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Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction-diffusion systems. By using Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka-Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka-Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.


Introduction
In studying the nonlinear dynamics of delayed reaction-diffusion systems, one of the important topics is the existence of traveling wave solutions because of their significant roles in biological invasion and epidemic spreading (we refer to [2,4,[6][7][8]10,[12][13][14][15][16][17][18]20,21,24,27,29,[31][32][33][34]37]). It has been shown that delay may induce some differences of traveling wave solutions between the delayed and undelayed systems, for example, the minimal wave speed (see Schaaf [24], Zou [39]) and the monotonicity of traveling wave solutions in scalar equations (see an example by Faria and Trofimchuk [7]). In particular, the asymptotic behavior of traveling wave solutions, which is often formulated by the asymptotic boundary conditions, plays a very crucial role since it describes the propagation processes in different natural environments. For example, with proper asymptotic boundary value conditions, the traveling wave solutions of two species competition-diffusion systems may reflect the coinvasion-coexistence of two invaders (see [1,13,15,28]), or exclusion between an invader and a resident (see [4,11]). Therefore, understanding the asymptotic behavior is a fundamental issue in the study of traveling wave solutions. Moreover, very precise asymptotic behavior of traveling wave solutions has also been studied due to the development in mathematical theory of delayed reaction-diffusion systems, such as in the asymptotic stability and uniqueness of traveling wave solutions (see [20,30,34]).
To obtain the asymptotic behavior of traveling wave solutions in delayed reactiondiffusion systems, there are several methods. The first is based on the monotonicity of traveling wave solutions, which is often considered under the assumption of quasimonotonicity in the sense of proper ordering [10,16,32,37]. The second is to construct proper auxiliary functions, such as the upper and lower solutions [13,15]. When these methods fail, a fluctuation technique is utilized to study the asymptotic behavior of traveling wave solutions if the system satisfies the locally quasimonotone condition near the unstable steady state [17,31]. Some other results have also been presented for (fast) traveling wave solutions of scalar equations [6][7][8]12].
Of course, before considering the asymptotic behavior of traveling wave solutions of delayed systems, we must establish the existence of nontrivial traveling wave solutions. To obtain the existence of traveling wave solutions of delayed systems, Wu and Zou [37] used a monotone iteration scheme if a delayed system is cooperative in the sense of proper ordering, see also Ma [16]. Li et al. [13] further considered the existence of traveling wave solutions in competition systems by a cross iteration technique. Very recently, Ma [17] studied the traveling wave solutions of a locally monotone delayed equation by constructing auxiliary monotone equations, see also Wang [31]. Moreover, by regarding the time delay as a parameter, the existence of traveling wave solutions was also studied by the perturbation method or Banach fixed point theorem, see Gourley and Ruan [4], Ou and Wu [21].
In this paper, we first study the existence and asymptotic behavior of traveling wave solutions in the following delayed reaction-diffusion system is the continuous functional space defined on [−τ, 0] and valued in R n , which is a Banach space equipped with the supremum norm.
To overcome the difficulty arising from the deficiency of comparison principle, we introduce the definition of generalized upper and lower solutions of corresponding wave system of delayed system (1.1). By Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Motivated by the idea of contracting rectangles in evolutionary systems, we establish an abstract conclusion on the asymptotic behavior of positive traveling wave solutions in general partial functional differential equations. Subsequently, we consider two scalar delayed equations with diffusion in population dynamics (see Ma [17] and Zou [39]), which implies that our methods can also be applied to some well studied models.
In population dynamics, the following Lotka-Volterra reaction-diffusion system with distributed delay has been widely studied in which i ∈ I = {1, 2, · · · , n}, x ∈ Ω ⊆ R k , k ∈ N, t > 0, u = (u 1 , u 2 , · · · , u n ) ∈ R n , u i (x, t) denotes the density of the i−th competitor at time t and in location x ∈ Ω, d i > 0, r i > 0, c ii > 0 and c ij ≥ 0 are constants for i, j ∈ I, i = j. We also suppose that which will be imposed throughout the paper. Let and set Clearly, a i > 0 implies the existence of instantaneous self-limitation effect in population dynamics.
The dynamics of (1.2) has been studied by several authors, for example, Gourley and Ruan [4], Fang and Wu [5], Li et al. [13], Lin et al. [15], Martin and Smith [19] and Ruan and Wu [23]. More precisely, when Ω is a bounded domain and (1.2) is equipped with the Neumann boundary condition, Martin and Smith [19] proved that if the initial values of (1.2) are positive and then the unique mild solution to (1.2) satisfies hereafter u * = (u * 1 , u * 2 , · · · , u * n ) is the unique spatially homogeneous positive steady state of (1.2), of which the existence can be obtained by (1.3). It is well known that a i < 1 implies the existence of time delay in intra-specific competition, which often leads to some significant differences between the dynamics of delayed and undelayed models if the time delay is large. For example, the following Logistic and Hutchinson equations exhibit dramatically different dynamics, we refer to Ruan [22] for detailed analysis on these two equations and Hale and Verduyn Lunel [9] and Wu [35] for fundamental theories on delayed equations. However, (1.4) does not depend on the size of delays and the distribution of η ii (s) for s ∈ [−τ, 0). Similar phenomena can be founded in the corresponding functional differential equations, see Smith [26,Section 5.7]. In particular, Li et al. [13] and Lin et al. [15] have established the existence of traveling wave solutions to (1.2), which models the invasion-coexistence scenario of multiple competitors. However, these results only hold if there exists τ 0 ∈ [0, τ ] small enough satisfying 0 −τ 0 dη ii (s) = 1, i ∈ I, which ensures the so-called exponentially monotone condition such that the upper and lower solutions are admissible. If τ 0 is large, then we cannot apply these techniques and results to study the existence and asymptotic behavior of traveling wave solutions of (1.2). In this paper, we shall consider the existence and further properties of nontrivial traveling wave solutions of (1.2) with a i > 0, i ∈ I.
Using the results on the existence and asymptotic behavior of traveling wave solutions in system (1.1), model (1.2) with Ω = R is studied by presenting the existence, nonexistence and asymptotic behavior of positive traveling wave solutions. In particular, due to less requirements for auxiliary functions, we obtain some sufficient conditions on the existence of nonmonotone traveling wave solutions of (1.2) with a i = 1, i ∈ I. Note that these conclusions remain true if τ = 0, we thus confirm the conjecture about the existence of nonmonotone traveling wave solutions of competitive systems, which was proposed by Tang and Fife [28, the last paragraph].
The rest of this paper is organized as follows. In Section 2, we list some preliminaries. Using contracting rectangles, the asymptotic behavior of traveling wave solutions of general partial functional differential equations (1.1) is established in Section 3, and is applied to two examples considered by Ma [17] and Zou [39]. In Section 4, we introduce the generalized upper and lower solutions and study the existence of traveling wave solutions in (1.1). In Section 5, we investigate the traveling wave solutions of the Lotka-Volterra system (1.2), including the existence, nonexistence, asymptotic behavior and monotonicity. This paper ends with a brief discussion of our methods and results.

