Propagation of monostable traveling fronts in discrete periodic media with delay

This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.


1.
Introduction. In the past decades, there has been great progress in modelling and investigating the dynamical behaviour of population systems with age structure, see e.g. Gourley and Kuang [8], Smith and Thieme [28], So et al. [29], Weng et al. [37], and Weng and Zhao [39]. In particular, Smith and Thieme [28] developed an approach to derive an age-structured population model with two age classes (i.e. immature and mature) and fixed maturation delay. Their approach is mainly based on the technique of integration along characteristics and the Fourier transform. Following their approach, Weng [37] further derived a delayed lattice differential model (DLDM for short) for a single species in a one-dimensional patchy environment based on the following system: for j ∈ Z, t > 0 and a ∈ (0, τ ), where v j (t, a) is the density of individuals at location j ∈ Z, time t ≥ 0 and age a ≥ 0; u j (t) is the density of the mature population; d(a) and µ(a) are the diffusion and death rate of the immature population at age a, respectively; D is the diffusion rate of the mature individuals; d(u) and b(u) are the mortality rate and birth rate of the mature population, respectively; v j (t, τ ) is the adults recruitment term for those of maturation age τ . In [37], the authors studied the well-posedness of the initial-value problem of the DLDM derived from (1) and obtain the existence of monotone travelling waves for wave speeds greater than the minimal wave speed. They further showed that the minimal wave speed is also the asymptotic speed of propagation, which depends on the maturation period and the diffusion rate of mature population monotonically. However, due to the natural phenomena or exposure to artificial distributions, the real environment is generally heterogeneous. For example, a simple but realistic and important form of heterogeneous environment is the periodic habitat. Therefore, in this paper we will consider the following spatially periodic version of system (1): where j ∈ Z, t > 0 and a ∈ (0, τ ). For simplicity, we assume that the immature individuals have the same diffusion and death rates in different patches, and hence, the equation for the immature individuals is homogeneous. In contrast to (1), the mortality rate d j (·), birth rate b j (·) and diffusion rates D j for the mature individuals of (2) are spatially dependent.
Similarly as in [28] (see also [37]), by integrating along characteristics and using the discrete Fourier transform, we can obtain Then the age-structured population model (2) is reduced to the following lattice periodic differential equation with delay and nonlocal interaction for mature individuals: where j ∈ Z, t > 0, and It is very important to understand how the heterogeneity of the medium and the delay influence the dynamics for such periodic and delayed systems. Based on this prototype, in this paper we will consider the spatial dynamics of the general spatially periodic lattice dynamical system with delay and nonlocal PROPAGATION OF MONOSTABLE PULSATING TRAVELING FRONTS 2989 interaction: where τ ≥ 0 is a constant; the kernel function J(·) satisfies the following assumption: (A0): J(k) ≥ 0, J(0) > 0, J(−k) = J(k), ∀k ∈ Z; k∈Z J(k) = 1 and k∈Z J(k)e λk < ∞, ∀λ ≥ 0. We also assume that the constants D j and nonlinearities f j (·, ·) and S j (·), j ∈ Z satisfy the following assumptions: (A1): [Periodicity] f j (·, ·) ∈ C 2 (R 2 + , R + ), S j (·) ∈ C 2 (R + , R + ), ∀j ∈ Z, and there exists a N ∈ N such that D j = D j+N > 0, f j (·, ·) = f j+N (·, ·) and S j (·) = S j+N (·); (A2): [Monostability] f j (0, 0) = S j (0) = 0, and there exists a K > 0 such that f j K, k∈Z J(j − k)S k (K) ≤ 0, ∀j ∈ Z; (A3): [Monotonicity] ∂ 2 f j (u, v) ≥ 0 and S j (u) ≥ 0 for all j ∈ Z, u ∈ [0, K] and v ∈ [0, S K ], where S K := max j∈Z S j (K); (A4): [Sub-homogeneity] For any γ ∈ (0, 1), j ∈ Z and u, v ∈ (0, K], it holds that It is clear that (4) is a generalization of (3). Under the assumptions (A0)-(A4), we shall show that (4) admits a unique positive and periodic equilibrium β = (β(j)) j∈Z (see Lemma 6). In particular, when τ = 0, J(0) = 1 and J(j) = 0 for j = 0, (4) becomes the following periodic lattice differential system: where g j (u) = f j (u, S j (u)) for j ∈ Z. Let's mention that the front dynamics of (5) have been studied by many researchers, see e.g. [4,9,10,41]. The purpose of this work is to study the propagation phenomena of (4), including the asymptotic speed of spread (spreading speed for short) and the pulsating traveling fronts (also called spatially periodic traveling waves in the literature) connecting the uniform zero state (denoted by 0) and β. The concept of the spreading speed was first introduced by Aronson and Weinberger [1] for reaction-diffusion equations and has been an important biological metric in a wide range of biological applications. In addition, the traveling wave fronts connecting 0 and β can describe the biological invasion of the 0 state by the heterogeneous state β. In the past decades, there were many works devoted to the spreading speed and traveling wave solutions for evolution systems, see e.g. [12,[15][16][17][18][19][21][22][23][24]31,[35][36][37]39] and the references therein. Under some reasonable assumptions, the spreading speed always coincides with the minimal wave speed of the traveling wave solutions for various monostable dynamics.
Recently, Liang and Zhao [19] generalized the earlier works in [18,21,22,35,36] on spreading speed and traveling wave fronts to some abstract monostable evolution systems with spatial structure. In this article, by applying the Liang and Zhao's theory [19], we first establish the existence of the rightward and leftward spreading speeds c ± * for system (4) (see Theorems 1). We emphasize that such an application non-local and periodic eigenvalue problem of delayed type, we study the dynamics of the spatially periodic initial value problem of (4).
2.1. Existence, positivity and comparison theorems. Let's consider the initial value problem of (4) with initial data: The definitions of supersolution and subsolution of (4) are given as follows. where According to Definition 1, we have the following result on the existence, eventual strong positivity, and comparison theorem for solutions of (4).
Proof. (1)-(2) Note that u j (t) = K is a supersolution of (4). The assertions of these two parts can be proved by using the similar arguments as those in [23,Lemma 4.1] and [24,Lemma 2.2]. We omit it here.

SHI-LIANG WU AND CHENG-HSIUNG HSU
This contradiction implies that u j (t) > 0 for all j ∈ Z and t > τ . The proof is complete.
Next, we establish the following comparison theorem for the solutions of the linear system of (4) about the trivial equilibrium 0, which will be used in investigating the stability of pulsating traveling fronts (see Section 6).
for all j ∈ Z, and t > 0.
The proof is similar to that of [40, Lemma 2.6] and omitted here.

2.2.
Dynamics of periodic initial value problem. Let's consider the following periodic initial value problem of (4): Some useful notations and definitions are introduced in the sequel.
Notation. (1) We denote P := P (Z, R) by the set of all N -periodic functions from Z to R with the maximum norm · P and P + := {φ ∈ P : φ(j) ≥ 0, ∀j ∈ Z}.
Note that P + and C + are closed cones of P and C, respectively.
is N -periodic for j ∈ Z such that (7) holds for any j ∈ Z and t ∈ [0, T ).
Proof. The existence and monotonicity of solutions of (8), i.e. assertions (1) and (2), can be proved by using a similar argument as that in [23,Lemma 4.1] or [24,Lemma 2.2]. It can also be proved by applying the theory of abstract functional differential equations (Martin and Smith [25,Corollary 5]). The proof of assertion (3) is similar to that of (3) in Lemma 1.
