Traveling waves in lattice differential equations with distributed maturation delay

In this paper we derive a lattice model with infinite distributed delay to describe the growth of a single-species population in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction. We consider the existence of traveling wave solutions when the birth rate is large enough that each patch can sustain a positive equilibrium. When the birth function is monotone, we prove that there exists a traveling wave solution connecting two equilibria with wave speed c > c∗(θ) by using the monotone iterative method and super and subsolution technique, where θ ∈ [0, 2π] is any fixed direction of propagation. When the birth function is non-monotone, we prove the existence of non-trivial traveling wave solutions by constructing two auxiliary systems satisfying quasi-monotonicity.


Introduction
In 1990, Aiello and Freedman [1] derived the following model to describe the growth of a single-species population: (1.1) Here α, β and γ are positive constants, u m and u i denote the number of immature (juvenile) and mature (adult) members of the population, respectively; the delay τ > 0 is the time taken from birth to maturity.In particular, they assumed that the maturation delay τ is known exactly and that all individuals take this amount of time to mature.They showed that the unique positive equilibrium of (1.1) is globally asymptotically stable.However, as reported by Al-Omari and Gourley [2], the individuals do not necessarily mature at the same time, and it has been observed that distributed delays are more reasonable than discrete delays in modeling maturation periods.Therefore, they proposed a more general model: where the probability density function f (s) ∈ L 1 ([0, ∞), R + ) describes the probability of maturing at each age s and satisfies ∞ 0 f (s)ds = 1, b(•) and d(•) are the more general birth rate and death rate functions, respectively.The authors also considered spatial effects and proposed the following nonlocal reaction-diffusion model with distributed delay x ∈ Ω, t > 0, (1.3) subject to homogeneous Neumann boundary conditions where n is an outward normal to ∂Ω, Ω ⊂ R N is bounded.They proved that the positive equilibria of system (1.2) and (1.3) are stable under some assumptions about EJQTDE, 2013 No. 44, p. 2 the functions b and d.In [3], Al-Omari and Gourley further studied the following system 4D i s u m (y, t − s)e −γs f (s)dyds, (1.4) where x ∈ (−∞, ∞).When f (s) = (s/τ 2 )e −s/τ , they proved that system (1.4) admits traveling waves connecting two equilibria.Weng and Wu [31] also studied the existence of traveling waves for the second equation of system (1.4).
Another important single-species model with diffusion and stage-structure is the following equation: which was derived by So et al. [25].In (1.5), w(x, t) is the total matured population at time t > 0 and position x ∈ R; D m and d m are the diffusion and death rates for the mature population respectively; r 0 is the maturation time and b(•) is the birth function.There have been many studies on the existence and stability of traveling waves of (1.5), see [7,12,20,23,28].However, the mature time of the individuals in (1.5) are the same, which is not realistic as mentioned in the previous.Therefore, Gourley and So [10] further proposed and studied the following model with distributed mature time For more details on the studies of traveling wave solutions of (1.4), (1.5) and (1.6), we refer to [7,9,10,12,16,18,22,23,27,28,29] and the references therein.
For the model (1.5), its discrete version was firstly proposed by Weng et al. [30].They considered the growth of a single-species population living in a patch environment consisting of all integer nodes of a 1D lattice.They divided the population into two ages classes: immature and mature, and assumed that the mature periods of all individuals are same as those in So et al. [25].By the discrete Fourier transform, EJQTDE, 2013 No. 44, p. 3 Weng et al. [30] obtained the following lattice differential equations where β α (l) = 2e −ν π 0 cos(lω)e ν cos ω dω for ν := 2α.They established the spreading speed and the existence of monotone traveling waves for (1.7) when the birth function is monotone.They further showed that the minimal wave speed coincides with the spreading speed.For more studies on (1.7), we refer to [6,11,12,15,16,19,21,22,30,32].
Cheng et al. [4] extended the work of Weng et al. [30] and proposed the following lattice equation which models the growth of a single-species population with two age classes distributed over a patchy environment consisting of all integer nodes of a 2D lattice.They studied the well-posedness of the initial-value problem and established the existence of monotone traveling waves for wave speed c ≥ c * (θ) > 0, where θ is any fixed direction of propagation.They further showed that the minimal wave speed c * (θ) coincides with the asymptotic speed of spread for any fixed direction θ ∈ [0, 2π].They showed that the asymptotic speed of propagation depends on not only the maturation period and the diffusion rate of mature population monotonically but also the direction of propagation, which is different from the case when the spatial variable is continuous.In Cheng et al. [5], the authors established the asymptotic stability of traveling wave fronts for equation (1.8) when immature population is not mobile.
The aim of the current paper is to modify (1.8) to allow for the fact that the time from birth to maturity may be rather imperfectly known, or it might vary from individual to individual, as done by Gourley and So [10].Therefore, in this paper we firstly derive a lattice differential equations with distribution mature delay in 2D lattice and then establish the well-posedness of the nontrivial traveling wave solutions for the equations.This paper is organized as follows.In Section 2, we derive the lattice differential equations (2.7) and show some properties of equation EJQTDE, 2013 No. 44, p. 4 (2.7).The existence of non-trivial traveling waves of equation (2.7) is obtained in Section 3, where we consider two cases, namely, the monotone birth function and the non-monotone birth function.

