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SPREADING SPEED AND TRAVELING WAVES FOR A NON-LOCAL DELAYED REACTION-DIFFUSION SYSTEM WITHOUT QUASI-MONOTONICITY

. A non-local delayed reaction-diﬀusion model with a quiescent stage is investigated. It is shown that the spreading speed of this model without quasi-monotonicity is linearly determinate and coincides with the minimal wave speed of traveling waves.


1.
Introduction. The notion of asymptotic speed of spread (spreading speed for short) was first introduced by Aronson and Weinberger [1,2] for reaction-diffusion equations. Since then, there have been many works to reveal the link between the traveling wave solutions and the spreading speeds for various types of evolution systems (see, e.g., [20,35] and references therein). A general theory of spreading speeds and traveling wave has been greatly developed in [14,15,16,17,25] for monotone semiflows so that it can be applied to various evolution equations admitting the comparison principle. For the results about the asymptotic speed of spread for non-monotone problems, we can refer to [5,6,12,22,23,27,29,34] and their references.
To describe a population where the individuals alternate between mobile and nonmobile states, Hadeler and Lewis [9] presented and discussed briefly the following single population model with mobile and non-mobile stages where u(x, t) and v(x, t) are the densities of the mobile and non-mobile subpopulations at location x and time t, respectively, D is the diffusion coefficient, γ 1 and γ 2 are the switching rates, and f is the reproduction function. All of the parameters in 4064 ZHENGUO BAI AND TINGTING ZHAO this model are positive constants. Zhang and Zhao [32] investigated the asymptotic behavior for system (1) in both unbounded and bounded spatial domains. Zhang and Li [31] further established the monotonicity and uniqueness of traveling waves for system (1). In model (1), it is assumed that the growth rate of the population acts instantaneously. However, due to the duration of gestation, maturation and hatching period, there may be a time delay in a population model. Thus, Wu and Zhao [28] considered the following delayed reaction-diffusion system where the reproduction function is f (u(x, t), u(x, t − τ )), τ > 0 is a constant. They established the existence of the minimal wave speed, monotonicity and uniqueness (up to translation) of the traveling wave fronts under the assumption that f (u, v) is monotone with respect to v. However, f (u, v) may not be monotone with respect to v in practical problems. For example, if f is chosen as f (u(x, t), u(x, t − τ )) = −du(x, t) + pu(x, t − τ )e −au(x,t−τ ) for d, p, a > 0, then it is not monotone with respect to the second variable. Recently, by using the method in [6,12,27], Zhao and Liu [34] obtained the spreading speed and its coincidence with the minimal wave speed of traveling wave solutions of system (2) in non-quasi-monotone case.
In ecology, since populations took time to move in space and usually were not at the same position in space at previous times, sometimes it is not sufficient to include only a discrete delay or a finite delay in a population model [13]. To address this difficulty, Britton [3,4] considered comprehensively the two factors and introduced the so-called nonlocal delay, that is, the delay term involves a weighted spatiotemporal average over the whole infinite spatial domain and the whole of previous times. Since then, there have been many works of studying the spatial dynamics for reaction-diffusions with nonlocal delay [13,24,27,29,30] (see also the references cited therein). Recently, Wu and Hsu [26] considered the general delayed non-local reaction-diffusion equation which can describe the evolution of the mature population of a single species. They investigated the entire solutions of (3) and extended the arguments to two specific non-quasi-monotone delayed differential equations. Note that the delay term in the first equation of (2) models the duration of gestation or hatching period, in which mobile subpopulations are not moving very much or not at all. Therefore, the use of a local time-delay term is probably reasonable. However, since model (2) is also applicable to many other species that have a maturation phase when the individuals may indeed move about. For such cases, nonlocal delays are indeed essential.
Here, and in what follows, we defineĪ :

ZHENGUO BAI AND TINGTING ZHAO
By means of these two inequalities, together with (C2), we have which imply that The remainder of this paper is organized as follows. In Section 2, we present preliminaries. In Section 3, we use comparison arguments and a fluctuation method to establish the existence of spreading speed c * of (4) with non-monotone reaction terms. In Section 4, we use Schauder's fixed point theorem to obtain the existence of traveling waves with c > c * , and get the existence of traveling waves with c = c * based on a limiting argument.

Preliminaries.
In this section, we study the spreading speed and traveling wave solutions of (4) in quasi-monotone case. For this, we make the following assumption: Note that if (C4) holds, then we can choose f ± = f , S ± = S and K ± = K. From the assumption (C2), we see that system (4) has two equilibria 0 = (0, 0) and K := (K,K) with γ 1 K = γ 2K .

