u (x, τ ) b u 2 (x, τ 2 ), u 2 (.3) 2(x, ) = d 2 x u 2(x, ) + r 2 2 u 2 (x, )[ b 2 u (x, τ 3 ) a 2 u 2 (x, τ 4 ). Thus, i is worhwhile o furher explore his opic for sysems wihou eiher QM or QM, and his consiues he purpose of his paper.

3 Travelling waves of delayed reacion diffusion sysems 255 In order o focus on he mahemaical ideas and for he sake of simpliciy, we consider a reacion diffusion sysem of wo equaions wih discree delays, ha is, u 2 (x, ) = d x u (x, ) + f 2 (u (x, τ ), u 2 (x, τ 2 )), u 2 (.4) 2(x, ) = d 2 x u 2(x, ) + f 2 2 (u (x, τ 2 ), u 2 (x, τ 22 )), where d i > 0,τ ij 0,f i : R 2 R is a coninuous funcion, x (, ), and (A) f i (0, 0) = f i (k,k 2 ) = 0 for i =, 2. (A2) There exis wo posiive consans L > 0 and L 2 > 0 such ha f (φ,ψ ) f (φ 2,ψ 2 ) L, f 2 (φ,ψ ) f 2 (φ 2,ψ 2 ) L 2 for = (φ,ψ ), = (φ 2,ψ 2 ) C([ τ, 0, R 2 ) wih 0 φ i (s), ψ i (s) M i,s [ τ, 0, M i >k i is posiive consan, i =, 2. Since he resuls of Huang and Zou [5, Ma [22 and Wu and Zou [33 do no apply o delayed reacion diffusion sysems (.2) and (.3), we mus search for new echniques ha can be applied o our delayed reacion diffusion sysem (.4), a leas for (.2) and (.3). To overcome he difficuly, we propose wo new condiions on he reacion erms, which are o be called he weak QM condiion (WQM) and he weak QM condiion (WQM ), respecively: (WQM) Two posiive numbers exis β > 0, β 2 > 0 such ha f (φ (s), ψ (s)) f (φ 2 (s), ψ (s)) + β [φ (0) φ 2 (0) 0, f (φ (s), ψ (s)) f (φ (s), ψ 2 (s)) 0, f 2 (φ (s), ψ (s)) f 2 (φ (s), ψ 2 (s)) + β 2 [ψ (0) ψ 2 (0) 0, f 2 (φ (s), ψ (s)) f 2 (φ 2 (s), ψ (s)) 0 for φ (s), φ 2 (s), ψ (s), ψ 2 (s) C([ τ,0, R) wih 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2,s [ τ,0. (WQM ) Two posiive numbers exis β > 0, β 2 > 0 such ha f (φ (s), ψ (s)) f (φ 2 (s), ψ (s)) + β [φ (0) φ 2 (0) 0, f (φ (s), ψ (s)) f (φ (s), ψ 2 (s)) 0, f 2 (φ (s), ψ (s)) f 2 (φ (s), ψ 2 (s)) + β 2 [ψ (0) ψ 2 (0) 0, f 2 (φ (s), ψ (s)) f 2 (φ 2 (s), ψ (s)) 0 for φ (s), φ 2 (s), ψ (s), ψ 2 (s) C([ τ,0, R) wih (i) 0 φ2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2,s [ τ,0; (ii) e βs [φ (s) φ 2 (s) and e β2s [ψ (s) ψ 2 (s) are non-decreasing in s [ τ,0. Since he nonlinear funcions f and f 2 in (.4) have differen monooniciy wih respec o he firs and second argumens in he firs and second equaions, respecively, following Pao [23 and Ye and Li [34, we inroduce definiions of he upper and lower soluions, and a new crossieraion scheme, which are differen from hose defined in Huang and Zou [5, Ma [22 and Wu and Zou [33. By using such a scheme, we will consruc a subse in he Banach space C(R, R 2 ) equipped wih he exponenial decay norm and reduce he exisence of ravelling wave soluions o he exisence of an admissible pair of upper and lower soluions which are easy o consruc in pracice. As applicaions, we shall show ha sysem (.2) saisfies he condiion (WQM) while sysem (.3) saisfies (WQM ) and esablish he exisence of ravelling wave soluions in boh models.

4 256 W-T Li e al We remark ha if τ = τ 2 = 0, hen (.2) reduces o he following Loka Volerra diffusion compeiion sysem: u 2 (x, ) = d x u (x, ) + r 2 u (x, )[ a u (x, ) b u 2 (x, ), (.5) u 2 2(x, ) = d 2 x u 2(x, ) + r 2 2 u 2 (x, )[ b 2 u (x, ) a 2 u 2 (x, ), where u and u 2 represen he densiies of wo compeiive species in a one-dimensional habia having infinie lengh and d,d 2,r,r 2,a,a 2,b,b 2 are some posiive consans. The exisence of ravelling wave soluions of (.5) has been exensively sudied in he lieraure (Conley and Gardner [5, Gardner [8, Gourley and Ruan [9, Kanel and Zhou [6, Kan-on [7, Tang and Fife [29 and van Vuuren [30). This model has a rivial (no species) equilibrium E 0 = (0, 0), wo semirivial (one species only) spaially homogeneous equilibria [9 ( ) ) E =, 0,E 2 = (0, a2 a and a posiive (wo coexising species) spaially homogeneous equilibrium ( ) E a2 b a b 2 =, a a 2 b b 2 a a 2 b b 2 provided ha a a 2 b b 2 and eiher (i) a 2 >b and a >b 2 or (ii) a 2 <b and a <b 2.By using phase space analysis for he ordinary differenial equaions, Tang and Fife [29 and van Vuuren [30 showed ha (.5) has ravelling fron soluions connecing he equilibria E 0 and E. Kanel and Zhou [6 furher proved ha (.5) has ravelling fron soluions connecing he equilibria E and E. Conley and Gardner [5 and Gardner [8 showed ha (.5) has ravelling fron soluions connecing he equilibria E and E 2, where Conley index and degree heory mehods have been developed. Oher relaed resuls can be found in Gourley and Ruan [9, Hosono [, 2, Kan-on [7, ec. We shall esablish he exisence of ravelling waves in sysem (.4), hus in sysems (.2) and (.3), ha connec he rivial equilibrium E 0 and he posiive equilibrium E. Thus our resuls can be regarded as a generalizaion of he resuls of Tang and Fife [29 and van Vuuren [30 o he diffusion compeiion models wih delays. This paper is organized as following. Secion 2 is devoed o some preliminary discussions. In secion 3, we esablish a new cross-ieraion scheme and apply i o obain he exisence of ravelling wave soluions if he nonlinear reacion erm saisfies he condiion (WQM). In secion 4, we use he non-sandard ordering of he profile se and prove ha similar resuls hold if he nonlinear reacion erm saisfies he condiion (WQM ). In secion 5, we apply our main resuls o he diffusion compeiion sysems (.2) and (.3) and prove he exisence of ravelling wave soluions. The paper ends wih a discussion in secion Preliminaries In his paper, we use he usual noaions for he sandard ordering in R 2. Tha is, for u = (u,u 2 ) and v = (v,v 2 ), we denoe u v if u i v i,i=, 2, and u<vif u v bu u v. In paricular, we denoe u v if u v bu u i v i,i =, 2. If u v, we also denoe (u, v =w R 2,u<w v}, [u, v) =w R 2,u w<v}, and [u, v =w R 2, u w v}. Le denoe he Euclidean norm in R 2 and denoe he supremum norm in C([ τ, 0, R 2 ). A ravelling wave soluion of (.4) is a special ranslaion invarian soluion of he form u (x, ) = φ(x + c), u 2 (x, ) = ψ(x + c), where (φ,ψ) C 2 (R, R 2 ) are he profiles of he wave ha propagaes hrough he one-dimensional spaial domain a a consan velociy