It is clear that the linearization of (8) at 0 can be represented by: (9) where j ∈ Z, t > 0. By substituting v j (t) = e λt ν j into (9), we obtain the following periodic and nonlocal eigenvalue problem of delayed type:
Lemma 5. Assume that (A0)-(A3) hold. Then there exists a principal eigenvalue λ * of (10) associated with a strictly positive eigenfunction (ν * j ) j∈Z with ν * j+N = ν * j , and for any τ > 0, λ * has the same sign as λ * 0 . In particular, λ * > 0 for any τ > 0. Proof. Arguing as in the proof of part (3) of Lemma 3, one can show that the solution of the linear equation (9) is strongly positive and compact for any t > 2τ . By applying a result about the sign of spectral bound of an operator on [14,Section 4], the first part of the assertion can be proved by using the same argument as in [30,Theorem 2.2]. From Lemma 4, we have λ * > 0 for any τ > 0. This completes the proof. Lemma 6. Assume that (A0)-(A4) hold. Then (4) admits a unique positive and periodic equilibrium β(j) such that β ∈ [0, K] C and for any ϕ and Φ t is monotone and strongly sub-homogeneous. Moreover, the semiflow generated by equation (9) is compact and strongly positive for any t > 2τ . Based on these observations, the assertion of this lemma follows from [44, Theorem 2.3.4].
3. Spreading speeds. For completeness and the readers' convenience, we first recall the general theory of Liang and Zhao [19] on the study of spreading speeds for monotone semiflows in a periodic habitat. Then, we apply their theory to prove the existence of the spreading speeds. We also give the formula and investigate the signs of the spreading speeds.
3.1. General theory for monotone semiflows in a periodic habitat. We first give the following definitions. (2) Let C := B(Z, X) be the set of all bounded functions from Z to X with the compact open topology.
Clearly, any element in X can be regarded as a constant function in C and (C, · C ) is a normed space. Moreover, the topology generated by · C and the compact open topology on C are equivalent on any uniformly bounded subset of C. For convenience, we also identify an element ϕ ∈ C as a function form In addition, we denote the interval [a, b] Z := {a, a + 1, · · · , b − 1} for any a, b ∈ Z with a < b, which has length b−a. Let I := [a, b] Z be a bounded and closed interval, ϕ ∈ C, U ⊆ C and h ∈ Z, we further define the function ϕ I : I → X, the set U| I and translation operator H is a sub-lattice of Z. We say that u ∈ C is periodic with respect toH (or, more briefly, N -periodic) if T h [u] = u for all h ∈H. According to [19], we set K = M := C β , Y = X, D = C. Let β ∈ D be strictly positive and N -periodic and Q[·] be a map from M to M. Assume that Q[·] satisfies the assumptions: compact open topology, and there is a equivalent norm · * X in X such that for any r ≥ 0, there exists k = k(r) ∈ [0, 1) such that for any interval I = [a, b] of the length r and any U ⊂ K with U(0, ·) being pre-compact in C(Z, R), we have α(Q[U]| I ) ≤ k(s)α(U| I ), where α denotes the Kuratowski measure of non-compactness (see [20]) on C| I with (X, · X ) replaced by (X, · * X ); admits exactly two N -periodic fixed points 0 and β in M, and for any N -periodic function Then it follows from [19, Theorem 5.1] that the discrete time semigroup {Q n } ∞ n=0 (in short, the map Q) on M admits rightward and leftward spreading speeds c ± * . Furthermore, a family of mappings {Q t } t≥0 is said to be a semiflow on a metric space (X ,d) provided that the following properties hold: It is clear that the property (iii) holds if Q(·, ϕ) is continuous on [0, ∞) for each ϕ ∈ X , and Q(t, ·) is uniformly continuous for t in bounded intervals in the sense that for any ϕ 0 ∈ X , bounded interval I, and > 0, there exists δ = δ(ϕ 0 , I, Suppose that Q t satisfies assumptions (E1)-(E5), and c + * and c − * are the rightward and leftward spreading speeds of Q 1 , respectively. Then c + * and c − * are the rightward and leftward spreading speeds of {Q t } t≥0 in the following senses: β, and ϕ(·, j) = 0 for j outside a bounded interval, then and ϕ ∈ M, then r σ can be chosen independently of σ.