Proof. For any
Sincef is continuous in φ ∈ W , andf is Lipschitz in φ on each compact subset of W , it then follows that system (5) has a unique solution w(t, φ) with w 0 = φ on its maximal interval [0, σ φ ) of existence (see, e.g., [ The assumption (C2) implies that [0, lK] W is positively invariant for system (5).
Since l can be chosen as large as we wish, this prove the positivity and boundedness of solutions in W , and hence, σ φ = ∞.

2.2.
Results for monotone delayed system. Since system (4) is cooperative and its solution maps are monotone, we can use the general theory developed in [16] to study the spreading speeds for (4). We start with some basic notations. Let X := BUC(R, R 2 ) be the Banach space of all bounded and uniformly continuous functions from R into R 2 with the usual supremum norm · X , and X + := {φ ∈ X : φ(x) ≥ 0, ∀ x ∈ R}. The space BUC(R, R) is defined similarly. Clearly, any vector in R 2 can be regarded as a function in X.
Let Y be the space of all continuous functions from [−τ, 0] to R 2 with the usual supremum norm · Y (i.e., Y = C([−τ, 0], R 2 )), and . Let C be the set of all bounded and continuous functions from R to Y equipped with the compact open topology, that is, u m → u in C means that the sequence of Define the reflection operator R by R[u](x, θ) = u(−x, θ), and the translation operator T y by T y [u](x, θ) = u(x − y, θ) for each y ∈ R. Let Q : C r → C r be a given map. The following assumptions on map Q will be referred to: u v in C r . (A5) Q : Y r → Y r admits exactly two fixed points 0 and r and lim n→∞ Q n [y] = r for any y ∈ Y r with y 0.

ZHENGUO BAI AND TINGTING ZHAO
Let T (t) = diag(T 1 (t), T 2 (t)) be a family of linear operators defined for t ≥ 0 with T (0) = I and It is convenient to write (4) into the following integral form . By using the theory of abstract functional differential equations developed in [19], we can establish the following result.
It is easy to see that Q 0 = I, and Q t+s = Q t • Q s for all t, s 0. By arguments similar to those in [7, Lemma 4.3], we can show that {Q t } t≥0 is a monotone semiflow on C K with time-one map Q 1 satisfying (A1)-(A5). It then follows from [16, Theorems 2.11 and 2.15] that Q 1 admits a spreading speed c * > 0. The following result shows that c * is also the spreading speed of (4).
Moreover, it is easy to see that ∂∆(c * ,λ * ) ∂λ = 0, where λ * := λ(c * ). Combining the above properties of the function ∆(c, λ), the conclusion follows. In the remainder of the paper, we will use c * instead of c * .
Following the idea in [18,Theorem 2.2], the existence of solutions of (8) can be reduced to the existence of a pair of super-solution and sub-solution of (8). By Lemma 2.4, we will construct a pair of super-solution and sub-solution of (8).
Next, we show that Φ − c (ξ) is a sub-solution of (8) provided q > b(c)/B(c, µ) is large enough. Let We now consider the case ξ < ξ − 1 < 0. It is easy to see that By using the Taylor' formula, there exist positive constants D 1 and D 2 such that Since S (0) exists, there is D 3 > 0 such that In view of (13)-(15), we have, for ξ < ξ − 1 < 0, According the definition of ∆(c, λ), if q > b(c)/B(c, µ) is large enough, then

ZHENGUO BAI AND TINGTING ZHAO
On the other hand, if ξ < ξ − 2 < 0, then is a sub-solution of (8). With the aid of the super-solution and sub-solution of (8), we have the following existence and non-existence of traveling wave fronts of (4).
We start with the well-posedness for the initial-value problems of (4).

Lemma 3.2. (Comparison principle) For any
w(x, t; φ) and w + (x, t; φ + ) be the solution of systems (18), (4) and (17) through φ − , φ and φ + , respectively. Then Proof. The proof is similar to that of [34, Lemma 3.1] and thus omitted. Now, we state the result on the spreading speed for (4) without quasi-monotonicity. In particular, to obtain the upward convergence of spreading speed, we further make the following assumption.
4. Traveling waves. In this section, we establish the existence of traveling waves of (4) without monotonicity by using the Schauder's fixed point theorem. Further, we employ the properties of the monotone traveling wave solutions of the lower auxiliary system (18) to obtain the asymptotic behavior of the wave profile.