Travelling wave fronts in reaction diffusion systems with spatio-temporal delays

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1 J. Differential Equations Travelling wave fronts in reaction diffusion systems with spatio-temporal delays Zhi-Cheng Wang a,b, Wan-Tong Li a,,, Shigui Ruan c,2 a School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 73, People s Republic of China b Department of Mathematics, Hexi University, Zhangye, Gansu 734, People s Republic of China c Department of Mathematics, University of Miami, P.O. Box 24985, Coral Gables, FL , USA Received 9 December 24; revised 8 August 25 Available online 27 September 25 Abstract This paper deals with the existence of travelling wave fronts in reaction diffusion systems with spatio-temporal delays. Our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system. New iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions. Following this, the existence of travelling wave fronts for reaction diffusion systems with spatio-temporal delays is established. Finally, we apply the main results to a single-species diffusive model with spatio-temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels. 25 Elsevier Inc. All rights reserved. MSC: 35K57; 34K; 35B2 Keywords: Travelling wave fronts; Reaction diffusion systems; Integral operator; Iterative techniques; Quasimonotonicity; Spatio-temporal delays Corresponding author. address: wtli@lzu.edu.cn W.-T. Li. Partially supported by the NNSF of China and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. 2 Partially supported by NSF Grant DAMS-4247 and the University of Miami /$ - see front matter 25 Elsevier Inc. All rights reserved. doi:.6/j.jde.25.8.

2 86 Z.-C. Wang et al. / J. Differential Equations Introduction The theory of travelling wave solutions of parabolic differential equations is one of the fastest developing areas of modern mathematics and has attracted much attention due to its significant nature in biology, chemistry, epidemiology and physics, see 4,8,23,27,3,32,34]. Travelling wave solutions are solutions of special type and can be usually characterized as solutions invariant with respect to transition in space. From the physical point of view, travelling waves describe transition processes. These transition processes from one equilibrium to another usually forget their initial conditions and the properties of the medium itself. Among the basic questions in the theory of travelling waves, the existence of travelling wave solutions is an important objective. The case of a scalar reaction diffusion equation has been rather well studied, basically due to applicability of comparison theorems of a special kind for parabolic equations and of phase space analysis for the ordinary differential equations, see 4,8,23,3,34]. For systems of reaction diffusion equations modeling various biological phenomena, many results have been established in 23,27,3]. Since comparison theorems are, in general, not applicable for reaction diffusion systems and the phase space analysis becomes more complicated, some new approaches, such as the Conley index and degree theory methods, have been developed in 23,27,3]. Recently, many researchers have paid attention to travelling wave solutions for reaction diffusion equations with time delays, for example, see,,7 9,22,26,29,32, 33,36]. In a pioneering work, Schaaf 26] systematically studied two scalar reaction diffusion equations with a single discrete delay for the so-called Huxley nonlinearity as well as Fisher nonlinearity by using the phase space analysis, the maximum principle for parabolic functional differential equations and the general theory for ordinary functional differential equations. For reaction diffusion systems with quasimonotonicity and a single discrete delay, Zou and Wu 36] established the existence of travelling wave fronts by first truncating the unbounded domain and then passing to a limit. Wu and Zou 33] further considered more general reaction diffusion systems with a single delay and obtained some results on the existence of travelling wave fronts, where the wellknown monotone iteration techniques for elliptic systems with advanced arguments in 2,24] are used. The results are applicable to delayed Fisher-KPP equation, Belousov Zhabotinskii model with delay, and some other models, see,,7,29], etc. Following Wu and Zou 33], Ma 22] employed the Schauder s fixed point theorem to an operator used 33] in a properly chosen subset of the Banach space CR, R n equipped with the so-called exponential decay norm. The subset is constructed in terms of a pair of upper-lower solutions, which is less restrictive than the upper-lower solutions required in 33]. This makes the searching for the pair of upper-lower solutions slightly easier. Since Ma 22] only considered delayed systems with quasimonotone reaction terms, Huang and Zou 8] extended the results of Ma 22] to a class of delayed systems with nonquasimonotone reaction terms. In ecology, since populations take time to move in space and usually were not at the same position in space at previous times, sometimes it is not sufficient only to include a discrete delay or a finite delay in a population model. Motivated by this, Britton

3 Z.-C. Wang et al. / J. Differential Equations ,6] considered comprehensively the two factors and introduced the so-called spatiotemporal delay or nonlocal delay, that is, the delay term involves a weighted spatiotemporal average over the whole of the infinite spatial domain and the whole of the previous times. Since then, great progress has been made on the existence of travelling wave fronts in reaction diffusion equations with spatio-temporal delays, see 3,9,, 2 6,2,25,28,3,35]. There are three methods which have been used to prove the existence of travelling wave solutions in these works. The first one is the perturbation theory of ordinary differential equations coupled with Fredholm alterative, see ] for an age-structured reaction diffusion model with nonlocal delay and Gourley 9] for a nonlocal Fisher equation. The second one is the geometric singular perturbation theory of Fenichel 7], see 2,,6,25,3]. More precisely, if the corresponding undelayed system under consideration has a travelling wave solution, then, by choosing special kernels and applying the geometric singular perturbation theory, the reaction diffusion system with spatio-temporal delay also has a travelling wave solution when the delay is sufficiently small. The third one is the monotone iteration approach of Wu and Zou 33], we refer to 2,28,35] for several special reaction diffusion models with distributed delay or spatio-temporal delay, where the quasimonotonicity condition is required. However, as pointed out by Al-Omari and Gourley ], the approach of Wu and Zou 33] cannot be applied directly to reaction diffusion systems with distributed delay or spatio-temporal delay since it requires that the delayed term remains local in space. It is natural to ask if the above results of Lan and Wu 9] and Wu and Zou 33] can be extended to general reaction diffusion equations with distributed delays and spatio-temporal delays. A prototype of such equations takes the form ut, x t D 2 ut, x x 2 + f ut, x, g ut, x,. where t, x R, D diag d,...,d n,d i >, i,...n, n N; u t,x u t,x,...,u n t,x T,f C R 2n, R n, and or g ut,x t g ut,x t g t s, x y u s, y dy ds.2 g t s u s, x ds..3 The purpose of this paper is to establish the existence of travelling wave fronts of.. A travelling wave front is a solution ut, x φx + ct, where c>isagiven constant and φ BC 2 R, R n see Section 2 is an increasing function satisfying the

4 88 Z.-C. Wang et al. / J. Differential Equations following functional differential system: Dφ t + cφ t f φ t, g φt, t R,.4 and the conditions φ and φ + K with < s K,.5 where the notation < s is defined in Section 2 and or g φt g φt g s, y φ t y cs dy ds.6 g s φ t cs ds..7 Our main idea is to change the existence problem for the functional differential system.4 into a fixed point problem for an integral operator of the form Aφ t k t,sf φs + γφ s ds φ t, t R,.8 where F : BC, K] BCR, R n L R, R n is a suitable map. The symbols and the precise definitions of conceptions mentioned in this section will be given later in this paper. New iterative techniques are established for the map A, which can be used to treat the existence of travelling waves for.. The main difficulty in establishing the theory is that the map A may not be continuous and it is not clear if AE is compact in BCR, R n for all bounded sets E in BC, K]. This idea was first used by Lan and Wu 9] to study the existence of travelling wave solutions of the scalar reaction diffusion equation with and without delay of the form ut, x t D 2 ut, x x 2 + f ut, x, ut r,x,...,ut r n,x..9 Since the approach of Lan and Wu 9] is only applicable to scalar reaction diffusion equations without and with discrete delays, we must search for new techniques for our reaction diffusion systems with spatio-temporal delays. To overcome the difficulty, we introduce and employ the so-called M-continuity for F and show that the closure of Pa bae is compact in C a,b], R n for each bounded subset E, where Pa b maps each element in BCR, R n to its restriction to a,b], which are extensions of Lan and