5 Travelling waves of delayed reacion diffusion sysems 257 c > 0. Subsiuing u (x, ) = φ(x + c), u 2 (x, ) = ψ(x + c) ino (.4) and denoing φ (s) = φ( + s),ψ (s) = ψ( + s) and x + c by, we find ha (.4) has a pair of ravelling wave soluions if and only if he following wave equaions d φ cφ + f c (φ,ψ ) = 0, d 2 ψ cψ + f2 c (φ (2.),ψ ) = 0 wih asympoic boundary condiions lim φ() = φ, lim φ() = φ +, lim ψ() = ψ, lim ψ() = ψ + (2.2) + + have a pair of soluions (φ(), ψ()) on R, where fi c (φ, ψ) : C([ τ, 0, R) R, i =, 2, is given by fi c (φ, ψ) = f i(φ c,ψ c ), φ c (s) = φ(cs), ψ c (s) = ψ(cs), s [ τ,0, where τ = max i,j 2 τ ij }, (φ,ψ ) and (φ +,ψ + ) are wo equilibria of (2.). Wihou loss of generaliy, we le φ = 0,φ + = k > 0,ψ = 0 and ψ + = k 2 > 0. Then boundary condiions (2.2) become lim φ() = 0, lim Le φ() = k, + lim ψ() = 0, lim ψ() = k 2. (2.3) + C [0,M (R, R 2 ) =(φ, ψ) C(R, R 2 ) :0 φ(s) M, 0 ψ(s) M 2,s R}. Define he operaor H = (H,H 2 ) : C [0,M (R, R 2 ) C(R, R 2 ) by H (φ, ψ)() = f c(φ,ψ ) + β φ(), H 2 (φ, ψ)() = f2 c(φ,ψ ) + β 2 ψ(). Then (2.) can be rewrien as following: d φ () cφ () β φ() + H (φ, ψ)() = 0, d 2 ψ () cψ () β 2 ψ() + H 2 (φ, ψ)() = 0. Le λ = c c 2 +4β d, λ 2 = c + c2 +4β d, 2d 2d λ 3 = c c 2 +4β 2 d 2, λ 4 = c + c2 +4β 2 d 2. 2d 2 2d 2 Then λ < 0 <λ 2,λ 3 < 0 <λ 4, d λ 2 i cλ i β = 0,i =, 2 and d 2 λ 2 i cλ i β 2 = 0,i = 3, 4. (2.4) (2.5) Define he operaor F = (F,F 2 ) : C [0,M (R, R 2 ) C(R, R 2 ) by [ F (φ, ψ)() = e λ( s) H (φ, ψ)(s)ds + e λ2( s) H (φ, ψ)(s)ds, d (λ 2 λ ) [ F 2 (φ, ψ)() = e λ3( s) H 2 (φ, ψ)(s)ds + e λ4( s) H 2 (φ, ψ)(s)ds. d 2 (λ 4 λ 3 ) (2.6) We can see ha he operaor F is well defined and for any (φ, ψ) C [0,M (R, R 2 ), d (F (φ, ψ)) () c(f (φ, ψ)) () β F (φ, ψ)() + H (φ, ψ)() = 0, d 2 (F 2 (φ, ψ)) () c(f 2 (φ, ψ)) (2.7) () β 2 F 2 (φ, ψ)() + H 2 (φ, ψ)() = 0.

6 258 W-T Li e al Thus, a fixed poin of F is a soluion of (2.5), which is a ravelling wave soluion of (.4) connecing 0 = (0, 0) and K = (k,k 2 ) if i saisfies (2.3). In he following, we inroduce he exponenial decay norm. Le µ > 0 such ha µ<min λ,λ 2, λ 3,λ 4 }. Define B µ (R, R 2 ) = C(R, R 2 ) : sup () e µ < } R and µ = sup () e µ. R Then i is easy o check ha (B µ (R, R 2 ), µ ) is a Banach space. 3. The case (WQM) In his secion, we consider he exisence of ravelling wave soluions of (2.) when he delayed reacion erms f and f 2 saisfy he condiion (WQM). We sar our cross-ieraion wih a pair of upper and lower soluions of (2.) defined as follows. Definiion 3.. A pair of wice coninuously differeniable funcions = ( φ, ψ) and = (φ,ψ) C(R, R 2 ) are called an upper and a lower soluion of (2.), respecively, if and saisfy d φ () c φ () + f c( φ,ψ ) 0onR, d 2 ψ () c ψ () + f2 c(φ, (3.) ψ ) 0onR, and d φ () cφ () + f c (φ, ψ ) 0onR, d 2 ψ () cψ () + f c 2 ( φ,ψ ) 0onR. (3.2) In wha follows, we assume ha an upper soluion = ( φ, ψ) and a lower soluion = (φ,ψ) of (2.) are given so ha (P) 0 (φ,ψ) ( φ, ψ) M = (M,M 2 ); (P2) lim ( φ, ψ) = 0, lim + (φ,ψ) = lim + ( φ, ψ) = K = (k,k 2 ). For he operaor H = (H,H 2 ) defined in secion 2, we have he following resul. Lemma 3.2. Assume ha (WQM) holds. Then H (φ 2,ψ )() H (φ,ψ 2 )(), H 2 (φ,ψ 2 )() H 2 (φ 2,ψ )() for (φ i,ψ i ) C [0,M (R, R 2 ) wih 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2,s R. Proof. By (WQM), direc calculaion shows ha H (φ,ψ 2 )() H (φ 2,ψ )() = H (φ,ψ 2 )() H (φ 2,ψ 2 )() + H (φ 2,ψ 2 )() H (φ 2,ψ )() H (φ,ψ 2 )() H (φ 2,ψ 2 )() = f c (φ,ψ ) f c (φ 2,ψ ) + β (φ () φ 2 ()) 0, R. The inequaliy for H 2 can be esablished similarly. The proof is complee.

