Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat

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Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat Guo-Bao Zhang a,b,∗ , Xiao-Qiang Zhao b a College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China b Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada

Received 18 March 2019; revised 8 August 2019; accepted 18 September 2019

Abstract This paper is devoted to the study of the propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat. We first show that this equation admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method combined with a pair of generalized super- and sub-solutions. Then we establish the nonexistence, uniqueness and global exponential stability of periodic forced waves by applying the sliding technique and the comparison argument. Finally, we obtain the spreading properties for a large class of solutions. © 2019 Elsevier Inc. All rights reserved. MSC: 35K57; 35C07; 35B40; 92D25 Keywords: Nonlocal dispersal; Periodic shifting habitat; Forced traveling waves; Spreading properties

1. Introduction In this paper, we consider the following nonlocal dispersal Fisher-KPP equation in a timeperiodic shifting habitat ut = d[J ∗ u − u] + u[r(t, x − ct) − u], * Corresponding author.

E-mail address: [email protected] (G.-B. Zhang). https://doi.org/10.1016/j.jde.2019.09.044 0022-0396/© 2019 Elsevier Inc. All rights reserved.

(1.1)

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where x ∈ R, t > 0, d > 0, r(t, x) is ω-periodic in t , i.e., r(t + ω, x) = r(t, x) for some ω > 0, and J ∗ u is a spatial convolution defined by  (J ∗ u)(t, x) =

 J (x − y)u(t, y)dy =

R

J (y)u(t, x − y)dy. R

Here the variables u(t, x) stands for the population density of the species under consideration at time t and location x, d[J ∗ u − u] models nonlocal dispersal processes [17,20,30], d is the dispersal rate, r(t, x −ct) is the growth rate which assumed to be space-time dependent, changing in the form of shifting with constant speed c ∈ R. Throughout this paper, we always assume that (J1) (J2) (H1)

 J ∈ C 1 (R), J (x) = J (−x) ≥ 0, x ∈ R, R J (x)dx = 1; J is compactly supported; The function r(t, z) is continuous in (t, z), non-decreasing in z and bounded on R2 with r(t, −∞) < 0 < r(t, +∞).

From the assumption (H1), we see that r(t, x − ct) divides the spatial domain into two parts: the region with good-quality habitat suitable for growth (i.e., r(t, x − ct) > 0), and the region with poor-quality habitat unsuitable for growth (i.e., r(t, x − ct) < 0). The edge of the habitat suitable for species growth is shifting at a speed c. The sign of the shifting speed c determines whether the favorable habitat can invade the unfavorable one or the reverse. In recent years, there has been an increasing interest in the study of the effect of climate change on the survival of ecological species [1,6,15,24,26,37]. The works in this aspect can trace back to Berestycki et al. [6], who proposed a mathematical model that involves a reaction-diffusion equation on the real line ut = Duxx + f (x − ct, u),

(1.2)

where c is the speed of the climate change. The authors in [6] defined the minimum size of a moving habitat necessary to sustain a non-zero species population and derived criteria for the persistence of a species in any region with a moving and spatially varying habitat. The higher dimensional versions with more general type of f were studied later in [7,8]. When f in (1.2) takes a special form f (x − ct, u) = u(r(x − ct) − u), Li et al. [24] investigated the spreading dynamics of solutions of (1.2). Later, Fang, Lou and Wu [15] established the existence and nonexistence of traveling waves and pulse waves of (1.2). Meanwhile, the existence of traveling waves was also obtained by Hu and Zou [19]. Recently, Berestycki and Fang [9] obtained the existence and multiplicity of forced traveling waves as well as their attractivity for (1.2) with the general f satisfying a sublinearity condition. Vo [37] investigated the persistence of species facing a forced time periodic and locally favorable environment in a cylindrical or partially periodic domain and established the existence and uniqueness of the forced waves. More recently, Fang, Peng and Zhao [16] studied a reaction-diffusion equation in a time-periodic shifting environment, and obtained existence, uniqueness and stability of periodic forced waves. For other related results, we refer to [13,18,23] and references therein. Note that the classical reaction-diffusion equation like (1.2) is based on the assumption that the internal interaction of species is random and local, i.e., individuals move randomly between the adjacent spatial locations. However, it is not always the case in reality. The movements and