5 Z.-C. Wang et al. / J. Differential Equations Wu 9]. In order to describe the monotonicity of reaction nonlinear terms, in addition to γ-increasing introduced in 9], we also introduce two new concepts: γ -increasing and γ -increasing, which include the cases that the reaction terms do not satisfy the quasimonotonicity condition, in particular, the later is new since it was not considered 9,33]. These, together with Lebesgue s dominated convergence theorem and uniform convergence of the integral gt, x dx in t,a], where a>, enable us to prove that the iterative sequences involved are convergent in some sense. Thus, we can apply the monotone iteration technique coupled with the upper-lower solutions and a nonstandard ordering in the profile set to deal with the existence of travelling wave fronts of reaction diffusion systems with spatio-temporal delays. Here we need to point out that uniform convergence of the integral gt, x dx in t,a], a>, is not a more restrictive condition. In fact, many known kernel functions gt, x, such as g t,x τ e t τ δ x, τ >,. g t,x δ t 4πρ e x2 4ρ, ρ >,. and g t,x τ e t τ 4πt e x2 4t, τ >.2 satisfy this condition, which will be verified in Section 5. The rest of this paper is organized as follows. Section 2 is devoted to some preliminary discussions. We introduce a class of maps, which are bounded, M-continuous, γ-increasing, γ -increasing and γ -increasing, respectively, and provide some basic properties. In Section 3, we develop a monotone iteration scheme and apply it to establish the existence of solutions for the second-order system of functional differential equations if the nonlinear term satisfies one of the γ-increasing, γ -increasing or γ -increasing conditions. Following this, we establish the existence of travelling wave fronts in Section 4. In the last section, we apply our results to a diffusive single-species model with spatio-temporal delay. By constructing a pair of the upper and lower solutions, the existence of travelling wave front are obtained by choosing different kernel functions, such as Preliminaries In this section, we introduce some definitions and lemmas, which will be needed in the sequel. We denote by CR, R n, C,b], R n and Ca,b], R n the space of all continuous vector functions defined on R,,b] and a,b] with sup-norm, respectively.

6 9 Z.-C. Wang et al. / J. Differential Equations Let and BCR, R n BC 2 R, R n { x CR, R n : x x t,...,x n t T },x i t CR, max i n sup{ x i t : t R} < { x BCR, R n : x,x BCR, R n }, x x t,...,x n tt,x x t,...,x n t T. Obviously, BCR, R n and BC 2 R, R n are Banach spaces with the norms and x BCR,R n max sup { x i t : t R} i n x BC 2 R,R n max { x BCR,R n, x BCR,R n, x BCR,R n }, respectively. For simplicity, we write BCR,R n. We also denote by with the norm L R, R n L R L R x L R,R n max x i L i n R. The following lemma provides relations between BCR, R n and Ca,b], R n, which is an extension of Lemma 2. in 9]. Its proof is straightforward and omitted. Lemma 2.. Let {x m } be a sequence in the Banach space BCR, R n, where m N. i x sup { x Ca,b],R n :<a<b< } for x BCR, R n. ii If {x m } {x} BCR, R n and x m x m, then x m x Ca,b],R n m for a,b R with a<b. iii If {x m } {x} BCR, R n and x m x Ca,b],R n m for a,b R with a<b, then x m t x t m for each t R. In the rest of this paper, we use the usual notations for the standard ordering in R n. That is, for α α,...,α n T R n and β β,...,β n T R n, we denote α β if α i β i,i,...,n and α < β if α β but α β. In particular, we denote α < s β if α i < β i,i,...,n. For u, v L R, R n, we denote u v if u i t v i t, i, 2,...,n, a.e. on R and u<v if u v. For given α, β R n with α < s β, let

7 Z.-C. Wang et al. / J. Differential Equations BC α, β ] { x BCR, R n : α xt β,t R }.IfTx Ty for x,y BC α, β ] with x y, we say the map T : BC α, β ] BCR, R n L R, R n is increasing. For any μ R n, let μ denote a constant vector function on t R taking the vector μ. Now, we introduce several concepts of γ-increasing, γ -increasing and γ -increasing maps, where the concept of a γ-increasing map was introduced by Lan and Wu 9]. Definition 2.2. A map T : BC α, β ] BCR, R n L R, R n is said to be γ- increasing if there exists a matrix γ diag γ,...,γ n with γi >,i,...n, such that Ty + γy Tx+ γx for x,y BC α, β ] with x y. Definition 2.3. A map T : BC α, β ] BCR, R n L R, R n is said to be γ - increasing if there exists a matrix γ diag γ,...,γ n with γi >,i,...n, such that Ty+γy Tx+γx, where x,y BC α, β ] with x y satisfy that e γt y t x t] is increasing in t R. Definition 2.4. A map T : BC α, β ] BCR, R n L R, R n is said to be γ - increasing if there exists a matrix γ diag γ,...,γ n with γi >,i,...n, such that Ty+γy Tx+γx, where x,y BC α, β ] with x y satisfy that e γt y t x t] is increasing in t R and e γt y t x t] is decreasing in t R. In order to establish our iterative techniques, we need the so-called M-continuity for a map. Definition 2.5. A map T : BC α, β ] BCR, R n L R, R n is said to be M-continuous on BC α, β ] if {x m } {x} BC α, β ] and x m x Ca,b],R n m for a,b R with a<bimply Tx m t Txt m a.e. on R. 3. Systems of the second-order functional differential equations In this section, we consider the existence of solutions for systems of the second-order functional differential equations of the form Dφ t + cφ t Φ φ t a.e. on R, 3. where D diag d,d 2,...,d n,d i >, i, 2,...,n; c R, Φ : BCα, β] BCR, R n R n is bounded and φ t BCα, β] is defined by φ t s φ t + s, φ BCα, β], s R; α, β R n with α < s β. Let Y { x BCR, R n : x,x L R, R n}. Then Y is a Banach space with the norm x Y max { x, x L R,R n, x L R,R n }.

8 92 Z.-C. Wang et al. / J. Differential Equations In particular, if we set Y { x BCR : x,x L R }, then Y Y Y. Now, by a solution to 3. we mean a function φ Y and satisfies 3.. Let γ i >,i, 2,...,n. We write λ i c c 2 + 4γ i d i, λ i2 c + c 2 + 4γ i d i. 2d i 2d i Then λ i < < λ i2 and d i λ 2 ij + cλ ij + γ i, where i, 2,...,n; j, 2. Let ρ i d i λ i2 λ i. Define a matrix map k t,s by where k t,s diagk t,s,...,k n t,s, { k i t,s ρ e λ i t s for s t, i e λ i2t s for s t, and i, 2,...,n. We now consider the linear integral operator L : L R, R n Y defined by where and Lφt k t,s φ s ds, 3.2 Lφt L φ t, L2 φ 2 t,..., Ln φ n t T Li φ i t k i t,s φ i s ds. By Lan and Wu 9, Theorem 3., p. 79], we know L i maps L R onto Y, so we obtain the following theorem: Theorem 3.. The map L defined in 3.2 maps L R, R n onto Y and is linear, bounded, and one to one. Moreover, L maps BCR, R n onto BC 2 R, R n and is linear, bounded, and one to one. Similar to that of 9], we now introduce the concept of G-compactness and show that the map L is G-compact. The concept of G-compactness is sufficient for us to establish our iterative scheme.