7 Travelling waves of delayed reacion diffusion sysems 259 As a direc consequence of lemma 3.2, he following lemma is rue. Lemma 3.3. Assume ha (WQM) holds. Then F (φ 2,ψ )() F (φ,ψ 2 )(), F 2 (φ,ψ 2 )() F 2 (φ 2,ψ )() for (φ i,ψ i ) C [0,M (R, R 2 ) wih 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2,s R. Now, we assume ha here exiss an upper soluion () = ( φ, ψ) and a lower soluion = (φ,ψ) of (2.) saisfying (P) and (P2). Define he following profile se. Ɣ([φ,ψ, [ φ, ψ) =(φ, ψ) C [0,M (R, R 2 ), φ() φ() φ(),ψ() ψ() ψ(), R}. Obviously, Ɣ([φ,ψ, [ φ, ψ) is non-empy. Lemma 3.4. Assume ha (A2) holds. Then F = (F,F 2 ) : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ in B µ (R, R 2 ). Proof. We firs prove ha H : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ.if = (φ,ψ ), = (φ 2,ψ 2 ) C [0,M (R, R 2 ) saisfy hen µ = sup () () e µ <δ, R H (φ,ψ )() H (φ 2,ψ 2 )() e µ = f c (φ,ψ )() f c (φ 2,ψ 2 )() + β (φ () φ 2 ()) e µ L C([ cτ,0,r 2 )e µ + β φ φ 2 µ = L sup s [ cτ,0 ( + s) ( + s) e µ + β φ φ 2 µ L sup (θ) (θ) e µ θ e µcτ + β µ θ R (L e µcτ + β ) µ. For any fixed ε>0, le δ < ε/(l e µcτ + β ). If = (φ,ψ ), = (φ 2,ψ 2 ) C [0,M (R, R 2 ) saisfy µ <δ, hen H (φ,ψ )() H (φ 2,ψ 2 )() e µ (L e µcτ + β ) µ <ε. Therefore, H (φ,ψ ) H (φ 2,ψ 2 ) µ ε. Tha is, H : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ. Now, we show ha F : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ.

8 260 W-T Li e al For 0, we find F (φ,ψ )() F (φ 2,ψ 2 )() [ e λ( s) H (φ,ψ )(s) H (φ 2,ψ 2 )(s) ds d (λ 2 λ ) + e λ2( s) H (φ,ψ )(s) H (φ 2,ψ 2 )(s) ds [ = e λ( s)+µ s H (φ,ψ )(s) H (φ 2,ψ 2 )(s) e µ s ds d (λ 2 λ ) + e λ2( s)+µ s H (φ,ψ )(s) H (φ 2,ψ 2 )(s) e µ s ds [ 0 e λ( s)+µs ds + e λ( s) µs ds d (λ 2 λ ) 0 + e λ2( s)+µs ds H (φ,ψ ) H (φ 2,ψ 2 ) µ [ λ 2 λ 2µ = d (λ 2 λ ) (µ λ )(λ 2 µ) eµ + λ 2 eλ µ2 H (φ,ψ ) H (φ 2,ψ 2 ) µ. Hence, we have F (φ,ψ ) () F (φ 2,ψ 2 ) () e µ d (λ 2 λ ) [ λ 2 λ (µ λ )(λ 2 µ) + 2µ λ 2 e(λ µ) µ2 H (φ,ψ ) H (φ 2,ψ 2 ) µ [ λ 2 λ d (λ 2 λ ) (µ λ )(λ 2 µ) + 2µ λ 2 µ2 H (φ,ψ ) H (φ 2,ψ 2 ) µ. (3.3) Similarly, for 0, we have F (φ,ψ ) () F (φ 2,ψ 2 ) () e µ d (λ 2 λ ) [ λ 2 λ (µ + λ )(λ 2 + µ) + 2µ λ 2 2 µ2 H (φ,ψ ) H (φ 2,ψ 2 ) µ. (3.4) Thus, i follows from (3.3) and (3.4) ha F : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ in B µ (R, R 2 ). By using a similar argumen as above, we can also prove ha F 2 : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ in B µ (R, R 2 ). This complees he proof. Lemma 3.5. Assume ha (WQM) holds. Then F : Ɣ([φ,ψ, [ φ, ψ) Ɣ([φ,ψ, [ φ, ψ). Proof. For any (φ, ψ) Ɣ([φ,ψ, [ φ, ψ), by lemma 3.3, i is easy o see ha F (φ, ψ) F (φ, ψ) F ( φ,ψ), F 2 ( φ,ψ) F 2 (φ, ψ) F 2 (φ, ψ).

9 Travelling waves of delayed reacion diffusion sysems 26 Now, we only need o prove φ F (φ, ψ) F ( φ,ψ) φ, ψ F 2 ( φ,ψ) F 2 (φ, ψ) ψ. (3.5) According o he definiion of he operaor F and he lower soluion, we have [ ( ) F (φ, ψ)() = e λ( s) + e λ 2( s) H φ, ψ (s)ds [ ( ) e λ( s) + e 2( s) λ β φ(s) + cφ (s) d φ (s) ds = φ(), R. In a similar way, we can prove ha (3.5) holds for R. The proof is complee. Lemma 3.6. Assume ha (A2) and (WQM) hold. Then F : Ɣ([φ,ψ, [ φ, ψ) Ɣ([φ,ψ, [ φ, ψ) is compac. Proof. We firs esablish an esimae for F. For any (φ, ψ) Ɣ([φ,ψ, [ φ, ψ), wehave So (F (φ, ψ)) () = λ e λ d (λ 2 λ ) + λ 2 e λ 2 d (λ 2 λ ) (F (φ, ψ)) µ sup [e µ λ e λ R d (λ 2 λ ) +e µ λ 2 e λ 2 d (λ 2 λ ) e λ s H (φ, ψ)(s)ds e λ 2s H (φ, ψ)(s)ds. e λs H (φ, ψ)(s)ds e λ2s H (φ, ψ)(s)ds λ d (λ 2 λ ) sup e λ µ e λs e µ s e µ s H (φ, ψ)(s)ds R λ 2 + d (λ 2 λ ) sup e λ 2 µ e λ2s e µ s e µ s H (φ, ψ)(s)ds R λ d (λ 2 λ ) H (φ, ψ) µ sup e λ µ e λs+µ s ds R λ 2 + d (λ 2 λ ) H (φ, ψ) µ sup e λ 2 µ e λ2s+µ s ds. R Therefore, for >0, we have (F (φ, ψ)) µ λ d (λ 2 λ )( λ µ) H (φ, ψ) µ λ 2 + d (λ 2 λ )(λ 2 µ) H (φ, ψ) µ [ λ = d (λ 2 λ ) (λ + µ) + λ 2 H (φ, ψ) µ. (λ 2 µ)