Preliminaries
In this paper, we shall use the standard partial order and interval notations in R n .
Moreover, X will be interpreted as follows X = {u : u is a bounded and uniformly continuous function from R to R n }, which is a Banach space equipped with the supremum norm · . If a, b ∈ R n with a ≤ b, then is a nonnegative, positive and strictly positive function iff u(x) ≥ 0, u(x) > 0 and u(x) ≫ 0, respectively. Let σ > 0 be a constant and | · | denote the supremum norm in R n . Define Then it is easy to check that (B σ (R, R n ) , |·| σ ) is a Banach space.

Consider the Fisher equation
with d > 0, r > 0 and n = 1 in the definition of X.
Lemma 2.1 For (2.1), we have the following conclusions.
then z(x, t) and z(x, t) are upper and lower solutions to (2.1), respectively. Furthermore, we have For Lemma 2.1, we refer to Aronson and Weinberger [3], and Ye et al. [38].

Asymptotic Behavior of Traveling Wave Solutions
Consider the following functional differential equation corresponding to (1.1) in which l = (l 1 , l 2 , · · · , l n ) ∈ R n , f i is defined by (1.1) and satisfies the following assumptions: (H4) there exists a one-parameter family of order intervals given by where a(y) and b(y) are continuous in y ∈ [0, 1]; (H5) (y) is a strict contracting rectangle, namely, let a(y) = (a 1 (y), a 2 (y), · · · , a n (y)), b(y) = (b 1 (y), b 2 (y), · · · , b n (y)), then for any y ∈ (0, 1) and u ∈ (y), we have To continue our discussion, we now introduce the following definition of traveling wave solutions of (1.1).
Were the statement false, then Φ + > Φ − holds and (3.3) implies that there exists y ∈ (0, 1) such that In particular, let y 1 ∈ (0, 1) be the largest y such that the above is true, then y 1 is well defined. Without loss of generality, we assume that with a(y) = (a 1 (y), a 2 (y), · · · , a n (y)). Due to the uniform boundedness of φ ′ 1 and φ ′′ 1 , the fluctuation lemma of continuous functions implies that there exists At the same time, (H5) leads to and we obtain a contradiction between The proof is complete.