9 Z.-C. Wang et al. / J. Differential Equations We define a map P b a : BCR, Rn Ca,b], R n by P b a x t x t a,b], where x t a,b] denotes the restriction of x t to a,b]. Definition 3.2. A map T : L R, R n BCR, R n is said to be G-compact P b a T E is compact in Ca,b], Rn for all a,b R with a<band every bounded subset E L R, R n. Lemma 3.3. Assume T i : L R BCR is G-compact, i, 2,...,n, then the map T : L R, R n BCR, R n defined by is G-compact. Txt T x t, T 2 x 2 t,...,t n x n t T Proof. Let E L R, R n be a bounded subset and P i : x t x i t be a project operator. Let E i P i E. Then E E E n and E i is a bounded subset in L R. For x t E, wehave and Thus, and Txt T x t, T 2 x 2 t,...,t n x n t T T E T 2 E 2 T n E n T Pa P b Txt a b T x t,pa b T 2x 2 t,...,pa b T nx n t P b a T E P b a T 2 E 2 P b a T n E n. P b a T E P b a T E P b a T 2 E 2 P b a T n E n P b a T E P b a T E P b a T 2 E 2 P b a T n E n. Since T i : L R BCR is G-compact, P b a T i E i is compact in Ca,b], R, then P b a T E P b a T 2 E 2 P b a T n E n is compact in Ca,b], R n and Pa bt E is compact in Ca,b], Rn too. By Definition 3.2, we know that T : L R, R n BCR, R n is G-compact. The proof is complete. if

10 94 Z.-C. Wang et al. / J. Differential Equations Theorem 3.4. The map L defined in 3.2 maps L R, R n into BCR, R n and is G-compact. Proof. By Lan and Wu 9, Theorem 3.2, p. 79], we know that L i : L R BCR is G-compact. Following 3.3 and Lemma 3.3, L is G-compact. The proof is complete. Now we define an integral operator A by Aφt k t,s F φ s + γφ sds, t R, where F : BCα, β] L R, R n is defined by F φt Φφ t. Then we have the following result: Lemma 3.5. i Let Y and φ BCα, β]. Then Aφ if and only if D + c + γ F φ + γφ. ii φ is a solution of 3. if and only if φ Y and φ Aφ. Proof. We only prove i, the proof of part ii is similar and omitted. Suppose that for Y and φ BCα, β]. Inviewof we have Let Then Thus, D + c + γ F φ + γφ D Aφ t + c Aφ t + γ Aφt F φt + γφ t, D Aφ t + c Aφ t + γ Aφt. Aφt w t w t,...,w n t T. d i w i t + cw i t + γ iw i t. w i t a i e λ it + a i2 e λ i2t.

11 Z.-C. Wang et al. / J. Differential Equations Since w i t BCR, it follows that a i a i2, i,...,n. Hence, w t for t R, that is, Aφ. The converse is obvious. The proof is complete. Lemma 3.6. i Assume that F : BCα, β] L R, R n is γ-increasing and bounded. If φ t BCα, β] is increasing, then Aφt is also increasing. ii Assume that F : BCα, β] L R, R n is γ -increasing and bounded. If φ t BCα, β] is increasing such that e γt φ t + s φ t ] is increasing in t R for every s>, then Aφt is also increasing in t R and for c> min { γ i d i ; i,...,n }, e γt Aφt + s Aφt ] is increasing in t R for every s>. iii Assume that F : BCα, β] L R, R n is γ -increasing and bounded, where γ satisfies min { γ i d i ; i,...,n } >. If φ t BCα, β] is increasing such that e γt φ t + s φ t ] is increasing in t R and e γt φ t + s φ t ] is decreasing in t R for every s>, then Aφt is increasing in t R and for c with min { γ i d i ; i,...,n } <c<min { γ i d i ; i,...,n }, 3.4 e γt Aφt + s Aφt ] is increasing and e γt Aφt + s Aφt ] is decreasing in t R for every s>. Proof. We only show iii, the proofs of i and ii are similar. Let Δt >. Noting that F φ s + Δt F φ Δt s and employing a change of variable, for t R, we have Aφ t + Δt Aφ t t + t t k t,sf φ s + Δt + γφ s + Δt ds k t,sf φ s + Δt + γφ s + Δt ds k t,sf φ s + γφ s ds t k t,sf φ s + γφ s ds k t,s F φ s + Δt + γφ s + Δt F φ s + γφ s] ds k t,s F φ Δt s + γφ Δt s F φ s + γφ s ] ds. Since φ t BCα, β] is increasing and satisfies that e γt φ t + Δt φ t ] is increasing and e γt φ t + Δt φ t ] in t R, then e γt φ Δt t φ t ] is increasing, e γt φ Δt t φ t ] is decreasing and φ Δt t φ t in t R. Note that F is γ - increasing, it follows that F φδt s + γφ Δt s F φ s + γφ s. Hence, Aφ t + Δt Aφ t, which implies that Aφt is also increasing.

12 96 Z.-C. Wang et al. / J. Differential Equations Let s> and Then for i {,...,n}, d + {e γ i t dt { d dt H i φt F i φ t + γ i φ i t, i {,...,n}. k i t,ξ F i φ ξ + s + γ i φ i ξ + s F i φ ξ + γ i φ i ξ ] } dξ ρ i e γ i +λ it { t e λ iξ H i φξ + s H i φξ ] } dξ e λ i2ξ H i φξ + s H i φξ ] } dξ + d ρ i e γ i +λ i2t dt t t γ i + λ i ρ i e γ i +λ it e λ iξ H i φξ + s H i φξ ] dξ +ρ i e γ i +λ it e λ it H i φt + s H i φt ] + γ i + λ i2 ρ i e γ i +λ i2t ρ i e γ i +λ i2t e λ i2t H i φt + s H i φt ] γ i + λ i ρ i e γ i +λ it t + γ i + λ i2 ρ i e γ i +λ i2t t t e λ i2ξ H i φξ + s H i φξ ] dξ e λ iξ H i φξ + s H i φξ ] dξ e λ i2ξ H i φξ + s H i φξ ] dξ. Noting that for c<min { γ i d i ; i,...,n } and each i {,...,n}, γ i + λ i γ i + c c 2 + 4γ i d i 2d i <, and γ i + λ i2 γ i + c + c 2 + 4γ i d i 2d i γ i γi d i c 2d i γ i c + <, c 2 + 4γ i d i H i φξ + s H i φξ F i φ s ξ + γ i φ is ξ F i φ ξ + γ i φ i ξ,

13 Z.-C. Wang et al. / J. Differential Equations we have that e γt Aφt + s Aφt ] is decreasing in t R for every s>. Similarly, we can show that for c> min { γ i d i ; i,...,n }, e γt Aφt + s Aφt] is increasing in t R for every s>. The proof is complete. Lemma 3.7. Assume that F : BCα, β] L R, R n is bounded and that, ψ BCα, β] Y satisfy: C D + c F and Dψ + cψ F ψ. Then, i A and Aψ ψ; ii for c < min { γ i d i ; i,...,n }, e γt ψ t Aψt ] and e γt A t t] are decreasing in t R; iii for c> min { γ i d i ; i,...,n }, e γt ψ t Aψt ] and e γt A t t ] are increasing in t R. Proof. i Let w A and Dw + cw + γw r t. Inviewof and we know r t a.e. on R. From we get D + c + γ F + γ D A + c A + γ A F + γ, Dw + cw + γw r t, w i t a i e λ it + a i2 e λ i2t + k i t,s r i s ds. Since w i t is bounded, we have a i a i2. Consequently, w i t k i t,s r i s ds, t R, i,...,n. Thus, we proved that A. Similarly, we can prove that ψ Aψ. ii By the proof of i, A t t w t, then d + ]} {e γ i t k i t,ξ r i ξ dξ dt d { ρ i e γ i +λ it dt t } e λiξ r i ξ dξ

