Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

: This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisﬁes monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ ﬁxed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c > c ∗ , where c ∗ > 0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.


Introduction
Very recently, we derived in [1] the following delayed temporally discrete non-local reaction-diffusion equation w(n + 1, x) = D∆ x w(n + 1, x) + (1 − d)w(n, x) + η +∞ −∞ K D i (r − 1, x − y)b(w(n + 1 − r, y))dy, (1) where w(n, x) denotes the density of the total matured population of a single species at time n ∈ N, location x ∈ R, N represents the set of non-negative integers. D > 0 is the diffusion rate of matured individuals. ∆ x is the Laplacian operator with respect to x ∈ R, r ≥ 2 is a positive integer, which denotes the maturation time for the species. 0 < d < 1 represents the death rate of matured individuals. 0 < η < 1 is the survival rate of an individual from birth to maturation. b(·) is the birth function. The integral kernel function K D i reads as where D i is the diffusion rate of immatured individuals. In particular, we use the convention that if m = 0 then m! = 1 and (r − 1)r · · · (r + m − 2) = 1. For the case r = 1, K D i (r − 1, x) = K D i (0, x) = δ(x), i.e., the Dulac δ function. Therefore, (1) is reduced to w(n + 1, x) = D∆ x w(n + 1, x) + (1 − d)w(n, x) + ηb(w(n, x)), Equation (1) reflects the changes of matured population of a single species in an unbounded habitat with dimension one. It can also be viewed as a non-standard discretization of the following well-known non-local reaction-diffusion equation model with delay, which was derived in 2001 by So, Wu and Zou [2]. The model (4) describes the adult population's evolution of a single species that have two age classes in an unbounded spatial domain R 1 . In above Equation (4), D > 0 is the diffusion rate of the adult population and d > 0 is the death rate of the adult population. ε > 0 reflects the impact of the death rate of the immature on the matured population. b(·) is the birth function. r > 0 is the maturation time of the species. α ≥ 0 also reflect the impact of the dispersal rate of the immature on the matured population. The integral kernel function is 4α . When α approaches to 0, that is, the immature do not disperse, (4) is reduced to and the non-local effect disappears. In our previous paper [1], we have already given a detailed derivation of Equation (1). Especially, when D i = 0 (the immature individuals do not disperse), (1) becomes w(n + 1, x) = D∆ x w(n + 1, x) + (1 − d)w(n, x) + ηb(w(n + 1 − r, x)).
Above equation is a non-standard temporally discrete version of (5). It is well known that the continuous-time reaction-diffusion equation has been widely used to describe diffusive phenomena in physics, engineering, chemistry, biology, and so on (see, for example, [3][4][5][6][7][8][9][10]). In general, the dynamical behaviors of solutions to a nonlinear reaction-diffusion equation are very complicated, and it is often very difficult to find exact solutions. For the sake of understanding the properties of the solutions numerically, we need to study its discrete analogue. However, the basic principle of constructing appropriate discretization of differential equations is to preserve the properties of the corresponding differential equations. Since many classical (standard) discretizations cannot achieve dynamic consistency, non-standard discretizations are usually used to ensure it. Thus, Mickens first introduced the concept of dynamical consistency in [11,12] for ordinary differential equations and since then, some dynamical consistent discrete schemes have been constructed. See, for example, Refs. [13][14][15][16][17], and references therein.
In [1], we established the existence of traveling wavefronts of Equation (6) by using upper and lower solution methods and iterate techniques. We found that (6) possesses the dynamical consistency with its time continuous counterpart (5) at least in the sense of the existence of traveling wave solutions. In the sense of propagation, Equation (6) is also a good approximation of corresponding continuous time model (5).
From the perspective of mathematical biological modeling, almost all of the data collected are discrete in time because observations are always discontinuous. For example, satellite photographs used for scientific research are usually taken periodically, but the spatial distribution can be seen as continuous. Temporally discrete and spatially continuous diffusion model will be more suitable than its corresponding time continuous diffusion model to study the dynamic behavior of a single species that living in a spatially continuous habitat in population ecology. In 2002, Weinberg, Lewis, and Li in [18] gave some reasons for studying discrete-time models rather than just reaction diffusion models. They also pointed out the advantage of a discrete-time model over a reaction-diffusion model. We note that in [18], the authors studied the discrete-time recursion system where u n (x) denotes the population distributions of species and Q is an operator that models the growth, migration, and interaction of the species. Although Equations (1) and (6) are different from (7) formally, we have proved in [1] that the following general non-linear equation is equivalent to an integral-difference equation as below, where Clearly, (9) is a special case of (7) with As for the general integro-difference equations, Kot and Schaffer [19] are the first to apply it modeling temporally discrete and spatially continuous dispersal phenomena, and studying the dispersal of a single species with nonoverlapping generations. They showed that, the above model will has more complex dynamic behavior than its corresponding time-continuous one. Moreover, even chaos could occur.
Since the selection of kernel function k(n, x, y) plays a key role in the dynamical behavior of (10), using such a model to describe some biological phenomena will have some uncertainty. Especially, when we discuss dynamical behaviors of populations of some species living in a bounded domain, the choice of suitable integral kernels is very difficult because we may cope with various different boundary value problems.
In contrast with integral difference equations, we found that there is no such problems for temporally discrete reaction diffusion equations like (8) (or (1)). Furthermore, from the point of view of the mathematical modeling, (8) (or (1)) has the same biological explanations as those for integral difference equations. In fact, in our previous paper [1], the life cycle of individuals of the population is divided into relatively sedentary and dispersal stages. This coincides with the explanations by Kot and Schaffer [19] in establishing Equation (7). To distinguish the difference of dynamical behaviors which occurs at different stages, we assume that the evolution (the relatively sedentary stage) occurs at time n and dispersal occurs at time n + 1.
For temporally discrete reaction diffusion models, there are only a few results in the literature. In 2006, Lin and Li [20] studied following equation with delay: They established the existence of traveling wavefronts and showed that (11) is a good approximation of its continuous time model in the sense of propagation. For more researches on this topic see [4,21,22]. However, we note that in the existing research literature, researchers simply assumed that the non-standard discretizations preserve the dynamical consistency of the continuous-time reaction-diffusion equations, but they do not provide reasonable biological explanations for the modeling process.
Although (1) has been derived in [1], the existence of traveling wave solutions is proved without non-local effect, and monotonic condition for birth function is assumed. In order to better understand the dynamical behaviors of (1) with non-local diffusion caused by immature individuals dispersion, we will study traveling wave solutions whenever the birth function is monotonic or non-monotonic, respectively.
The rest of this paper is organized as follow. In Section 2, by using the upper-lower solutions and monotone iteration technique, we studied the existence of traveling wavefronts of (1) for the cases that the birth function b(w) is increasing in [0, w * ], where w * > 0 is the unique solution to the equation ηb(w) = dw. As for the case that birth function b(w) is non-monotone in [0, w * ], the theory of monotone dynamical system cannot be directly used. By using a similar idea as Ma in [23,24], we establish the existence of traveling waves of (1) in Section 3. Finally, we give a short discussion in Section 4.