Remark 3.3
In fact, the proof of existence of traveling wave solutions often implies the uniform boundedness of Φ ′ , Φ ′′ and we will not discuss the boundedness in the following two examples. Now, we recall two scalar equations to give a simple illustration of the theorem. We first consider the example in Zou [39].
Then traveling wave solutions of the corresponding partial functional differential equation has been established by Zou [39].
If the quasimonotone condition does not hold, Ma [17] studied the traveling wave solutions by constructing proper auxiliary systems. In particular, the traveling wave solutions of the following example have been well studied.
and denote f (w) = e 2 we −w .
Using the results in Smith [26, Section 5.2], we can give two auxiliary quasimonotone equations to study the dynamics of the corresponding functional differential equations of (3.5). In addition, Ma [17] presented two auxiliary quasimonotone equations of (3.5) and obtained the convergence of traveling wave solutions. Let be a traveling wave solution of (3.5) and denote lim inf then the auxiliary equations in Ma [17] imply that and f (w) is monotone decreasing for w ∈ [e 3−e , e]. If ρ + = e, then a discussion similar to that of (3.4) implies that f (ρ − ) ≥ e by the monotonicity of f, which is impossible by (3.6). By the monotonicity of f and (3.6), we further obtain that Let E = f (f (e 3−e )), E = f (f (e)), then [ E, E] defines an invariant region of the corresponding functional differential equation of (3.5).

Generalized Upper and Lower Solutions
In this section, we shall establish the existence of traveling wave solutions of delayed systems (1.1), where f satisfies (H1)-(H3) in Section 3. We first introduce the generalized upper and lower solutions of (3.2) as follows.
Now, to prove the existence of (3.2), it is sufficient to seek after a fixed point of F (see [37]).
are a pair of generalized upper and lower solutions of (3.2) such that It is clear that Γ is nonempty and convex. Moreover, it is closed and bounded with respect to the decay norm | · | σ . Choose Φ = (φ 1 , φ 2 , · · · , φ n ) ∈ Γ, then for each fixed ξ ∈ R, the definition of β implies that

Traveling Wave Solutions of the Lotka-Volterra System
(1.2) In this section, we study the existence and nonexistence of traveling wave solutions of (1.2) with Ω = R, a i > 0, i ∈ I. We first introduce some notations.
Proof. By the monotonicity, it suffices to prove that if ψ i (ξ) and ψ i (ξ), i ∈ I, are differentiable.
and this completes the proof of (5.5).
and the monotonicity of ψ j indicates that Therefore, we only need to verify that Since q > 1, we have ξ < 0 and e ηγ i1 ξ > e 2γ i1 ξ , e ηγ i1 ξ > e (γ i1 +γ j1 )ξ , which imply that (5.7) is true if then (5.6) is true and we complete the proof. From Lemma 5.2, Theorem 4.3 and Remark 4.4, we obtain the following result. (5.8)

Asymptotic Behavior of Traveling Wave Solutions
The following is the main conclusion of this subsection. Proof. Note that Ψ(x + ct) is a special classical solution of the following initial value problem The boundedness and smoothness of Ψ(x + ct) imply that ψ i (x + ct) is an upper solution to the following Fisher equation Thus Lemma 2.1 asserts that then there exists s ∈ (0, 1] such that a(s) = (a 1 (s), a 2 (s), · · · , a n (s)), b(s) = (b 1 (s), b 2 (s), · · · , b n (s)).
By Smith [26,Lemma 7.4], there exists a constant ǫ > 0 such that [a(s), b(s)] defines a strictly contracting rectangle of the corresponding functional differential equations of (1.2). Applying Theorem 3.2, we complete the proof. Furthermore, from the proof of Theorem 5.4, we obtain the following result.
Clearly, we have the following generalization. Therefore, it also improves some results of undelayed systems by confirming the nonexistence of traveling wave solutions without the requirement of monotonicity, for example, Ahmad and Lazer [1] and Tang and Fife [28].