14 98 Z.-C. Wang et al. / J. Differential Equations d { ρ i e γ i +λ i2t dt t } e λi2ξ r i ξ dξ t γ i + λ i ρ i e γ i +λ it e λiξ r i ξ dξ + γ i + λ i2 ρ i e γ i +λ i2t t e λ i2ξ r i ξ dξ, i,...,n. This implies that e γt A t t ] is decreasing in t R. The other cases are similar and omitted. The proof is complete. Lemma 3.8. Assume that F : BCα, β] L R, R n is bounded and, ψ BCα, β] with ψ. i If F is γ-increasing, then Aψt A t in t R; ii If F is γ -increasing and e γt ψ t t ] is increasing in t R, then for c> min { γ i d i ; i,...,n }, Aψt A t and e γt Aψt A t ] is increasing in t R; iii If F is γ -increasing, e γt ψ t t ] is increasing and e γt ψ t t ] is decreasing in t R, where γ satisfies min { γ i d i ; i,...,n } >, then Aψt A t, e γt Aψt A t ] is increasing and e γt Aψt A t] is decreasing in t R, where c satisfies 3.4. Proof. We only show that iii holds. Let w t Aψt A t, t R. Then we have Dw t + cw t + γw t F ψt + γψ t F t γ t on R. Denote g t Dw t + cw t + γw t. Then g t a.e. on R and g t L R, R n. By an argument similar to that of Lemma 3.7, we get Aψt A t and d { e γt Aψt A t ]} on R for c<min { γ dt i d i ; i,...,n }.

15 Z.-C. Wang et al. / J. Differential Equations This implies that e γt Aψt A t ] is decreasing in t R. Similarly, we have that e γt Aψt A t ] is increasing in t R. The proof is complete. By Lemma 3.5, we see that a solution of 3. is a fixed point of A. Hence, we have the following main results in this section. Theorem 3.9. Assume that F : BCα, β] L R, R n is bounded and M-continuous. Assume further that, ψ Y BCα, β] with ψ satisfy C. i If F is γ-increasing, then 3. has two solutions, ψ BCα, β] with ψ ψ. If and ψ are increasing, then and ψ are increasing. Moreover, for a,b R with a<b, m Ca,b],R n and ψ m ψ Ca,b],R n, 3.5 where m A m, ψ m Aψ m and m ψ m ψ ψ ψ. 3.6 ii If F is γ -increasing and e γt ψ t t ] is increasing in t R, then for c> min { γ i d i ; i,...,n },3. has two solutions, ψ BCα, β] with ψ ψ, which satisfy 3.5 and 3.6. Furthermore, if and ψ are increasing such that e γt ψ t + s ψ t ] and e γt t + s t ] are increasing in t R for every s>, then and ψ are increasing. iii If F is γ -increasing, e γt ψ t t ] is increasing and e γt ψ t t ] is decreasing in t R, where γ satisfies min { γ i d i ; i,...,n } >, then for c satisfying 3.4, 3. has two solutions, ψ BCα, β] with ψ ψ, which satisfy 3.5 and 3.6. Furthermore, if and ψ are increasing in t R, e γt ψ t + s ψ t ] and e γt t + s t ] are increasing in t R for every s> and e γt ψ t + s ψ t ] and e γt t + s t ] are decreasing in t R for every s>, then and ψ are increasing. Proof. i By Theorem 3.4 and Lemma 3.8i, A maps BCα, β] into BC R, R n and is increasing, so Lemma 3.7 and condition C imply that 3.6 holds. Then there exist, ψ L R, R n such that m t t and ψ m t ψ t for each t R. Obviously, ψ ψ follow from 3.6. By Theorem 3.4, Pa blh { m} Pa ba { m} is compact in Ca,b], R n for a,b R with a < b. It follows that there exists y Ca,b], R n such that A m y Ca,b],R n. Hence, we have t y t for t a,b] and thus, BCR, R n. Since F is bounded and

16 2 Z.-C. Wang et al. / J. Differential Equations M-continuous, it follows that for each t R, k i t,s F i m s + γ i m i s k i t,s F i s + γi i s for s R, k i t,s F i m s + γ i m i s ηk i t,s for s R and some η >. It follows from Lebesgue s dominated convergence theorem that L i Fi m t + γ i m i t L i Fi t + γi i t for each t R and i L i Fi t + γi i t, i, 2,...,n. Hence, L F t + γ t A. A similar argument shows that ψ Aψ.If and ψ are increasing, by Lemma 3.6i, m and ψ m are increasing, m N. Consequently, and ψ are increasing. The proof of i is complete. ii Fix c> min { γ i d i ; i,...,n }. Let ψ m Aψ m and m A m, m N. By Lemmas 3.7 and 3.8, ψ Aψ Aψ and A A satisfy I t t ψ t ψ t for t R; II e γt ] t t is increasing in t R; ] III e γt ψ t IV e γt ψ t ψ t t is increasing in t R; ] is increasing in t R. { By induction and the above lemmas, we obtain two sequences of vector functions ψ m } m and { m} with the following properties: m a t m t m+ t ψ m+ t ψ m t ψ t for t R and m N; b for m N, e γt m+ t m ] t is increasing in t R; c for m N, e γt ψ m t m t ] is increasing in t R; d for m N, e γt ψ m t ψ m+ ] t is increasing in t R.

17 By Lemma 3.5, we have Z.-C. Wang et al. / J. Differential Equations D ψ m + c ψ m + γψ m F D m + c m + γ m F ψ m + γψ m m N, m + γ m m N and m ψ m ψ ψ ψ. Then the existence of and ψ follow from a similar argument to that in i. If and ψ are increasing and satisfy that e γt ψ t + s ψ t ] and e γt t + s t ] are increasing in t R for every s>, then from Lemma 3.6ii and by induction, we see that for m N, ψ m and m are increasing in t R and e γt ψ m t + s ψ m t ] and e γt m t + s m t ] are increasing in t R for every s>. Therefore, and ψ are increasing. The proof of ii is complete. iii Fix c with 3.4. Let ψ m Aψ m and m A m, m N. By Lemmas 3.7 and 3.8, ψ Aψ Aψ and A A satisfy I t t ψ t ψ t for t R; II e γt ] t t is increasing and e γt ] t t is decreasing in t R; III e γt ψ t ] t is increasing and e γt ψ t ] t is decreasing in t R; IV e γt ψ t ψ ] t is increasing and e γt ψ t ψ ] t is decreasing in t R. By { induction and the above lemmas, we obtain two sequences of vector functions ψ m } m and { m} which satisfy that for m N, m a t m t m+ t ψ m+ t ψ m t ψ t for t R; b e γt m+ ] t m t is increasing and e γt m+ ] t m t is decreasing in t R;