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interactions of many species in ecology and biology can occur between non-adjacent spatial locations [22,30]. Thus, nonlocal dispersal equations have been presented to investigate the evolution of species, see [12,14,20,25,29,32,43] and references therein. Correspondingly, the study of various properties of nonlocal dispersal operators, such as the spectral theory and the maximum principle, has been carried out. We refer to [10,11,31,36] and references therein. Recently, Li, Wang and Zhao [26] considered the following nonlocal dispersal population model ut = d[J ∗ u − u] + u[r(x − ct) − u],

(1.3)

to explore the species spread in the context of climate change. They showed that there exists a number c∗ such that the species will die out in the whole habitat if c > c∗ , while the species persists and spreads along the shifting habitat at an asymptotic spreading speed c∗ if c < c∗ . In addition, the authors of [26] proved the existence of forced traveling waves of (1.3) by constructing a pair of appropriate upper-lower solutions and using the method of monotone iteration. More recently, the uniqueness and global exponential stability of forced traveling waves of (1.3) were established by Wang and Zhao [38]. We should point out that the growth rate r(x − ct) of many populations may be influenced greatly by the time varying environments (e.g., due to seasonal variation). For example, in a one year period, the birth rate may be high in spring and summer and low in winter, while in winter more individuals might be at risk of death because of low temperature, lack of food, or some other reasons. Therefore, it is important to study a more general time-periodic nonlocal dispersal Fisher-KPP equation (1.1). The time-periodic reaction-diffusion equations without shifting habitat have been extensively studied, see, e.g., [2–5,21,27,31,39,42,44,45] for traveling waves and spreading speeds. Inspired by the recent work [16], we shall focus on the time periodic forced waves and spreading properties of solutions of (1.1). More precisely, we first prove the existence of periodic forced waves with speed c > −c∗ , where c∗ is the minimal wave speed of traveling waves of (1.1) with r(t, x − ct) replaced by r(t, +∞). Motivated by the works in [19,26,44,45], we adopt the monotone iteration technique combined with a pair of generalized super- and sub-solutions. Then by a sliding method [7,11,16,26] and some analytical skills, we establish the nonexistence of periodic forced waves with speed c ≤ −c∗ and the uniqueness of the periodic forced waves with speed c > −c∗ . Furthermore, we show that the periodic forced waves with speed c ∈ (−c∗ , c∗ ) are globally exponentially stable. Finally, by constructing various super-solutions and applying the spreading properties of solutions of (1.1) with r only depending on time t , we investigate the spreading properties of solutions to the corresponding initial value problem of (1.1). Compared with [26,38], we find that the time periodicity leads to many difficulties in the study of the periodic forced waves and spreading properties of (1.1). On the one hand, the corresponding wave profile equation of (1.1) is a partial differential equation, and hence, the integral operator for the fixed point and the super- and sub-solutions in [26] cannot be used to prove the existence of periodic forced waves of (1.1). As such, a new integral operator and a pair of generalized super- and sub-solutions should be introduced. On the other hand, it is difficult to apply the monotone semiflows approach in [38] to obtain the global stability of periodic forced waves of (1.1), and we will use the elementary comparison argument to prove it instead. With the help of the global stability, we further establish the exponential asymptotic stability by constructing a pair of appropriate super- and sub-solutions and using the comparison principle. Our main results can be summarized as follows.

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 Theorem 1.1 (Existence). Assume that (J1) and (H1) hold, and R J (x)eλx dx < +∞ for every λ > 0. Then for any given c > −c∗ , system (1.1) has a positive time ω-periodic solution V (t, x − ct) connecting 0 to p(t), where p(t) is the unique positive ω-periodic solution of (2.4).  Theorem 1.2 (Nonexistence). Assume that (J1) and (H1) hold, and R J (x)eλx dx < +∞ for every λ > 0. Then for any given c ≤ −c∗ , system (1.1) has no periodic forced wave V (t, x − ct) connecting 0 to p(t). Theorem 1.3 (Uniqueness). Assume that (J1)-(J2) and (H1) hold. Then for any given c > −c∗ , the periodic forced waves V (t, x − ct) of system (1.1) connecting 0 to p(t) are unique. In order to obtain the stability of the periodic forced waves V (t, x − ct), we need the following additional assumption: (H2)

d > maxt∈[0,ω] r(t, +∞).