10 262 W-T Li e al Similarly, for <0, we have (F (φ, ψ)) λ µ d (λ 2 λ )( λ µ) H (φ, ψ) µ [ λ 2 + d (λ 2 λ ) (λ 2 µ) (λ 2 + µ) + H (φ, ψ) µ (λ 2 + µ) [ λ d (λ 2 λ ) (λ + µ) + λ 2 H (φ, ψ) µ. (λ 2 µ) Since H : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ and he se Ɣ([φ,ψ, [ φ, ψ) is uniformly bounded, here exiss a consan C such ha (F (φ, ψ)) µ C. In a similar way, here exiss a consan C 2 such ha (F 2 (φ, ψ)) µ C 2. Hence F is equiconinuous on Ɣ([φ,ψ, [ φ, ψ) and FƔ([φ,ψ, [ φ, ψ) is uniformly bounded. We nex prove ha F : Ɣ([φ,ψ, [ φ, ψ) Ɣ([φ,ψ, [ φ, ψ) is compac. Define F n (φ, ψ) by F (φ, ψ)(), [ n, n; F n (φ, ψ)() = F (φ, ψ)(n), (n, ); F(φ,ψ)( n), (, n). Then, for any n, F n is equiconinuous and uniformly bounded. Ascoli Arzela lemma implies ha F n is compac. Since F n (φ, ψ)} 0 is a compac series, and sup F n (φ, ψ)() F (φ, ψ)() e µ R = sup F n (φ, ψ)() F (φ, ψ)() e µ (, n) (n, ) 2C 0 e µn 0asn, where C 0 is a posiive consan such ha (φ, ψ) C 0 for any (φ, ψ) Ɣ([φ,ψ, [ φ, ψ), by proposiion 2. in [35, we know ha F n } 0 converges o F in Ɣ([φ,ψ, [ φ, ψ) wih respec o he norm µ. Therefore F is compac. Theorem 3.7. Assume ha (A), (A2) and (WQM) hold. If (2.) has an upper soluion ( φ, ψ) C [0,M (R, R 2 ) and a lower soluion (φ,ψ) C [0,M (R, R 2 ) such ha (P) and (P2) are saisfied, hen (2.) has a ravelling wave soluion saisfying (2.3). Proof. From lemmas , we know ha FƔ([φ,ψ, [ φ, ψ) Ɣ([φ,ψ, [ φ, ψ) and F is compac. By Schauder s fixed poin heorem here exiss a fixed poin (φ,ψ ) Ɣ([φ,ψ, [ φ, ψ), which is a soluion of (2.). In order o prove ha his soluion is a ravelling wave soluion, we need o verify he asympoic boundary condiion (2.3). By (P2) and he fac ha 0 (φ(), ψ()) (φ (), ψ ()) ( φ(), ψ()) (M,M 2 ), we see ha lim (φ (), ψ ()) = (0, 0) and lim (φ (), ψ ()) = (k,k 2 ). Therefore, he fixed poin (φ (), ψ ()) saisfies he asympoic boundary condiion (2.3). The proof is complee.

11 Travelling waves of delayed reacion diffusion sysems 263 From heorem 3.7 we can see ha he exisence of soluions of (2.) and (2.3) is reduced o he exisence of an admissible pair of upper and lower soluions. However, i is difficul o consruc such a pair of upper and lower soluions in pracice because hey are required o be wice coninuously differeniable on R. In order o overcome he difficuly, we inroduce he following weaker definiions of upper and lower soluions of (2.) han definiion 3.. More precisely, we do no require ha he upper and lower soluions are wice coninuously differeniable on R and hey are easy o consruc in pracice. See examples 5. and 5.5 in secion 5. Definiion 3.8. A pair of coninuous funcions = ( φ, ψ), = (φ,ψ) C(R, R 2 ) is called a weak upper soluion and a weak lower soluion of (2.), respecively, if consans T i,i =,,m exis, such ha and are wice coninuously differeniable in R\T i : i =,,m} and saisfy d φ () c φ () + f c( φ,ψ ) 0, R\T i : i =,,m}, d 2 ψ () c ψ () + f2 c(φ, (3.6) ψ ) 0, R\T i : i =,,m}, and d φ () cφ () + f c (φ, ψ ) 0, R\T i : i =,,m}, d 2 ψ () cψ () + f c 2 ( φ,ψ ) 0, R\T i : i =,,m}. (3.7) Lemma 3.9. Assume ha (WQM) holds. If, C [0,M (R, R 2 ) are a weak upper soluion and a weak lower soluion of (2.), respecively, saisfying (P), (P2) and φ (+) φ ( ), ψ (+) ψ ( ), R, φ (+) φ ( ), ψ (+) ψ (3.8) ( ), R, hen (F (φ, ψ),f 2 ( φ,ψ)) (F ( φ,ψ), F 2 (φ, ψ)), (3.9) and (F (φ, ψ),f 2 ( φ,ψ)), (F ( φ,ψ), F 2 (φ, ψ)) C [0,M (R, R 2 ) are a lower and an upper soluion of (2.), respecively. Proof. Wihou loss of generaliy, we assume ha and are wice coninuously differeniable in R\T i : i =,,m} wih <T <T 2 < <T m < +. Denoe T 0 = and T m+ = +. I is easy o verify ha (F (φ, ψ),f 2 ( φ,ψ)) is well defined and wice coninuously differeniable. For any (T k,t k+ ), 0 k m, i follows from (2.6) and he definiion of weak lower soluion ha e λ 2( s) H (φ, ψ)(s)ds [ F (φ, ψ)() = e λ( s) + d (λ 2 λ ) [ e λ( s) + e λ 2( s) d (λ 2 λ ) ( ) β φ(s) + cφ (s) d φ (s) ds k = φ() + ( )) e λ 2( T i ) φ (T j +) φ (T j λ 2 λ j= m + ( )) e λ ( T i ) φ (T j +) φ (T j λ 2 λ j=k+ φ().

12 264 W-T Li e al Using a similar argumen, we can prove ha (3.9) holds. Furhermore, (2.7) ogeher wih lemma 3.2 yields 0 = d (F (φ, ψ)) () c(f (φ, ψ)) () β F (φ, ψ)() + H (φ, ψ)() d (F (φ, ψ)) () c(f (φ, ψ)) () β F (φ, ψ)() + H (F (φ, ψ),f 2 (φ, ψ))() = d (F (φ, ψ)) () c(f (φ, ψ)) () + f c (F (φ, ψ),f 2 (φ, ψ)). Similarly, we have d 2 (F 2 ( φ,ψ)) () c(f 2 ( φ,ψ)) () + f2 c (F ( φ,ψ), F 2 ( φ,ψ)) 0. Noe ha (F (φ, ψ),f 2 ( φ,ψ)) C [0,M (R, R 2 ) C 2 (R, R 2 ); we conclude ha i is a lower soluion of (2.). In a similar way, we can prove ha (F ( φ,ψ), F 2 (φ, ψ)) is an upper soluion of (2.). The proof is complee. Theorem 3.0. Assume ha (A), (A2) and (WQM) hold. If (2.) has a weak upper soluion ( φ, ψ) and a weak lower soluion (φ,ψ) saisfying (P), (P2) and (3.8), hen (2.) has a ravelling wave soluion saisfying (2.3). 4. The Case (WQM ) In his secion, we shall consider he exisence of ravelling wave soluions of (2.) when he delayed reacion erms f and f 2 saisfy he condiion (WQM ). In wha follows, we assume ha an upper soluion ( φ(), ψ()) and a lower soluion (φ(), ψ()) saisfy (P), (P2), and (P3) e βs [ φ(s) φ(s) and e β2s [ ψ(s) ψ(s) are non-decreasing for s R. To sar wih, we define he following profile se: (i) (φ,ψ) (φ, ψ) ( φ, ψ), Ɣ ([φ,ψ, [ φ, ψ) = (φ, ψ) C [0,M (R, R 2 (ii) e βs [φ(s) φ(s),e βs [ φ(s) φ(s), ) : e β2s [ψ(s) ψ(s),e β2s. [ ψ(s) ψ(s) are non-decreasing for s R I is easy o see ha Ɣ ([φ,ψ, [ φ, ψ) is non-empy. In fac, by (P3), we know ha e βs [ φ(s) φ(s) and e β2s [ ψ(s) ψ(s) are non-decreasing in s R, e βs [ φ(s) φ(s) = 0, and e β2s [ ψ(s) ψ(s) = 0. Thus ( φ, ψ) saisfies (ii) of Ɣ ([φ,ψ, [ φ, ψ). Similarly, (φ(), ψ()) saisfies (ii) of Ɣ ([φ,ψ, [ φ, ψ). Lemma 4.. Assume ha (WQM ) holds. Then H (φ 2,ψ )() H (φ,ψ 2 )(), H 2 (φ,ψ 2 )() H 2 (φ 2,ψ )(), where = (φ,ψ ), = (φ 2,ψ 2 ) C [0,M (R, R 2 ) wih (i) 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2 ; (ii) e βs [φ (s) φ 2 (s) and e β2s [ψ (s) ψ 2 (s) are non-decreasing for s R. Lemma 4.2. Assume ha (WQM ) holds. Then F (φ 2,ψ )() F (φ,ψ 2 )(), F 2 (φ,ψ 2 )() F 2 (φ 2,ψ )(), where = (φ,ψ ), = (φ 2,ψ 2 ) C [0,M (R, R 2 ) wih (i) 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2 ; (ii) e βs [φ (s) φ 2 (s) and e β2s [ψ (s) ψ 2 (s) are non-decreasing for s R.