18 22 Z.-C. Wang et al. / J. Differential Equations c e γt ψ m+ t m+ ] t is increasing and e γt ψ m t ψ m+ ] t is de- is decreasing in t R; d e γt ψ m t ψ m+ ] t creasing in t R. is increasing and e γt ψ m+ t m+ ] t The remainder of the proof is similar to that of ii and is omitted. The proof of iii is complete. 4. Existence of travelling wave fronts In this section, we shall consider the existence of travelling wave fronts in reaction diffusion systems with spatio-temporal delays of the form u t,x t D 2 u t,x x 2 + f u t,x, g ut,x,...,g m ut,x, 4. where t, x R, D diag d,...,d n,d i >, i,...n, n N; u t,x u t,x,...,u n t,x T,f C R m+n, R n, and gj u t,x t g j t s, x y u s, y dy ds, the kernel g j t,x is any integrable nonnegative function satisfying g j t, x g j t,x and g j s, y dy ds, j,...m, m N. 4.2 In the following, we shall apply our theory developed in Section 3 to establish the existence of travelling wave fronts for system 4. and give an iterative scheme to compute the travelling wave fronts. Assume u t,x φ x + ct and replace x + ct with t, then we can write 4. in the form Dφ t + cφ t f φ t, g φt,...,g m φt, t R, 4.3 where gj φ t g j s, y φ t y cs dy ds, j,...m. By a travelling wave front with a wave speed c>to4., we mean an increasing function φ BC 2 R, R n and a number c> which satisfy 4.3 and the following

19 boundary condition: Z.-C. Wang et al. / J. Differential Equations φ and φ + K K,...,K n T with < s K. 4.4 Now we make an assumption on the kernels g j t,x,j,...m, and then list several propositions of convolutions g j φ, j,...m. The assumption is as follows: H g j t,x dx is uniformly convergent for t,a], a>, j,...m.in other word, if given ε>, then there exists M>such that M g j t,x dx < ε for any t,a]. Proposition 4.. Assume that φ t BCR, R n and satisfies lim t φ t α and lim t + φ t β. If g j t,x satisfies H, then lim gj φ t α and lim gj φ t β t t +, j,...,m. Proof. Fix j {,...,m}. Let h t g j t,x dx. Then by 4.2, we have h t dt. Given ε>, there exists A> such that A h t dt < ε. Note that H holds, then there exists B> such that for any t,a], B g j t,x dx < ε A. By lim t + φ t β, there exists T>such that φ t β <ε for any t>t. Consequently, we have for any t>t+ B + ca that A +B g j s, y ] φ t y cs β dy ds ε B A +B B A +B B g j s, y φ t y cs β dy ds g j s, y dy ds ε.

20 24 Z.-C. Wang et al. / J. Differential Equations In terms of g j t, x g j t,x, it follows that g j φ t β A A B B A B A B 6 φ + ε, g j s, y φ t y cs β ] dy ds g j s, y φ t y cs β dy ds g j s, y φ t y cs β dy ds g j s, y φ t y cs β dy ds g j s, y φ t y cs β dy ds which implies that lim t + gj φ t β. Similarly, we can show the other. The proof is complete. Proposition 4.2. Assume that φ k t and φ t BCR, R n are uniformly bounded for k N and satisfy lim k + φ k φ Ca,b],R n for a,b R with a<b. If g j t,x satisfies H, then lim g j φ k t g j φ t k + for each t R, j,...,m. Proof. Fix j {,...,m} and t R, then there exists N> such that t N,N]. Let M > satisfy φ k < M and φ < M. By a similar argument to that of Proposition 4., for given ε>, there exists A> such that A h t dt < ε, and there exists B> such that for any t,a], B g j t,x dx < ε A. From lim k + φ k φ Ca,b],R n for any a,b R with a<b,

21 Z.-C. Wang et al. / J. Differential Equations we know that there exists K N such that φ k φ <ε Ca,b],R n for any k>k, where a b N + B + ca. Consequently, for any k>k,wehave A +B B A +B g j s, y φ k ] t y cs φ t y cs B dy ds g j s, y φ k t y cs φ t y cs dy ds ε. In view of g j s, x g j s, x, we know that for any k>k, g j φ k t g j φ t A A B B A B A B 6M + ε, g j s, y φ k ] t y cs φ t y cs dy ds g j s, y φ k t y cs φ t y cs dy ds g j s, y φ k t y cs φ t y cs dy ds g j s, y φ k t y cs φ t y cs dy ds g j s, y φ k t y cs φ t y cs dy ds which implies that lim k + gj φ k t g j φ t. Noting that t R and j {,...,m} are arbitrary, so the conclusion follows. The proof is complete. Proposition 4.3. If φ t BCR, R n and H holds, then g j φ t BCR, R n, j,...,m.

22 26 Z.-C. Wang et al. / J. Differential Equations Proof. Fix j {,...,m} and t R. Then there exist N> such that t N,N]. Let M> satisfy φ <M. As above, for given ε>, there exists A> such that A h t dt < ε, and there exists B> such that for any s,a], B g j s, x dx < ε A. Since φ s is uniformly continuous in s N + B + ca +,N + B + ca + ], there exists δ with < δ < such that φ s + θ φ s <εfor any θ δ, δ and any s N + B + ca,n + B + ca]. Following this, we have for any θ δ, δ that A +B B A +B g j s, y φ t + θ y cs φ t y cs ] dy ds B Now, for any θ δ, δ, g j s, y φ t + θ y cs φ t y cs dy ds ε. g j φ t + θ g j φ t g j s, y φ t + θ y cs φ t y cs ] dy ds 6M + ε, which implies that lim gj φ t + θ g j φ t. θ Noting that t R and j {,...,m} are arbitrary, we complete the proof. For the sake of convenience, we list some kernel functions which have been frequently used in the references. A If g j t,x δ t δ x, then g j u t,x u t,x, which is a local version without temporal delay, j {,...,m}, where δ is the Dirac delta function.

23 Z.-C. Wang et al. / J. Differential Equations B If g j t,x δ t p j x, then gj u t,x p j x y u t,y dy, which is a nonlocal version without temporal delay, j {,...,m}. C If g j t,x δ t τ j δ x, then gj u t,x u t τ j,x, which is a local version with a discrete temporal delay, j {,...,m}. D If g j t,x δ t τ j pj x, then gj u t,x p j x y u t τ j,y dy, which is a nonlocal version with a discrete temporal delay, j {,...,m}. E If g j t,x q j t δx, then gj u, t,x t q j t s u s, x dx, which is a local version with distributed temporal delay, j {,...,m}. Let F φt Φ φ t f φ t, g φt,...,g m φt, t R. 4.5 Following the argument in Section 3, 4.3 can be changed into the following integral equation φ t k t,sf φs + γφ s ds Aφt, t R, where k is the same as that in Section 3. Here we list some conditions, which will be used in our results. H There exists a matrix γ diag γ,...,γ n with γi >,i,...n such that f φ 2 t, g φ 2 t,..., gm φ 2 t + γφ2 t f φ t, g φ t,..., gm φ t + γφ t, where φ, φ 2 CR, R n satisfy φ t φ 2 t K in t R.