Theorem 1.4 (Exponential stability). Assume that (J1)-(J2) and (H1)-(H2) hold. Let V (t, x − ct) be the unique periodic forced wave of (1.1) with c ∈ (−c∗ , c∗ ) and connecting 0 to p(t). Then there exists a real number μ > 0 such that for any φ ∈ X+ \ {0} (X+ is defined in Section 2.1), the unique solution u(t, x; φ) of system (1.1) with u(0, ·; φ) = φ satisfies lim sup |u(t, x; φ) − V (t, x − ct)|eμt = 0.

t→+∞ x∈R

(1.4)

 Theorem 1.5 (Spreading properties). Assume that (J1) and (H1) hold, and R J (x)eλx dx < +∞ for every λ > 0. Let u(t, x; φ) be the unique solution of system (1.1) with u(0, ·; φ) = φ ∈ X+ \ {0}. Then the following statements are valid: (i) For c > −c∗ , if there exists a constant X1 such that φ(x) = 0 for all x ≤ X1 , then limt→+∞ supx≤(c−σ )t u(t, x; φ) = 0 for any σ > 0. (ii) For c > c∗ , if φ(x) = 0 for x outside a bounded interval, then limt→+∞ u(t, x; φ) = 0 uniformly in x ∈ R; and if there exists a constant X2 such that φ(x) = 0 for all x ≥ X2 , then limt→+∞ supx≥ct u(t, x; φ) = 0. ∗ (iii) For c ∈ (0, c∗ ) and each σ ∈ (0, c 2−c ), there holds lim

sup

t→+∞ (c+σ )t
|u(t, x; φ) − p(t)| = 0.

The rest of this paper is organized as follows. In Section 2, we first study the well-posedness of solutions of the corresponding initial value problem of (1.1), and then present some results on spreading speeds and periodic traveling waves of (1.1) with r(t, x − ct) replaced by r(t, +∞). In Section 3, we prove the existence of periodic forced waves. In Section 4, the nonexistence and uniqueness of periodic forced waves are obtained. In Section 5, we establish the globally exponential stability of periodic forced waves. Section 6 is devoted to the investigation of the spreading properties of solutions of (1.1).

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2. Preliminaries In this section, we first consider the existence, uniqueness of solutions of the corresponding initial value problem of (1.1), and then recall some results on the spreading speeds and the periodic traveling waves of (1.1) with r = r(t, +∞). Inthis section and Sections 3 and 6, we always assume that (H1) holds and J satisfies (J1) and R J (x)eλx dx < +∞ for every λ > 0. For brevity, we don’t repeat these assumptions below. 2.1. The well-posedness of solutions Let X = BC(R, R) be the set of all bounded and continuous functions from R to R, and X+ = {φ ∈ X : φ(x) ≥ 0, ∀x ∈ R}. For any r > 0, we define Xr := {φ ∈ X : 0 ≤ φ(x) ≤ r, ∀x ∈ R}. Consider the following initial value problem 

ut (t, x) = d[(J ∗ u)(t, x) − u(t, x)] + u(t, x)[r(t, x − ct) − u(t, x)], u(0, x) = φ(x),

(2.1)

where (t, x) ∈ [0, +∞) × R. In view of [40, Lemma 3.1], the solution semigroup of the following linear nonlocal dispersal equation 

ut (t, x) = (J ∗ u − u)(t, x), x ∈ R, t > 0, u(0, x) = φ(x), x ∈ R,

is given by P (t)[φ](x) = e−t

∞

tm m=0 m! am (φ)(x),

where

 a0 (φ)(x) = φ(x),

am (φ)(x) =

J (x − y)am−1 (φ)(y)dy, ∀m ≥ 1. R

Then we can rewrite (2.1) as an integral equation t u(t, x) =P (dt)[u(0, ·)](x) +

  P (d(t − s)) u(s, ·)[r(s, · − cs) − u(s, ·)] (x)ds, (2.2)

0

where (t, x) ∈ [0, +∞) × R. As usual, solutions of (2.2) are called mild solutions to equation (1.1). In order to obtain the existence and uniqueness of the solution to (2.1), we need a comparison theorem for super- and sub-solutions of (2.1). For this purpose, we first introduce the concept of super- and sub-solutions. Definition 2.1. A function w(t, x) ∈ C([0, T ), X+ ) with 0 < T ≤ +∞ is said to be a supersolution (sub-solution) of (2.1) if it satisfies

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t w(t, x) ≥ (≤)P (dt)[w(0, ·)](x) +

  P (d(t − s)) w(s, ·)[r(s, · − cs) − w(s, ·)] (x)ds.