Existence of Nonmonotone Traveling Wave Solutions
If τ = 0, Ahmad and Lazer [1], Tang and Fife [28] proved the existence of monotone traveling wave solutions of (1.2). Recently, Fang and Wu [5] also confirmed the existence of monotone traveling wave solutions if τ is small and n = 2. In particular, Tang and Fife [28] thought that the monotonicity was a technical requirement and conjectured the existence of nonmonotone traveling wave solutions if τ = 0 and n = 2.
In the previous section, we obtained the existence of traveling wave solutions connecting 0 with u * . Because our requirement for auxiliary functions was very weak, we can present some sufficient conditions of the existence of nonmonotone traveling wave solutions if a i = 1 for all i ∈ I.
By rescaling, it suffices to consider (1.2) with c ii = 1, i ∈ I. Then (1.3) implies that c ij < 1, i = j, i, j ∈ I, which further indicates that we can obtain a fixed q such that Lemma 5.2 holds for any fixed c (so η can be a constant) and for all c ij < 1, i = j, i, j ∈ I.
Therefore, for each fixed c, there exist m i > 0 (e.g., m i = sup ξ∈R ψ i (ξ)) independent of c ij such that sup Let u * be the function of c ij with i = j, i, j ∈ I, then there exist c ij with i = j, i, j ∈ I, such that ( To further illustrate our conclusions, we also give the numerical results. Take Then the nonmonotonic traveling wave solutions are presented in Figures 2-3.
In particular, we shall suppose that τ is the real maximum delay involved in (5.13), namely, there is no τ 0 < τ such that and (5.14) will be imposed in this subsection.
Let v(x, t) = φ(x + ct) be a traveling wave solution of (5.13). Then it satisfies Clearly, if c > 2 √ dr or c < 2 √ dr, then the existence or nonexistence of (5.15)-(5.16) has been addressed by the previous results. The following result deals with the case when c = 2 √ dr.
Proof. Let {c n } be a decreasing sequence with c n < 4 √ dr and c n → 2 √ dr, n → ∞. Then for each c n , (5.13) has a positive traveling wave solution connecting 0 with 1, denoted by φ n (ξ). It follows that 0 ≤ φ n (ξ) ≤ 1/b, ξ ∈ R, n ∈ N. Note that a traveling wave solution is invariant in the sense of phase shift, so we assume that By (5.16), we know that (5.17) is admissible. For n ∈ N, ξ ∈ R, it is evident that φ n (ξ) are equicontinuous and bounded in the sense of the supremum norm. By Ascoli-Arzela lemma and a nested subsequence argument, {φ n (ξ)} has a subsequence, still denoted by {φ n (ξ)}, such that for a continuous function φ(ξ). We see that the above convergence is pointwise on R, and is also uniform on any bounded interval of R.
Define lim sup It is clear that If φ − > 0, then the dominated convergence theorem in F implies that which is impossible by (5.17). Hence, φ − = 0 follows.

Discussion
Delay is an universal phenomena in the evolutionary process, and many important and realistic models with delay have been proposed, we refer to Hale and Verduyn Lunel [9] and Wu [35] for different delayed systems. However, the dynamics between the delayed and the undelayed systems may have significant differences, e.g., the delayed Logistic equation or Hutchinson equation may have a nontrivial periodic solution while all the nonnegative solutions of the Logistic equation converge to the positive steady state. Furthermore, to study delayed systems, more complex phase spaces than that of the corresponding undelayed systems are required. The investigation of traveling wave solutions of delayed systems is also more difficult than that of the corresponding undelayed systems, at least the phase plane method which is powerful in studying undelayed systems meets some difficulties in the study of delayed systems.
Of course, if a system is (local) quasimonotone, then the classical theory established for monotone semiflows is applicable and there are plentiful results. For example, the existence, nonexistence, minimal wave speed, uniqueness and stability of traveling wave solutions have been widely studied and many sharp results have been established, see [14,24,27,29,33,34]. If the system is not quasimonotone, then the study becomes harder and some new phenomena can occur, e.g., the existence of nonmonotone traveling wave solutions in scalar equations, see [7]. When the delay is small enough, some nice results on traveling wave solutions can also be obtained by different techniques such as exponential ordering, perturbation and so on, see [2,6,15,21,32,37].
However, if the delay is large, these results cannot deal with the traveling wave solutions of delayed systems including (1.2) and (5.13). In this paper, we applied generalized upper and lower solutions to seek after the positive traveling wave solutions. Since these systems do not satisfy the quasimonotone condition, the limit behavior of traveling wave solutions cannot be obtained by the monotonicity of traveling wave solutions and the dominated convergence theorem. In particular, for the case c = 2 √ dr in (5.13), the asymptotic behavior of traveling wave solutions cannot be considered by the techniques of monotone traveling wave solutions in [14,29]. In this paper, we studied the asymptotic behavior of traveling wave solutions of (1.2) and (5.13) by combining the idea of contracting rectangles with the theory of asymptotic spreading.
Our results imply that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competition terms may not affect the persistence of traveling wave solutions. However, very likely large delay may also lead to some significant differences between the traveling wave solutions of delayed and undelayed systems, e.g., the nonexistence of monotone traveling wave solutions of (5.13), which will be investigated in our future studies.