24 28 Z.-C. Wang et al. / J. Differential Equations H There exists a matrix γ diag γ,...,γ n with γi >,i,...n such that f φ 2 t, g φ 2 t,..., gm φ 2 t + γφ2 t f φ t, g φ t,..., gm φ t + γφ t, i φ t φ 2 t K in t R; where φ, φ 2 CR, R n satisfy ii e γt φ 2 t φ t ] is increasing in t R. H There exists a matrix γ diag γ,...,γ n with γi >,i,...n such that f φ 2 t, g φ 2 t,..., gm φ 2 t + γφ2 t f φ t, g φ t,..., gm φ t + γφ t, i φ t φ 2 t K in t R; where φ, φ 2 CR, R n satisfy ii e γt φ 2 t φ t ] is increasing and e γt φ 2 t φ t ] is decreasing in t R. H 2 f μ,...,μ for < μ < K; H 3 f μ,...,μ when μ or K. Lemma 4.4. If H holds and φ t BCR, R n, then F φt BCR, R n, where F is defined by 4.5. Lemma 4.5. Assume that H holds. i If f satisfies H, then F : BC, K] BCR, R n is γ-increasing, M-continuous and bounded. ii If f satisfies H, then F : BC, K] BCR, Rn is γ -increasing, M- continuous and bounded. iii If f satisfies H, then F : BC, K] BCR, Rn is γ -increasing, M- continuous and bounded. Lemma 4.6. Assume that H holds and φ BC 2 R, R n is a solution of 4.3 satisfying lim t φtα and lim t + φtβ. Then f α,...,α f β,...,β, that is, F α t F β t for any t R. We remark that Lemmas are obvious. In fact, Lemmas 4.4 and 4.5 follow from the continuity of f and Propositions 4.2 and 4.3. By Proposition 4. and a similar argument to that of 33, Proposition 2.], it is easy to see that Lemma 4.6 is also true. Now we give definitions of the upper and lower solutions of 4.3 see 33, Definition 3.2]. Definition 4.7. A continuous function φ : R R n is called an upper solution of 4.3 if φ and φ exist almost everywhere and are essentially bounded on R, and φ

25 Z.-C. Wang et al. / J. Differential Equations satisfies Dφ t + cφ t f φ t, g φt,...,g m φt, a.e. on R. 4.6 A lower solution of 4.3 is defined in a similar way by reversing the inequality in 4.6. Obviously, if ψ and are upper and lower solutions of 4.3, respectively, then ψ and satisfy C. Now we are in a position to state our main results in this section. Let Γ Γ { φ Y : φ Y : } iφ is increasing in R;, ii lim t φ t < K and lim t + φ t K. i φ is increasing in R; ii lim t φ t < K and lim t + φ t K; iii e γt φ t + s φ t ] is increasing in t R for every s>., and i φ is increasing in R; ii lim t φ t < K and lim t + φ t K; Γ φ Y : iii e γt φ t + s φ t ] is increasing in t R and e γt φ t + s φ t ]. is decreasing in t R for every s>. Theorem 4.8. Assume that H 2, H 3 and H hold. Assume further that and ψ, where BC, K] Y with, lim t t and ψ, are lower and upper solutions of 4.3, respectively. i If H holds and ψ Γ, then 4. has a travelling wave front ψ such that 4.4 holds and for a,b R with a<b, where and ψ m ψ Ca,b],R n, 4.7 D ψ m + c ψ m + γψ m F ψ m + γψ m m N 4.8 ψ ψ m ψ ψ ψ. 4.9

26 2 Z.-C. Wang et al. / J. Differential Equations ii If H holds, ψ Γ and e γt ψ t t ] is increasing in t R, then for c> min { γ i d i ; i,...,n },4. has a travelling wave front ψ such that 4.4 holds and for a,b R with a<b,4.7, 4.8 and 4.9 hold. iii If H holds, ψ Γ, e γt ψ t t] is increasing in t R and e γt ψ t t] is decreasing in t R, where min { γ i d i ; i,...,n } >, then for <c<min { γ i d i ; i,...,n }, 4. has a travelling wave front ψ such that 4.4 holds and for a,b R with a<b,4.7, 4.8 and 4.9 hold. In particular, if lim t ψ t, then ψ m ψ. Proof. i Let F : BC, K] BCR, R n be defined by 4.5. It follows from Lemma 4.5 that F : BC, K] BCR, R n is bounded, M-continuous and γ-increasing. By the definitions of upper and lower solutions, we have D + c F and Dψ + cψ F ψ. Then Theorem 3.9i implies that there exists ψ BC, K] such that 4.7, 4.8 and 4.9 hold. Since F ψ BCR, R n, it follows from Theorem 3. that that is, ψ Aψ BC 2 R, R n, D ψ + c ψ F ψ. Since ψ is increasing in R, ψ is increasing. Let lim t ψ t α and lim t + ψ t β, then f α,...,α f β,...,β follows from Lemma 4.6. So the conditions H 2 and H 3 mean that α, β {, K}. Since and ψ, we have β > and thus, β K. Since ψ t Γ and ψ ψ, lim t ψ t lim t ψ t < K. Hence, α. Thus, we complete the proof of i. ii It follows from Lemma 4.5ii that F : BC, K] BCR, R n is bounded, M-continuous and γ -increasing. By Theorem 3.9ii, Lemmas 3.6ii and 4.5ii, and employing an argument similar to that of i, we can complete the proof of ii. iii It follows from Lemma 4.5iii that F : BC, K] BCR, R n is bounded, M-continuous and γ -increasing. Noting that Theorem 3.9iii, Lemmas 3.6iii and 4.5iii, the conclusion can be proved following a similar argument to that of i. If lim t ψ t and lim t ψ t K, then for given ε>, there exists M> such that max sup i n t M max sup i n t M ψi t ε < 2, max sup i n t M ψ i t < ε 2, max i n sup t M ψi t K i ε < 2, ψ i t K i ε < 2,

27 so for any m N, Z.-C. Wang et al. / J. Differential Equations max sup i n t M ψ m i t < ε 2, max sup i n t M ψ m i t K i < ε 2. Consequently, max sup i n t M ψ m i t ψ i t <ε. By 4.7, there exists N N such that ψ m ψ C M,M],R n <ε for all m N. Thus, ψ m ψ <ε for all m N. Therefore, ψ m ψ. The proof is complete. Corollary 4.9. Assume that H 3 and H hold, and f μ,...,μ for <δ μ < K, where δ R n. Also assume that and ψ, where BC, K] Y with sup t R t δ, ψ and lim t ψ t, are lower and upper solutions of 4.3, respectively. Then i iii of Theorem 4.8 hold. 5. Applications In 5], Britton proposed a model for a single biological population of the form u t,x t u t,x + u t,x + aut,x + ag ut,x], where a>, g is a given function and g u represents a convolution in the spatial variable. In this equation, the term au with a> represents an advantage in local aggregation, the term + a g u represents a disadvantage in global population levels being too high because of the resultant depletion of resources. Under recognizing that animals take time to move, he proposed a spatio-temporal average model weighted toward the current time and position of the form u t,x t u t,x + u t,x + aut,x + ag ut,x], 5. where t g ut,x g t ξ,x y u ξ,y dy dξ, Ω g and g t,x g t, x, x Ω R.

28 22 Z.-C. Wang et al. / J. Differential Equations In 6], Britton considered three kinds of bifurcations from the uniform steady-state solution u, that is, i steady spatially periodic structures, ii periodic standing wave solutions, and iii periodic travelling wave solutions. In the following, we consider the existence of travelling wave fronts of 5., where Ω R. This system has two equilibria: u and u. Obviously, the travelling wave equation of 5. corresponding to 4.3 is φ t + cφ t φ t + aφ t + ag φt ], t R, 5.2 where g φt g ξ,y φ t cξ y dy dξ. Let K. Then the travelling wave front φ of 5. satisfies the asymptotic boundary condition Let lim φ t and lim φ t. 5.3 t t + λ c c a 2 and λ 2 c + c a 2 be two real positive roots of the equation λ 2 cλ + + a, where c 2 + a. Let λ 3 c + c 2 + 4a. 2 Then λ 3 satisfies λ 2 cλ a. Now we let ψ t + αe λ t and { } t min εe λ3t,ε, <ε< + α, 5.4 where α is a positive constant.