Traveling Wavefronts for the Monotone Case
In this section, we will consider the existence of traveling wavefronts to equation (1) for monotone case. We are interested in finding traveling waves w(n, x) = ϕ(x + cn) of following equation (12) For this purpose, we will find a solution ϕ(ξ) to (12) with ξ = x + cn. Clearly, ϕ(ξ) satisfies the following wave profile equation Let ξ = ξ + c and still denote it by ξ. Then, the above equation becomes Throughout this section, we always assume that Hypothesis 1 (H1). b(w) = wg(w), where g(w) is a continuously differentiable function and satisfying g(w) > 0, g (w) < 0 for w ≥ 0 and lim w→∞ g(w) → 0; Hypothesis 2 (H2). b(w) and b (w) are bounded; From (H1) and (H3), Equation (12) has only two constant equilibria w = 0 and w = w * , where w * is the unique solution of the equation ηg(w) = d.
We will study the existence of non-decreasing solutions to Equation (14) subject to the boundary value conditions Our approach is similar to that in [1], which is based on the monotonic iteration techniques combined with upper and lower solution methods that was developed in [2]. To this end, we further assume that To proceed further, for readers' convenience, we introduce some results on the following temporally discrete reaction-diffusion equation which will be used later, where f (·) is assumed to be a continuous function. For detailed proofs on these results, readers can refer to [1].
has a unique solution u(n, x) and where ϕ is a continuous and absolutely integrable function on R 1 satisfying lim In particular, we use the convention that if m = 0, then m! = 1 and n(n + 1) · · · (n + m − 1) = 1. The integral kernel function K(n, x) satisfies the following properties.