In the rest of this section, we always assume that (A0)-(A4) hold.
has the following properties.
(1) It is clear that Q t [·] satisfies the property (i). The semigroup property (ii) follows from the existence and uniqueness of solutions of (4). Now, we prove the property (iii). Given ϕ ∈ C β , it follows from (4) that d dt u j (t; ϕ) is bounded for all (t, j) ∈ Z × [0, ∞). Hence, there exists a constant L = L(ϕ) > 0 such that the compact open topology. Therefore, to prove the property (iii), it suffices to show that Q t [ϕ] is continuous in ϕ with respect to the compact open topology, uniformly for t in any bounded interval. Given anyφ,φ ∈ C β . For any given > 0 and t 0 > 0, we define where D := max j∈Z D j , L := 2D + L f + L f L S , L S := max j∈Z,u∈[0,K] S j (u) and Let η = 2e Lt 0 > 0, we consider the following two sub-cases: It then follows from Gronwall's inequality that Summarizing the above two cases, we conclude that for any > 0, t 0 > 0 and compact set A ⊆ [−τ, 0] × Z, there exist η > 0 and compact set Ω M (j * ) such that A ⊆ Ω M (j * ) and Using this conclusion, it is easy to show that Q t [ϕ] is continuous in ϕ with respect to the compact open topology, for all γ ∈ [0, 1] and ϕ ∈ C β . Thanks to (A4), it is easy to verify that the functions u j (t) := u j (t; γϕ) and u j (t) := γu j (t; ϕ) constitute a pair of sub-and supersolution of (4) withū j (s) = u j (s) for j ∈ Z and s ∈ [−τ, 0]. By comparison principle, (2) It is clear that Q t [·] satisfies (E1), (E2) and (E4); while the assumption (E5) follows from Lemma 6. To verify (E3), let's define a family of linear operator It then follows from [19,Remark 4.1] that for any given γ > 0, there is an equivalent norm · * X in X such that L (t) * X ≤ e −γt , ∀t ≥ 0. Moreover, we define , ∀φ ∈ C β , t ≥ 0. It is easy to verify that Q t [C β ](0, ·) = u(t, ·; C β ) is pre-compact in the space C(Z, where α is the Kuratowski measure of non-compactness on C| I with (X, · X ) replaced by (X, · * X ). Thus, for each t > 0, Q t [·] satisfies (E3) with k = e −γt . The proof is complete. By Lemma 7 and [19, Theorem 5.1], the map Q 1 : C β → C β admits a rightward spreading speed c + * and a leftward spreading speed c − * . Then Proposition 1 implies that c ± * are also the leftward and rightward spreading speeds for solutions of (4) provided that c + * + c − * > 0 which will be proved in Theorem 2. Theorem 1. Let c + * and c − * be the rightward and leftward spreading speeds of Q 1 [·], respectively. Then the following results hold.
3.3. Computation of spreading speeds. We first give the formula of the spreading speeds c ± * by considering the linearized equation of (4) with respect to the equilibrium 0: (12) where j ∈ Z, t > 0. Let {L(t)} ∞ t=0 be the solution maps associated with (12), i.e.

4.
Effects of periodicity and delay on c ± * . In this section, we analyze the dependency of the spreading speeds c ± * with respect to the periodicity and delay. Since the influence of these facts may be opposite, we shall investigate each of them separately. For simplicity, we only consider the issue for c + * , since results for c − * can be obtained similarly. Throughout this section, we always assume that (A0)-(A4) hold.

Effect of delay on c +
* . For simplicity, we call c + * = c + * (τ ), given in Theorem 2, the rightward invasion speed of (25). The following lemma means that the delay decreases the invasion speed, in comparison with the model without delay.