29 Z.-C. Wang et al. / J. Differential Equations Lemma 5.. i ψ t / + αe λ t is increasing in t R and satisfies the asymptotic boundary condition 5.3; ii ψ t t for t R. Proof. We only show ii. For t, ψ / + α, t ε. Since ψ t / + αe λ t is increasing, then ψ t / + α >ε t. For t<, ψ t / + αe λ t, t εe λ3t. Since λ 3 > λ >, then e λ3t < and e λ 3 λ t <. Consequently, ψ t t + αe λ t ε εα + αe λ t εe λ 3t εeλ 3t εαe λ 3 λ t + αe λ t >. The proof is complete. Lemma 5.2. Assume γ > λ. For sufficiently small α and ε, the following statements hold: i e γt ψ t t ] is increasing and e γt ψ t t ] is decreasing in t R; ii e γt ψ t + s ψ t ] is increasing and e γt ψ t + s ψ t ] is decreasing in t R for every s>. Proof. i First, we show that e γt ψ t t ] is increasing in t R. For t>, t ε and e λ t, by a direct calculation, we have d {e γt dt + αe λ t ]} ε e γt γ γε + α 2γε ] + γ γεαe λt + λ e λ t + αe λ t 2 ] e γt γ γε + α 2γε + γ γεα + λ e λ t + αe λ t 2. For t<, t εe λ 3t, e λ 3t and e λ 3 λ t, a direct calculation yields d {e γt dt + αe λ t ]} εe λ 3t

30 24 Z.-C. Wang et al. / J. Differential Equations ] e γt γ γ + λ 3 εe λ 3t + αe λ t 2 ] αe γ λ t 2γεe λ3t + γ γεαe λ 3 λ t 2λ 3 εe λ3t λ 3 εαe λ 3 λ t + λ + + αe λ t 2 ] e γt γ γε λ 3 ε + α 2γε + γ γεα 2λ 3 ε λ 3 εα + λ e λ t + αe λ t 2. If let ε and α, then we have d { e γt ψ t t ]}. dt Second, we show that e γt ψ t t ] is decreasing in t R. For t>, t ε and e λ t, by a direct calculation we have d {e γt dt + αe λ t ]} ε e γt γ + γε + α 2γε ] γ + γεαe λt + λ e λ t + αe λ t 2 ] e γt γ + γε + α 2γε γ + γεα + λ e λ t + αe λ t 2. For t<, t εe λ 3t, e λ 3t and e λ 3 λ t, a direct calculation implies that ]} d {e γt dt + αe λ εe λ 3t t ] e γt γ + γ λ 3 εe λ 3t + αe λ t 2 αe γ+λ t 2γεe λ3t γ + γεαe λ 3 λ t 2λ 3 εe λ3t λ 3 εαe λ 3 λ t + λ + + αe λ t 2 ] e γt γ + γε + α 2γε γ + γεα + λ e λ t + αe λ t 2.

31 Z.-C. Wang et al. / J. Differential Equations In view of γ > λ, ε and α, then we have d { e γt ψ t t ]}. dt ii First, we show that e γt ψ t + s ψ t ] is increasing in t R for every s>. Fix s>. Since e λ s <, by a direct calculation, it follows that d { e γt ψ t + s ψ t ]} dt d ]} {e γt dt + αe λ t+s + αe λ t α γ λ e γ λ t e λ s + αe λs e λ t + αe λ t + αe λ t+s 2 + αe λ t 2 α 2 λ e λt e γ λ t e λ s + e λs + 2αe λs e λ t + + αe λ t+s 2 + αe λ t 2 αe γ λ t e λ s γ λ + γαe λ t + e λ s + γ + λ α 2 e λ s e 2λ t ] + αe λ t+s 2 + αe λ t 2. Since γ > λ, we can see that d { e γt ψ t + s ψ t ]} >. dt Second, we show that e γt ψ t + s ψ t ] is decreasing in t R for every s>. Fix s>. Since e λ s <, we have d { e γt ψ t + s ψ t ]} dt d ]} {e γt dt + αe λ t+s + αe λ t

32 26 Z.-C. Wang et al. / J. Differential Equations α γ + λ e γ+λ t e λ s + αe λs e λ t + αe λ t + αe λ t+s 2 + αe λ t 2 α 2 λ e λt e γ+λ t e λ s + e λs + 2αe λs e λ t + + αe λ t+s 2 + αe λ t 2 αe γ+λ t e λ s γ + λ + γαe λ t + e λ s + γ λ α 2 e λ s e 2λ t ] + αe λ t+s 2 + αe λ t 2. Taking into account that γ > λ,wehave d { e γt ψ t + s ψ t ]} <. dt The proof is complete. Lemmas 5. and 5.2 imply that ψ t Γ, ψ t Γ, ψ t Γ and < sup t R t ε. Lemma 5.3. t defined by 5.4 is a lower solution of 5.2. Proof. For t>, t ε, t, t, we have t + c t f t, g t t + c t t + a t + ag t ] t a 2 t + + a tg t ε aε a ε 2 ε ε <. For t<, t εe λ 3t, t ελ 3 e λ 3t, t ελ 2 3 eλ 3t,wehave t + c t f t, g t t + c t t + a t + ag t ] t + c t t a 2 t + + a tg t t + c t t a 2 t + + a t t + c t + a t ε λ cλ 3 + a e λ3t. This implies that t is a lower solution of 5.2.

33 Z.-C. Wang et al. / J. Differential Equations Now we show that ψ t is an upper solution of 5.2 by choosing different kernel functions g. Here we consider three cases: i g t,x τ e t τ δ x, τ > ; ii g t,x δ t e x2 4ρ, ρ > ; 4πρ iii g t,x τ e t τ 4πt e x2 4t, τ >. 5.. The case g t,x τ e t τ δ x, τ > In this case, Eq. 5. becomes a reaction diffusion model with temporal delay. Obviously, g t,x τ e t τ δ x satisfies H. Then and g φt τ e ξ τ φ t cξ dξ ] f φ t, g φt φ t + aφ t + a τ e ξ τ φ t cξ dξ. Lemma 5.4. For sufficiently small τ >, f φ t, g φt satisfies H. Proof. Let φ, φ 2 C R, R with φ t φ 2 t K so that e γt φ 2 t φ t] is increasing in t R. It is easy to see that for any s R, e γt φ 2 s + t φ s + t] is increasing in t R, then for γ > 3a + 2 and sufficiently small τ > satisfying γcτ /2, f φ 2 t, g φ 2 t f φ t, g φ t φ 2 t + aφ 2 t + a ] g φ 2 t φ t + aφ t + a ] g φ t φ 2 t φ t ] ] + a φ 2 2 t φ2 t + a φ 2 t g φ 2 t φ t ] g φ t φ 2 t φ t ] + aφ 2 t + aφ t ] + a φ 2 t φ t ] g φ 2 t + a φ t ] g φ 2 t g φ t a φ 2 t φ t ] + a ] g φ 2 t g φ t

34 28 Z.-C. Wang et al. / J. Differential Equations a φ 2 t φ t ] + a τ e ξ τ φ2 t cξ φ t cξ ] dξ a φ 2 t φ t ] { + a τ e ξ τ e γcξ e γcξ φ 2 t cξ φ t cξ ]} dξ a φ 2 t φ t ] + a τ e ξ τ e γcξ φ 2 t φ t ] dξ ] a + + a τ e ξ τ e γcξ φ2 dξ t φ t ] a + + a φ2 t φ γcτ t ] 3a + 2 φ 2 t φ t ] γ φ 2 t φ t ]. The proof is complete. Lemma 5.5. For sufficiently small τ >, ψ t defined by 5.4 is an upper solution of 5.2. Proof. Since ψ t αλ e λ t + αe λ t 2, ψ t αλ2 e λt + α 2 λ 2 e 2λ t + αe λ t, 3 and for τ > such that 2cλ τ >, g ψt τ e ξ τ + αe λ t cξ ] e ξ + τ + αe λ + t cξ + αe λ t cαλ e λ t dξ e ξ τ d + αe λ t cξ de + αe λ t cξ ξ cλ e τ + αe λ t cξ ] 2 dξ ξ τ + αe λ t + cαλ τe λ t cλ cλ τ e τ ξ + αe λ t cξ ] 2 +