Model derivation
Let u k,j (t, a) ≥ 0 denote the population density of the species of the (k, j) − th patch at time t ≥ 0 and age a ≥ 0. Assume that the patches are located at the integer nodes of a 2D lattice and spatial diffusion occurs only at the nearest neighborhood.From Metz and Diekmann [24], we can obtain the following model: where D(a) and d(a) are the diffusion and death rates at age a, respectively.Assume that u k,j (t, ∞) = 0 for t ≥ 0, (k, j) ∈ Z × Z.We want an expression for w k,j (t), that is the total matured population at time t and (k, j) ∈ Z × Z.Let f (r)dr = probability of maturing between the ages r and r + dr, where f (r) is the probability of maturing at each age r.Note that the probability of maturing before age a is Since f is a probability density function, we assume f (r) ≥ 0, and of course, Of the total number of mature adults, the number that matured between age r and r + dr is (number of age at least r)×(probability of having matured between ages r and r+dr), Thus, the total number of matures is Reversing the order of integration yields the following alternative expression: where F (a) is given by (2.3).
In the following, we want to find a differential equation satisfied by w k,j (t).Differentiating (2.4) with respect to t, together with (2.1), one sees that In this paper we shall assume that the diffusion coefficient and death rate are age independent.i.e.D(a) = D and d(a) = d for a ∈ [0, ∞), where D and d are positive constants.From (2.5), Integrating by parts on the first term, and using F (0) = 0 and u k,j (t, ∞) = 0, we obtain (2.6) By an argument similar to that of Cheng et al. [4], using discrete Fourier transformation and inverse discrete Fourier transformation(see [8,26]), we obtain a closed system as follows: for any l ∈ Z.

Existence of traveling waves
In this section, we establish the existence of traveling waves for the lattice differential equation (2.7) when it has a positive equilibrium.We will consider two cases: (a) the birth function b(•) is monotone; (b) the birth function b(•) is nonmonotone.

Monotone birth functions
We assume that the birth function b : R + → R + satisfies (H b ).Substituting it into (2.7),we have that φ(•) satisfies In view of the symmetry and periodicity in equation Define two operators A and H on C by everywhere on R and satisfies the following inequality A lower solution can be defined similarly by reversing the inequality above.Let Since S(0) = 1 and we have Similarly, Using the expressions of S(α) and T (α), we rewrite ∆(λ, c, θ) as It is obvious that lim By simple computations, we have EJQTDE, 2013 No. 44, p. 9 Then ∆(λ, c, θ) is convex with respect to λ. Differentiating ∆(λ, c, θ) with respect to c > 0, we obtain that − (cλ + d)a af (a)da < 0 for λ > 0. Furthermore, it is easy to show that ∆(λ, 0, θ) > 0 and lim c→+∞ ∆(λ, c, θ) = −∞ for any given λ > 0.
(3) F(φ) ∈ S for any φ ∈ S; (4) There exists K S > 0 such that for any φ ∈ S, it holds that By Lemma 3.3 and the monotonicity of H, we can easily obtain (1)-(3) of Lemma 3.4, see also Ma [17].Differentiating with F(φ)(s) and using the Lipschitz continuity of b, we can prove Lemma 3.4 (4).
Define an iteration sequence φ n = Fφ n−1 , n ≥ 1, φ 0 = φ + .It follows from Lemma 3.4 that, (3.4) By (3.4) and Lemma 3.4 (4), there exists a function It follows from the Lebesgue dominated convergence theorem that φ * is a fixed point of F, which is also a solution of (3.1).Furthermore, φ * is nondecreasing and satisfies

Nonmonotone birth functions
In this subsection, we establish the existence of nontrivial traveling waves when the birth function b(•) is nonmonotone.The main method is to construct two auxiliary lattice differential equations with monotone birth functions and then apply Schauder's fixed-point theorem.
At first, let b satisfy the following assumption: The traveling wave equations of (3.5) and (3.6) are The following lemma is the immediate consequence of Theorem 3.5.Proof.It is obvious that traveling wave equation φ(s) of (3.1) is a solution of the following equation: where Similarly, define T + and T − by replacing H in (3.9) with H + and H − , respectively.It is not difficult to show that T ± are nondecreasing and satisfy where φ − (s) is a traveling wave solution of equation (3.6).By Lemma 3.4, it is easy to show T − (φ)(s) = φ(s), T + ( φ)(s) ≤ φ(s).