Lemma 2 ([1]
). The function u(n, x) is a solution of (15) if, and only if, it is a solution of following non-linear integro-difference equation For ϕ ∈ C(R, R), define By (H2) and Proposition 1, H(ϕ)(ξ) is well defined. Then, Equation (14) becomes Then from the definition of H and Proposition 1, we can easily obtained the following result.
By direct computations, we have has a unique solution ϕ ∈ Γ satisfying Further define the mapping F : Γ → Γ, The following theorem is a direct consequence of Lemmas 3 and 4. (14) if, and only if, it satisfies

Theorem 1. ϕ ∈ Γ is a solution of Equation
In other words, ϕ is a fixed point of mapping F in Γ.
Above theorem shows that the existence of traveling wavefronts of Equation (12) is equivalent to the existence of fixed points to mapping F. Therefore, in what follows, we will construct upper and lower solutions of (22) to prove that the mapping F has a unique fixed point in Γ, as long as c is greater than a certain constant c * .
The linearized equation of (14) at ϕ = 0 is as below.
Then, its characteristic equation is given by Define Note that for |λ| < 1 By direct computations, we have That is, In fact, one can give another proof of the equality as follows. By Lemma 1, for |λ| < 1 has a unique solution u(n, x) and On the other hand, one can easily verified that is a solution to initial value problem (26). Therefore, which leads to the equality (25). Then and It is easy to see that for any given c ≥ 0 and |λ| < 1 Then, for any If ∆(c, λ(c)) > 0, the characteristic Equation (24) has no positive real roots; If ∆(c, λ(c)) = 0, the characteristic Equation (24) has only one positive real root; If ∆(c, λ(c)) < 0, the characteristic Equation (24) has exactly two positive real roots.

Proof. Consider the function
which implies that (24) will has no positive real roots.

Definition 1. A continuous bounded function
A lower solution of (22) is defined in a similar way by reversing the inequality in (29).
Choose sufficiently large constant q > 1 to be determined later, define the functionsφ and φ bȳ
We have already obtained an upper and a lower solution of (22). Using the classical upper and lower solution method together with iteration techniques, we find the following existence result.

Traveling Wave Solution for the Non-Monotone Case
This section is devoted to the existence of traveling wave solutions to (1) when the birth function b(w) is not increasing in [0, w * ]. Our approach is to construct two auxiliary temporally discrete diffusion equations with birth functions satisfying monotonic conditions. Then, by using a similar method developed in [23] and applying the results obtained in the Section 1 to these auxiliary equations, we can prove that (1) possesses a traveling wave solution connecting equilibrium 0 and w * .
Throughout of this section, we suppose that b(w) satisfies assumptions (H1)-(H3). By (H3) and the continuity of g(w) at zero, there exists a small constant δ > 0, such that for any w ∈ (0, δ), g(w) > d η . Then, for any On the other hand, when w > w * , g(w) < d η . This means that for w > w * , b(w) − d η w < 0. Then For sufficiently small ε < W * , we define two auxiliary functions b * (w) and b (w) as follows.
Lemma 6. The following statements hold true: (i) b * and b ε are continuous on [0, +∞) and non-decreasing on [0, W * ]; The proof of Lemma is a direct verification, we omit it. Now we consider the following two auxiliary temporally discrete diffusion equations and Then, the corresponding wave equations of (30) and (31) read as respectively. They are equivalent to the following two integral equations: Moreover, the traveling wavefronts of (30) and (31) are fixed points of operators F * and F ε , where By Lemma 6, it is easily to verify that b * and b ε satisfy all the assumptions (H1)-(H4). Therefore, the results below follow from Theorem 2.
In the sequel, we will always assume c > c * . Therefore, there exist two positive roots In order to proceed further, we also need the following assumptions.