35 Z.-C. Wang et al. / J. Differential Equations cαλ τe λ t cλ τ cλ e τ ξ d + αe λ t cξ ] 2 + αe λ t + 2c2 α 2 λ 2 τe 2λ t cλ τ cαλ τe λ t cλ τ + αe λ t 2 2cλ τ ξ e + αe λ t cξ ] 3 dξ + αe λ t cαλ τe λ t cλ τ + αe λ t 2, we have ψ t + cψ t f ψ t, g ψt ψ t + cψ t ψ t + aψ t + ag ψt ] ψ t + cψ t + a ψ t + + a ψ tg ψt αλ2 e λ t + α 2 λ 2 e 2λ t + αe λ t 3 + cαλ e λ t + αe λ t 2 + a + αe λ t + a + + αe λ t 2 + a cαλ τe λ t cλ τ + αe λ t 3. Let the right part of the last inequality be C/ + αe λ 3 t. Then C αλ 2 e λ t α 2 λ 2 e 2λ t + cαλ e λ t + cα 2 λ e 2λ t 2 + a αe λ t + a α 2 e 2λ t + + a αe λ t + a cαλ τ e λ t cλ τ ] α 2 λ 2 + cλ + a e 2λ t +α λ 2 + cλ + a + a cλ τ cλ τ α 2λ 2 + a cλ ] τ e λt. cλ τ ] e λ t

36 22 Z.-C. Wang et al. / J. Differential Equations Obviously, for sufficiently small τ, which implies that C>. Thus 2λ 2 + a cλ τ cλ τ >, ψ t + cψ t f ψ t, g ψt. The proof is complete. Now by Theorem 4.8ii, the following result is true. Theorem 5.6. For any c 2 + a, there exists τ c > such that for any τ < τ c, system 5. has a travelling wave front which satisfies The case g t,x δ t e x2 4ρ, ρ > 4πρ In this case, Eq. 5. becomes a reaction diffusion model with a nonlocal spatial delay. It is easy to see that the kernel g satisfies H, g φt 4πρ e y2 4ρ φ t y dy and ] f φ t, g φt φ t + aφ t + a e y2 4ρ φ t y dy. 4πρ Lemma 5.7. Assume that γ >a+ 2 + a e ργ2. Then H. f φ t, g φt satisfies Proof. Assume that φ, φ 2 C R, R and satisfy that φ t φ 2 t K, e γt φ 2 t φ t ] is increasing in t R and e γt φ 2 t φ t ] is decreasing in t R. Then f φ 2 t, g φ 2 t f φ t, g φ t a φ 2 t φ t ] + a ] g φ 2 t g φ t a φ 2 t φ t ] + a 4πρ e y2 4ρ φ 2 t y φ t y ] dy

37 Z.-C. Wang et al. / J. Differential Equations a φ 2 t φ t ] + a e y2 4ρ e γy { e γy φ 4πρ 2 t y φ t y ]} dy + a e y2 4ρ e γy { e γy φ 4πρ 2 t y φ t y ]} dy a φ 2 t φ t ] + a e y2 4ρ e γy φ 4πρ 2 t φ t ] dy + a e y2 4ρ e γy φ 4πρ 2 t φ t ] dy a φ 2 t φ t ] + a + a e y+2γρ2 4ρ e ργ2 φ2 t φ 4πρ t ] dy e y 2γρ2 4ρ e ργ2 φ2 t φ 4πρ t ] dy a φ 2 t φ t ] + a e ργ2 φ2 t φ t ] e y 2γρ2 4ρ dy 4πρ + a e ργ2 φ2 t φ t ] e y+2γρ2 4ρ dy 4πρ a a e ργ2] φ2 t φ t ] γ φ 2 t φ t ], The proof is complete. Lemma 5.8. For sufficiently small ρ >, ψ t defined by 5.4 is an upper solution of 5.2. Proof. Let Then F y,ρ y 4πρ e ξ2 4ρ dξ. y F y,ρ 4πρ e y2 4ρ, F, ρ, F, ρ 2.

38 222 Z.-C. Wang et al. / J. Differential Equations Since lim y e λ y F y,ρ, we have Now define Then e λ y F y,ρ dy λ F y,ρ de λ y ] e λy F y,ρ λ + λ + e ρλ2 2λ λ + 2ρλ e ρλ2 2λ λ 2λ + 2λ e ρλ2 + λ e ρλ2 e y+2ρλ 2 4ρ dy 4πρ 4πρ e y2 4ρ dy 2ρλ 2λ + 2λ e ρλ2 + λ π e ρλ2 G y,ρ y 4πρ e y2 4ρ λ y dy 4πρ e y2 4ρ dy ρλ e y2 dy. e λ ξ F ξ, ρ dξ. G, ρ + e ρλ2 + ρλ e ρλ2 e y2 dy, 2λ 2λ λ π G, ρ. Similarly, we can define and obtain G + y,ρ y e λ ξ F ξ, ρ dξ G +, ρ e ρλ2 + ρλ e ρλ2 e y2 dy, 2λ 2λ λ π G +, ρ.

39 Z.-C. Wang et al. / J. Differential Equations Furthermore, e λ y G y,ρ dy < and e λ y G + y,ρ dy <. Consequently, it follows that g ψt + 4πρ e y2 4ρ + + αe λ t y dy e y2 4ρ 4πρ + αe λ t y dy e y2 4ρ 4πρ + αe λ t+y dy e y2 4ρ 4πρ + αe λ t y dy + αe λ df y,ρ + t+y + αe λ df y,ρ t y F y,ρ + αe λ t+y ] F y,ρ + + αe λ t y ] αλ e λ t + αλ e λ t 2 + αe λ t αλ e λ t αe λ t + αλ e λ t + αe λ t +2α 2 λ 2 e 2λ t e λ y F y,ρ + αe λ t+y ] 2 dy αλ e λ G t y,ρ + αe λ t+y ] 2 e λ y G y,ρ + αe λ t+y ] 3 dy + αλ e λ G t + y,ρ + αe λ t+y ] 2 ] e λ y F y,ρ + αe λ t y ] 2 dy + αe λ t+y ] 2 dg y,ρ + αe λ t y ] 2 dg + y,ρ ]

40 224 Z.-C. Wang et al. / J. Differential Equations α 2 λ 2 e 2λ t e λ y G + y,ρ + αe λ t y ] 3 dy + αe λ t + αe λ t + αλ e λ t + αe λ t ] G+, ρ G 2, ρ ] + αe λ t e ρλ2 + αe λ t ]. 2 Therefore, we have ψ t + cψ t f ψ t, g ψt ψ t + cψ t + a ψ t + + a ψ tg ψt αλ2 e λ t + α 2 λ 2 e 2λ t + αe λ t 3 + a α + a e λ t + + αe λ t αe λ t 3 ] α 2 λ 2 cλ + + a e 2λ t + αe λ t 3 α + α λ 2 + cλ + a e ρλ2 + αe λ t 3 λ 2 + cλ + a e ρλ2 + αe λ t 3 + cαλ e λ t + αe λ t 2 + a + αe λ t e ρλ2 ] e λ t ] e λ t ] α 2λ a e ρλ2 e λ t + αe λ t. 3 Since 2λ 2 > and lim e ρλ2 ρ for sufficiently small ρ >, we finally have ψ t + cψ t f ψ t, g ψt. This completes the proof.