13 Travelling waves of delayed reacion diffusion sysems 265 Lemma 4.3. Assume ha (A2) and (WQM ) hold. Then F : C [0,M (R, R 2 ) C(R, R 2 ) is coninuous wih respec o he norm µ in B µ (R, R 2 ). The proofs of lemmas are similar o hose of lemmas and are omied here. By (P3), i is easy o see he following lemma. Lemma 4.4. Ɣ ([φ,ψ, [ φ, ψ) is a closed, bounded and convex subse of B µ (R, R 2 ). Lemma 4.5. Assume ha (WQM ) holds. If c> minβ d,β 2 d 2 }, hen FƔ ([φ,ψ, [ φ, ψ) Ɣ ([φ,ψ, [ φ, ψ). Proof. For any (φ, ψ) Ɣ ([φ,ψ, [ φ, ψ), repeaing he argumen in he proof of lemma 3.5, we have φ F (φ, ψ) φ and ψ F 2 (φ, ψ) ψ. This implies ha (i) of Ɣ ([φ,ψ, [ φ, ψ) holds. We now prove (ii) of Ɣ ([φ,ψ, [ φ, ψ). Le F (φ, ψ) = φ for (φ, ψ) Ɣ, hen e β [ φ() φ () e β [ = e λ( s) + d (λ 2 λ ) + e λ 2( s) [ β φ(s) + c φ (s) d φ (s) [ β φ (s) + cφ (s) d φ (s)} ds e β [ + = e λ( s) + e λ 2( s) d (λ 2 λ ) [ β φ(s) + c φ (s) d φ (s) H (φ, ψ)(s) [ β φ (s) + cφ (s) d φ (s) H (φ, ψ)(s) } ds e β [ + = e λ( s) + e λ 2( s) d (λ 2 λ ) [ β φ(s) + c φ (s) d φ (s) H (φ, ψ)(s) ds. Hence, we have d d eβ [ φ() φ ()} = (β + λ ) e β e λ ( s) [ β φ(s) + c φ (s) d φ (s) H (φ, ψ)(s) ds d (λ 2 λ ) + (β + λ 2 ) e β d (λ 2 λ ) (β + λ ) e β d (λ 2 λ ) + + (β + λ 2 ) e β d (λ 2 λ ) + e λ 2( s) [ β φ(s) + c φ (s) d φ (s) H (φ, ψ)(s) ds [ e λ ( s) β φ(s) + c φ (s) d φ (s) H ( φ,ψ)(s) ds [ e λ 2( s) β φ(s) + c φ (s) d φ (s) H ( φ,ψ)(s) ds 0, R. Similarly, we can prove ha e β2 [ ψ() F 2 (φ, ψ)(),e β [F (φ, ψ)() φ() and e β2 [F 2 (φ, ψ)() ψ()

14 266 W-T Li e al are non-decreasing in R. Thus, F (φ, ψ) saisfies (ii) of Ɣ ([φ,ψ, [ φ, ψ). The proof is complee. Similar o lemma 3.6 we have he following lemma. Lemma 4.6. Assume ha (A2) and (WQM ) hold. Then F : Ɣ ([φ,ψ, [ φ, ψ) Ɣ ([φ,ψ, [ φ, ψ) is compac. Now we sae our main resul in his secion; is proof is similar o ha of heorem 3.7. Theorem 4.7. Assume ha (A), (A2) and (WQM ) hold. Assume furher ha ( 2.) has an upper soluion ( φ, ψ) C [0,M (R, R 2 ) and a lower (φ,ψ) C [0,M (R, R 2 ) saisfying (P) (P3). Then, for any c> minβ d,β 2 d 2 },(2.) has a ravelling wave soluion saisfying (2.3). Similar o ha in secion 3, we have he following. Theorem 4.8. Assume ha (A), (A2) and (WQM ) hold. If (2.) has a weak upper soluion ( φ, ψ) C [0,M (R, R 2 ) and a weak lower soluion (φ,ψ) C [0,M (R, R 2 ) such ha (P) (P3) and (3.8) hold, hen, for any c> minβ d,β 2 d 2 },(2.) has a ravelling wave soluion saisfying (2.3). Remark 4.9. If (WQM ) is saisfied, hen we can always choose β i > 0 sufficienly large such ha c> minβ d,β 2 d 2 }. 5. Applicaions As menioned in he inroducion, in his secion, we employ our conclusions in secions 3 and 4 o esablish he exisence of ravelling wave soluions for sysems (.2) and (.3). Example 5.. We consider he exisence of he ravelling wave soluions for he delayed diffusion compeiion sysem (.2), ha is, u 2 (x, ) = d x u (x, ) + r 2 u (x, )[ a u (x, ) b u 2 (x, τ ), u 2 (5.) 2(x, ) = d 2 x u 2(x, ) + r 2 2 u 2 (x, )[ b 2 u (x, τ 2 ) a 2 u 2 (x, ). Assume ha c>0. Le u (x, ) = u (x + c) = φ(s), u 2 (x, ) = u 2 (x + c) = ψ(s), s R, and denoe he coordinae s as, hen he corresponding wave sysem is d φ () cφ () + r φ()[ a φ() b ψ( cτ ) = 0, d 2 ψ () cψ (5.2) () + r 2 ψ()[ b 2 φ( cτ 2 ) a 2 ψ() = 0. We are ineresed in soluions of (5.2) saisfying lim φ() = 0, lim where provided ha φ() = k, + lim ψ() = 0, lim k = a 2 b a a 2 b b 2 > 0,k 2 = a b 2 a a 2 b b 2 > 0, ψ() = k 2, + a >b 2,a 2 >b. (5.3)