0

Remark 2.2. If a function w ∈ C([0, T ), X+ ) is C 1 in t ∈ (0, T ), and satisfies wt (t, x) ≥ (≤)d[(J ∗ w)(t, x) − w(t, x)] + w(t, x)[r(t, x − ct) − w(t, x)] for (t, x) ∈ [0, T ) × R, then w(t, x) is a super-solution (sub-solution) of (1.1). The following comparison principle can be proved by an argument similar to that in [21, Theorem 2.3], so we omit the proof here. Lemma 2.3. Let u(t, x) and u(t, x) be super- and sub-solutions of (1.1), respectively. If u(0, ·) ≥ u(0, ·), then u(t, ·) ≥ u(t, ·) for all t ≥ 0. Now we present the result on the existence and uniqueness of solutions of (2.1). Lemma 2.4. For any φ ∈ X+ , (2.1) admits a unique nonnegative and bounded classical solution u(t, x; φ) with u(0, x; φ) = φ, ∀t ≥ 0. Proof. By a classical monotone iteration method combined with super- and sub-solutions, we can obtain that when φ ∈ X+ , system (2.1) has a unique nonnegative mild solution u ∈ C(R+ , X+ ) for all t ≥ 0. We refer readers to [26, Theorem 2.3] for (1.1) with r(t, x − ct) replaced by r(x − ct). It is easy to calculate that ∂[P (t)φ](x) = −[P (t)φ](x) + ∂t

 J (y)[P (t)φ](x − y)dy. R

Hence, the right side of (2.2) is differentiable with respect to t . Thus, u(t, x; φ) is a classical solution of (2.1). 2 2.2. The nonlocal Fisher-KPP equation without shifting habitat In this subsection, we study the spreading speed and periodic traveling waves of the following time periodic Fisher-KPP equation with nonlocal dispersal ut (t, x) = d[(J ∗ u)(t, x) − u(t, x)] + u(t, x)[r(t, +∞) − u(t, x)].

(2.3)

The spatially homogeneous system of (2.3) is u (t) = u(t)[r(t, +∞) − u(t)].

(2.4)

ω Define r¯ (x) = ω1 0 r(t, x)dt. It then follows from [46, Theorem 3.1.2] that if r¯ (+∞) > 0, then (2.4) admits a unique positive ω-periodic solution p(t), which is globally asymptotically stable for positive initial value. Moreover, p(t) can be explicitly given by

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p0 e 0 r(s,+∞)ds p(t) = ,  t s 1 + p0 0 e 0 r(τ,+∞)dτ ds

7

ω

e 0 r(s,+∞)ds − 1 p0 =  ω  s > 0. 0 r(τ,+∞)dτ ds 0 e

In [21], the authors considered the following periodic evolution equation  ut (t, x) = a(t) k(x − y)u(t, y)dy + F (t, u(t, x)).

(2.5)

(2.6)

R

By applying the theory of asymptotic speeds of spread and traveling waves for monotone periodic semi-flows developed in [27,28,41], they established the existence of the spreading speed, the nonexistence of continuous periodic traveling wave solutions and the existence of left-continuous periodic traveling waves. Note that if we choose a(t) = d, k = J and F (t, u) = u[r(t, +∞) − d − u], then (2.6) reduces to (2.3). Let g(t, u) := r(t, +∞) − d − u and uˆ be some constant such that uˆ > maxt∈[0,ω] r(t, +∞). Then it is easy to verify that (H1) in [21] holds. Moreover, the assumptions on J imply (H2) in [21] holds. Let  d( R J (y)eλy dy − 1) + r¯ (+∞) cω∗ ∗ c := = inf , (2.7) λ>0 ω λ where cω∗ is the spreading speed for the Poincaré map Qω = u(ω, x; φ). The subsequent result shows that c∗ is the spreading speed of solutions of (2.3), which is a straightforward consequence of [21, Theorem 3.5]. Lemma 2.5. Assume that r¯ (+∞) > 0, and let u(t, x; φ) be the solution of (2.3) with initial function φ ∈ Xp(0) . Then the following statements are valid: (i) For any c > c∗ , if φ(x) = 0 for x outside a bounded interval, then lim

t→∞,|x|≥ct

u(t, x; φ) = 0.