Lemma 11 implies that increasing the amplitude of the effective birth rate of the species increases the invasion speed. Now, we call c + * = c + * (ρ), given in Theorem 2, the rightward invasion speed of (25) with g j (0) = ρ j > 0, where ρ j+N = ρ j , ∀j ∈ Z. The following lemma means that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium.

5.
Existence and uniqueness of pulsating traveling fronts. This section is devoted to the existence, non-existence, asymptotic behavior and uniqueness of the pulsating traveling fronts. As a direct consequence of Lemma 7 and Liang and Zhao [19,Theorem 5.3], we can show that the rightward and leftward spreading speeds c ± * are exactly the minimal wave speed of the rightward and leftward pulsating traveling fronts, respectively. However, we can not obtain any information about the decay rates of the wave profiles at −∞. So, in the sequel, we reprove the existence result of the pulsating traveling fronts for c > c ± * by a different method. The main idea is to extend the technique of monotone iteration scheme coupled with the method of sub-super solutions to periodic system (4). Based on constructing a pair of explicit sub-and supersolutions, we show that (4) has two rightward and leftward pulsating traveling fronts with speed c > c + * and c > c − * , respectively, which have exponential decay rates at −∞. Further, we show that the noncritical pulsating traveling front with given speed and satisfying the exponential decay rates at −∞ is unique (up to a translation).
We first give the definition of the pulsating traveling wave solution.
Definition 4. A leftward (or rightward) pulsating traveling wave solution of (4) connecting 0 and β refers to a solution where c represents the wave speed, and Ψ(·) = (ψ j (·)) j∈Z is called the wave profile. Moreover, we say that Ψ(·) is a pulsating traveling (wave) front if Ψ(·) is monotone.
As mentioned above, the following result on minimal wave speeds of the leftward and rightward pulsating traveling fronts is a direct consequence of Lemma 7 and [19, Theorem 3. Assume that (A0)-(A4) hold. Then, system (4) has a rightward (or leftward) pulsating traveling front with speed c connecting 0 and β if and only if c ≥ c + * (or c ≥ c − * ). We now consider the characteristic problems of wave profiles' equations. Let Ψ + (ξ) = ψ + j (−j + ct) j∈Z and Ψ − (ξ) = ψ − j (j + ct) j∈Z be the rightward and leftward wave profiles of (4), respectively. By (29), it is clear that The characteristic problems for (30) with respect to the equilibrium 0 can be represented by respectively. Let λ(µ) be the principal eigenvalue of (31) and ν j (µ) be the corresponding strongly positive eigenfunction. Thenλ(µ) = λ(−µ) andν j (µ) = ν j (−µ) are the principal eigenvalue and strongly positive eigenfunction of (32) respectively. By Lemma 8, we have the following result.
Based on Lemma 13, we can construct a pair of supersolution and subsolution of (30).
By (36) and (37), we see that φ + j (ξ) is a supersolution of (30). Similarly, we can show that φ − j (ξ) is a subsolution of (30) provided that q is sufficiently large. The proof is complete.
By Theorem 5 and Lemma 14, we are ready to prove the results of Theorem 4.

5.3.
Uniqueness of noncritical pulsating traveling fronts. Theorem 4 guarantees that system (4) has two rightward and leftward pulsating traveling fronts with speed c > c + * and c > c − * , respectively, which have exponential decay rates at −∞. In this section, we shall show that the noncritical pulsating traveling front with given speed and satisfying the exponential decay rates at −∞ is unique (up to a translation). We only prove the uniqueness of the leftward pulsating traveling front, since the same issue for the rightward pulsating traveling front can be discussed similarly. In addition to the assumptions of Theorem 4, we also need the following assumption: (A5): ∂ i f j (u, v) ≤ ∂ i f j (0, 0), i = 1, 2, and S j (u) ≤ S j (0) for all j ∈ Z.