15 Travelling waves of delayed reacion diffusion sysems 267 For φ,ψ C([ τ,0, R), where τ = maxτ,τ 2 }, denoe f (φ, ψ) = r φ(0)[ a φ(0) b ψ( τ ), f 2 (φ, ψ) = r 2 ψ(0)[ b 2 φ( τ 2 ) a 2 ψ(0). Obviously, (A) and (A2) are saisfied. We now verify ha f = (f,f 2 ) saisfies (WQM). Lemma 5.2. The funcion f saisfies (WQM). Proof. For any (s) = (φ (s), φ 2 (s)), (s) = (ψ (s), ψ 2 (s)) C([ τ,0, R 2 ) wih 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2, in view of M >k and M 2 >k 2,we have 2a M + b M 2 > 2a k + b k 2 = a k > 0 and f (φ,ψ ) f (φ 2,ψ ) = r φ (0)[ a φ (0) b ψ ( τ ) r φ 2 (0)[ a φ 2 (0) b ψ ( τ ) = r [φ (0) φ 2 (0) r a φ 2 (0) φ2 2 (0) r b ψ ( τ )[φ (0) φ 2 (0) = r [φ (0) φ 2 (0)[ a (φ (0) + φ 2 (0)) b ψ ( τ ) r [φ (0) φ 2 (0)[ 2a M b M 2 = r (2a M + b M 2 )[φ (0) φ 2 (0) = β [φ (0) φ 2 (0) and f (φ,ψ ) f (φ,ψ 2 ) = r φ (0)[ a φ (0) b ψ ( τ ) r φ (0)[ a φ (0) b ψ 2 ( τ ) = r b φ (0)(ψ ( τ ) ψ 2 ( τ )) 0. In a similar argumen, we can prove ha f 2 saisfies (WQM). The proof is complee. In order o apply heorem 3.0, we need o consruc a weak upper soluion and a weak lower soluion for (5.2). If c>max2 d r, 2 d 2 r 2 }, hen here exis 0 <λ <λ 2 such ha d λ 2 i cλ i + r = 0,i =, 2, and 0 <λ 3 <λ 4 such ha d 2 λ 2 i cλ i + r 2 = 0,i = 3, 4. For fixed ( η, min 2, λ 2, λ }) 4 (5.4) λ λ 3 and large consan q > 0, we consider he funcions l () = e λ qe ηλ and l 2 () = e λ3 qe ηλ3. I is easy o see ha l () and l 2 () have global maximum m > 0 and m 2 > 0, respecively. Define = max : l () = m } and 3 = max : l 2 () = m Then for any given λ>0, here exiss ε 2 > 0 and ε 4 > 0 such ha k ε 2 e λ = l ( ) = m 2 and k 2 ε 4 e λ 3 = l 2 ( 3 ) = m 2 2. }.

16 268 W-T Li e al Noe ha (5.3) holds, we have ε 0 > 0,ε > 0 and ε 3 > 0 such ha a ε b ε 4 >ε 0,a 2 ε 3 b 2 ε 2 >ε 0, (5.5) a ε 2 b ε 3 >ε 0,a 2 ε 4 b 2 ε >ε 0. For he above consans and suiable consans 2, 4, we define he coninuous funcions as follows: and φ() = φ() = e λ, 2, k + ε e λ, 2, e λ qe ηλ,, k ε 2 e λ,, ψ() = e λ 3, 4 k 2 + ε 3 e λ, 4 ψ() = e λ 3 qe ηλ3, 3, k 2 ε 4 e λ, 3, where q>0is large enough and λ>0is small enough. I is easy o see M = sup R φ() > k, M 2 = sup R ψ() > k 2, φ(), ψ(),φ() and ψ() saisfy (P), (P2), (3.8) and min 2, 4 } maxτ,τ 2 } max, 3 } for sufficienly large q>0and sufficienly small λ>0. We now prove ha he coninuous funcions ( φ(), ψ()) and (φ(), ψ()) are a weak upper soluion and a weak lower soluion of (5.2), respecively. Lemma 5.3. Assume ha (5.3) and (5.5) hold. Then () = ( φ(), ψ()) is a weak upper soluion and () = (φ(), ψ()) is a weak lower soluion of (5.2). Proof. For 2,inviewofψ( cτ ) 0 for R and d λ 2 cλ + r = 0, we have d φ () c φ () + r φ()[ a φ() b ψ( cτ ) d φ () c φ () + r φ() = [d λ 2 cλ + r (k + ε )e λ = 0. For 2, since ψ( cτ ) = k 2 ε 4 e λ( cτ), hen d φ () c φ () + r φ()[ a φ() b ψ( cτ ) = e λ d ε λ 2 + cε λ + r (k + ε e λ )(b ε 4 e λcτ a ε )}. Le I (λ) = d ε λ 2 + cε λ + r (k + ε e λ )(b ε 4 e λcτ a ε ). Then, a ε b ε 4 >ε 0 implies ha I (0) = r (k + ε )(b ε 4 a ε )<0, and here exiss a λ > 0 such ha I (λ) < 0 for λ (0,λ ). Thus, we have d φ () c φ () + r φ()[ a φ() b ψ( cτ ) 0. Similarly, here exiss a λ 2 > 0 such ha for λ (0,λ 2 ) we have d 2 ψ () c ψ () + r 2 ψ()[ b 2 φ( cτ 2 ) a 2 ψ() 0. Taking λ (0, min(λ,λ 2 )), we see ha our conclusion is rue. A similar argumen applies o () = (φ(), ψ()). The proof is complee.

17 Travelling waves of delayed reacion diffusion sysems 269 By heorem 3.0, we have he following resul. Theorem 5.4. Assume ha (5.3) holds. Then for every c>max2 d r, 2 d 2 r 2 },(5.) has a ravelling wave soluion (φ(x + c), ψ(x + c)) wih wave speed c, which connecs (0, 0) and (k,k 2 ). Furhermore, lim ξ (φ(ξ)e λξ, ψ(ξ)e λ3ξ ) = (, ), where ξ = x + c. Example 5.5. We now consider he delayed diffusion compeiion sysem (.3), ha is u 2 (x, ) = d x u (x, ) + r 2 u (x, ) [ a u (x, τ ) b u 2 (x, τ 2 ), u 2 (5.6) 2(x, ) = d 2 x u 2(x, ) + r 2 2 u 2 (x, ) [ b 2 u (x, τ 3 ) a 2 u 2 (x, τ 4 ). The corresponding ravelling wave sysem is d φ () cφ () + r φ()[ a φ( cτ ) b ψ( cτ 2 ) = 0, d 2 ψ () cψ (5.7) () + r 2 ψ()[ b 2 φ( cτ 3 ) a 2 ψ( cτ 4 ) = 0. For φ,ψ C([ τ,0, R) wih τ = maxτ,τ 2,τ 3,τ 4 }, we denoe f (φ, ψ) = r φ(0)[ a φ( τ ) b ψ( τ 2 ), f 2 (φ, ψ) = r 2 ψ(0)[ b 2 φ( τ 3 ) a 2 ψ( τ 4 ). Obviously, (A) and (A2) are saisfied. We now verify ha f = (f,f 2 ) saisfies (WQM ). Lemma 5.6. Assume ha τ and τ 4 are small enough. Then he funcion f saisfies (WQM ). Proof. For any (s) = (φ (s), φ 2 (s)), (s) = (ψ (s), ψ 2 (s)) C([ τ, 0, R 2 ) wih (i) 0 φ 2 (s) φ (s) M, 0 ψ 2 (s) ψ (s) M 2 and (ii) e βs [φ (s) φ 2 (s) and e β2s [ψ (s) ψ 2 (s) are non-decreasing in s [ τ, 0, we have f (φ,ψ ) f (φ 2,ψ ) = r φ (0)[ a φ ( τ ) b ψ ( τ 2 ) r φ 2 (0)[ a φ 2 ( τ ) b ψ ( τ 2 ) = r [φ (0) φ 2 (0) r a [φ (0)φ ( τ ) φ 2 (0)φ 2 ( τ ) r b ψ ( τ 2 )[φ (0) φ 2 (0) (r r b M 2 )[φ (0) φ 2 (0) r a φ (0)[φ ( τ ) φ 2 ( τ ) r a φ 2 ( τ )[φ (0) φ 2 (0) r ( b M 2 a M )[φ (0) φ 2 (0) r a φ (0)e β τ e β τ [φ ( τ ) φ 2 ( τ ) r ( b M 2 a M a M e β τ )[φ (0) φ 2 (0). If τ > 0 is small enough, hen we can choose β > 0 such ha r ( b M 2 a M a M e β τ )> β. We have f (φ,ψ ) f (φ,ψ 2 ) = r φ (0)[ a φ ( τ ) b ψ ( τ 2 ) r φ (0)[ a φ ( τ ) b ψ 2 ( τ 2 ) = r b φ (0)[ψ ( τ 2 ) ψ 2 ( τ 2 ) 0. Thus f (φ, ψ) saisfies (WQM )ifτ is small enough. In a similar way, we can prove ha f 2 (φ, ψ) saisfies (WQM )ifτ 4 is small enough. The proof is complee.

18 270 W-T Li e al Remark 5.7. From he proof of lemma 5.6 we can see ha if τ and τ 4 are small enough, hen we can always choose β i > 0 sufficienly large such ha c> minβ d,β 2 d 2 }. Now we define φ(), ψ(),φ() and ψ() as in example 5.. φ(), ψ(),φ() and ψ() saisfy (P) (P3) and (3.8). I is easy o see ha Lemma 5.8. Assume ha (5.3) and (5.5) hold. If τ and τ 4 are small enough, hen () = ( φ(), ψ()) is a weak upper soluion and () = (φ(), ψ()) is a weak lower soluion of (5.7). Proof. For φ(), we need o prove ha d φ () c φ () + r φ()[ a φ( cτ ) b ψ( cτ 2 ) 0. (5.8) For > 2 + cτ or < 2, he proof of (5.8) is similar o ha of lemma 5.3: we omi i here. For 2 << 2 + cτ, we noe ha we can choose λ small enough such ha d φ () c φ () + r φ()[ a φ( cτ ) b ψ( cτ 2 ) < 0, for = 2 + cτ. Since τ is small enough and independen of φ,ψ, and φ (), φ (), φ() and ψ() are uniformly bounded and uniformly coninuous for R\ 2, 3 }, i follows ha (5.8) holds for 2 << 2 + cτ. Similarly, we can prove ha ψ saisfies d 2 ψ () c ψ () + r 2 ψ()[ b 2 φ( cτ 3 ) a 2 ψ( cτ 4 ) 0 and () = (φ(), ψ()) is a weak lower soluion. The proof is complee. Theorem 5.9. Assume ha (5.3) holds and τ and τ 4 are sufficienly small. Then for every c>max2 d r, 2 d 2 r 2 },(5.6) has a ravelling wave soluion (φ(x + c), ψ(x + c)) wih wave speed c, which connecs (0, 0) and (k,k 2 ). Furhermore, lim ξ (φ(ξ)e λ ξ, ψ(ξ)e λ3ξ ) = (, ), where ξ = x + c. Remark 5.0. We noe ha he delay of example 5. does no affec he exisence of ravelling wave soluions. However, he delays (τ and τ 4 ) of example 5.5 do. Remark 5.. If τ = 0, hen (5.6) reduces o (.5), ha is, u 2 (x, ) = d x u (x, ) + r 2 u (x, )[ a u (x, ) b u 2 (x, ), u 2 (5.9) 2(x, ) = d 2 x u 2(x, ) + r 2 2 u 2 (x, ) [ b 2 u (x, ) a 2 u 2 (x, ). Tang and Fife [29 and van Vuuren [30 proved ha (5.9) has a bounded ravelling wave fron soluion connecing (0, 0) and (k,k 2 ) if and only if c max2 d r, 2 d 2 r 2 }. Our resuls cerainly include heir resuls when c>max2 d r, 2 d 2 r 2 }.Ifc = max2 d r, 2 d 2 r 2 }, we can ge he exisence of ravelling wave soluion only by changing he definiion of φ and ψ. However, our resuls canno ensure ha he bounded ravelling wave soluions of sysems (5.) and (5.6) connecing (0, 0) and (k,k 2 ) are monoonic. Numerical simulaions indicae ha he ravelling wave soluions are monoonic. I would be ineresing o sudy he monooniciy of he ravelling wave soluions in he delayed diffusion compeiion models.

19 Travelling waves of delayed reacion diffusion sysems Discussion Coexisence of compeing species is very common in naural sysems (Durre and Levin [6,7, Silverown e al [28). In he classical compeiion models (described by ordinary differenial equaions and used quaniies averaged over space) spaial heerogeneiy is usually negleced, while in real sysems species are disribued spaially; organisms experience differen local environmens, hey consume resources locally and move around. I is known now ha spaial heerogeneiy is crucial o he dynamics of biological sysems, in paricular for he coexisence of some compeing species (Durre and Levin [6). Differen ypes of mahemaical models have been used o describe spaial heerogeneous environmens. For example, in meapopulaion models (Hanski and Gilpin [0) space is represened as a se of paches wih local ineracions. Reacion diffusion equaion models (Canrell and Cosner [4) ake accoun of space explicily. Sochasic spaial models (Durre and Levin [6, 7) are a combinaion of hese wo approaches. In his paper we considered a class of delayed reacion diffusion sysems wihou monooniciy. By using Schauder s fixed poin heorem, a new cross-ieraion scheme was given o esablish he exisence of ravelling wave soluions. More precisely, by using such a new cross-ieraion, we reduced he exisence of ravelling wave soluions o he exisence of an admissible pair of upper and lower soluions which are easy o consruc in pracice. The general resuls were hen applied o sudy he exisence of ravelling wave soluions in delayed wo-species diffusion compeiion sysems. I is ineresing o noe ha he delays appearing in he inerspecific compeiion erms do no affec he exisence of ravelling waves while he delays appearing in he inraspecific compeiion erms do. The exisence of ravelling wave soluions which connec he rivial equilibrium (0, 0) and he posiive equilibrium (k,k 2 ) indicaes ha here is a ransiion zone moving from he seady sae wih no species o he seady sae wih he coexisence of boh species. I is also known ha he dynamics of reacive sysems on suppors of resriced geomery may deviae subsanially from he predicions of mean-field descripions (Provaa e al [24). To couple microscope level processes and he evoluion of he macroscope observables, nonlinear models on regular laices have also been proposed (Kowalik e al [8, Provaa e al [24, Rauch and Bar-Yam [25 and he references cied herein). The delayed laice differenial equaions version of our model (.4) akes he following form: du n m = a j [u n+j () 2u n () + u n j ()+f (u n ( τ ), v n ( τ 2 )), d j= (6.) dv n m = b j [v n+j () 2v n () + v n j ()+f 2 (u n ( τ 2 ), v n ( τ 22 )), d j= where n Z, m is an ineger, a j > 0, b j > 0, j m, τ 0. I would be ineresing o combine he echniques and resuls in Huang e al [3 and he presen paper o esablish he exisence of ravelling waves for sysem (6.) and apply he resuls o wo species laice compeiion models. Acknowledgmens The auhors are very graeful o an anonymous referee and he handling edior for heir helpful commens and suggesions. W-T Li acknowledges suppor by he NNSF of China (057078) and he Teaching and Research Award Program for Ousanding Young Teachers in Higher

20 272 W-T Li e al Educaion Insiuions of Minisry of Educaion of China. S Ruan s research was parially suppored by NSF gran DMS and he Universiy of Miami. References [ Ashwin P B, Baruccelli M V, Bridges T J and Gourly S A 2002 Travelling frons for he KPP equaion wih spaio-emporal delay Z. Angew. Mah. Phys [2 Al-Omari J and Gourly S A 2002 Monoone raveling frons in age-srucured reacion diffusion model of a single species J. Mah. Biol [3 Billingham J 2004 Dynamics of a srongly nonlocal reacion diffusion populaion model Nonlineariy [4 Canrell R S and Cosner G C 2003 Spaial Ecology via Reacion Diffusion Equaions (Chicheser, UK: Wiley) [5 Conley C and Gardner R 984 An applicaion of he generalized Mores index o raveling wave soluions of a compeiion reacion diffusion model Indiana Univ. Mah. J [6 Durre R and Levin S 994 The imporance of being discree (and spaial) Theor. Pop. Biol [7 Durre R and Levin S 998 Spaial aspecs of inerspecific compeiion Theor. Pop. Biol [8 Gardner R 982 Exisence and sabiliy of raveling wave soluions of compeiion models: a degree heoreic approach J. Diff. Eqns [9 Gourley S A and Ruan S 2003 Convergence and raveling frons in funcional differenial equaions wih nonlocal erms: a compeiion model SIAM J. Mah. Anal [0 Hanski I A and Gilpin M E 996 Meapopulaion Biology: Ecology, Geneics, and Evoluion (New York: Academic) [ Hosono Y 995 Traveling waves for a diffusive Loka Volerra compeiion model, II: a geomeric approach Forma [2 Hosono Y 2003 Traveling waves for a diffusive Loka Volerra compeiion model, I: singular perurbaion Discree Con. Dyn. Sys. B [3 Huang J, Lu G and Ruan S 2005 Traveling wave soluions in delayed laice differenial equaions wih parial monooniciy Nonlinear Anal. TMA [4 Huang J and Zou X 2002 Traveling wavefrons in diffusive and cooperaive Loka Volerra sysem wih delays J. Mah. Anal. Appl [5 Huang J and Zou X 2003 Exisence of raveling wavefrons of delayed reacion diffusion sysems wihou monooniciy Discree Con. Dyn. Sys [6 Kanel J I and Zhou L 996 Exisence of wave fron soluions and esimaes of wave speed for a compeiion diffusion sysem Nonlinear Anal. TMA [7 Kan-on Y 995 Parameer dependence of propagaion speed of raveling waves for compeiion diffusion equaions SIAM J. Mah. Anal [8 Kowalik M, Lipowski A and Ferreira A L 2002 Oscillaions and dynamics in a wo-dimensional prey predaor sysem Phys. Rev. E [9 Lan K and Wu J 2003 Traveling wavefrons of reacion diffusion equaions wih and wihou delays Nonlinear Anal. RWA [20 Leung A W 989 Sysems of Nonlinear Parial Differenial Equaions wih Applicaions o Biology and Engineering (Dordrech: Kluwer) [2 Li W T, Ruan S and Wang Z C 2005 On he diffusive Nicholson s Blowflies equaion wih nonlocal delays J. Nonlinear Sci. submied [22 Ma S 200 Traveling wavefrons for delayed reacion diffusion sysems via a fixed poin heorem J. Diff. Eqns [23 Pao C V 992 Nonlinear Parabolic and Ellipic Equaions (New York: Plenum) [24 Provaa A, Nicolis G and Baras F 999 Oscillaory dynamics in low-dimensional suppors: a laice Loka Volerra model J. Chem. Phys [25 Rauch E M and Bar-Yam Y 2006 Long-range ineracions and evoluionary sabiliy in a predaor prey sysem Phys. Rev. E [26 Ruan S and Zhao X Q 999 Persisence and exincion in wo species reacion diffusion sysems wih delays J. Diff. Eqns [27 Schaaf K W 987 Asympoic behavior and raveling wave soluions for parabolic funcional differenial equaions Trans. Am. Mah. Soc [28 Silverown J, Holier S, Johnson J and Dale P 992 Cellular auomaion models of inerspecific compeiion for space he effec of paern on process J. Ecol

21 Travelling waves of delayed reacion diffusion sysems 273 [29 Tang M M and Fife P 980 Propagaing frons for compeing species equaions wih diffusion Arch. Raion. Mech. Anal [30 van Vuuren J H 995 The exisence of raveling plane waves in a general class of compeiion diffusion sysems IMA J. Appl. Mah [3 Wang Z C, Li W T and Ruan S 2006 Traveling wave frons in reacion diffusion sysems wih spaio-emporal delays J. Diff. Eqns [32 Wu J 996 Theory and Applicaions of Parial Funcional Differenial Equaions (New York: Springer) [33 Wu J and Zou X 200 Traveling wave frons of reacion diffusion sysems wih delay J. Dyn. Diff. Eqns [34 Ye Q X and Li Z Y 994 Inroducion o Reacion Diffusion Equaions (Beijing: Science Press) [35 Zeilder E 986 Nonlinear Funcional Analysis and is Applicaions: I, Fixed-Poin Theorems (New York: Springer) [36 Zou X and Wu J 997 Exisence of raveling wave frons in delayed reacion diffusion sysems via he monoone ieraion mehod Proc. Am. Mah. Soc