(ii) For any c ∈ (0, c∗ ), if φ ≡ 0, then lim

t→∞,|x|≤ct

(u(t, x; φ) − p(t)) = 0.

An ω-periodic traveling wave of (2.3) is a special solution taking the following form u(x, t) = U (t, z),

U (t, z) = U (t + ω, z),

z = x − ct,

(2.8)

where c is a real number and denotes the wave speed, U : R × R → R formulates the wave profile. Plugging (2.8) into (2.3) leads to ⎛ ⎞  Ut = d ⎝ J (y)U (t, z − y)dy − U ⎠ + cUz + U (r(t, +∞) − U ) (2.9) R

for (t, z) ∈ R2 .

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By [21, Theorems 4.1 and 4.5], we have the following result on the existence and nonexistence of periodic traveling waves. Lemma 2.6. Assume that r¯ (+∞) > 0. Then for any c ≥ c∗ , (2.3) has a periodic traveling wave U (t, x − ct) connecting p(t) to 0 such that U (t, z) is left continuous and nonincreasing in z. Moreover, for any c < c∗ , (2.3) has no periodic traveling wave U (t, x − ct) connecting p(t) to 0. 3. The existence of periodic forced waves In this section, we study the existence of periodic traveling waves of (1.1) with the wave speed at which the habitat is shifting. We call such a wave as a periodic forced wave. The periodic forced wave of (1.1) takes the same form as that in (2.8). In order to distinguish the periodic traveling waves of equations (2.3) and (1.1), we let u(t, x) = V (t, z) with z = x − ct in (1.1), and then get the wave profile equation ⎛ Vt = d ⎝



⎞ J (y)V (t, z − y)dy − V ⎠ + cVz + V (r(t, z) − V ),

(3.1)

R

with the periodic constraints V (t, z) ≡ V (t + ω, z). We impose the following boundary conditions lim V (t, z) = 0 and lim V (t, z) = p(t) uniformly for t ∈ R.

z→−∞

z→+∞

(3.2)

For any given s ∈ R ∪ {±∞}, let V s (t, z; φ) be the unique solution of ⎛



Vt = d ⎝

⎞ J (y)V (t, z − y)dy − V ⎠ + cVz + V (r(t, z + s) − V ),

t > 0,

(3.3)

R

satisfying V (0, z) = φ(z) for z ∈ R. It is easy to see that if s = 0, then (3.3) reduces to (3.1), and if s = +∞, then (3.3) reduces to (2.9). By the uniqueness of solutions, we obtain V s (t, z; φ(· + s)) = V 0 (t, z + s; φ),

t ≥ 0, z, s ∈ R.

(3.4)

Definition 3.1. A function W ∈ C 1,1 ([0, +∞) × R, R) is said to be a super-solution of (3.3) if ⎛



Wt ≥ d ⎝

⎞ J (y)W (t, z − y)dy − W ⎠ + cWz + W (r(t, z + s) − W ).

R

It is called a sub-solution of (3.3) if the above inequality is reversed.

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Definition 3.2. A function W ∈ C([0, +∞) × R, R) is said to be a generalized super-solution of (3.3) if there exist super-solutions W1 , · · · , Wk of (3.3) such that W = min{W1 , · · · , Wk }. It is called a generalized sub-solution of (3.3) if there exist sub-solutions V1 , · · · , Vk of (3.3) such that W = max{V1 , · · · , Vk }. Let Qs : C(R, R+ ) → C(R, R+ ) be the Poincaré map associated with the time periodic equation (3.3), that is, Qs [φ] := V s (ω, ·; φ). We can easily prove that Qs admits the following properties. Lemma 3.3. The following statements are valid: (i) (ii) (iii) (iv)

If φ ≥ ψ , then Qs [φ] ≥ Qs [ψ]. If φ is nondecreasing, then Qs [φ] is also nondecreasing. If φn is uniformly bounded, then Qs [φn ], up to subsequence, converges locally uniformly. If W ≤ W are a pair of generalized sub- and super-solutions of (3.3) for t ∈ (0, T0 ) with T0 > ω, then W (0, ·) ≤ Qs [W (0, ·)] ≤ Qs [W (0, ·)] ≤ W (0, ·). If, in addition, W (0, ·) ≥ W (ω, ·), then V s (t, z; W (0, ·)) converges, as t → +∞, to a time periodic solution V s,∗ (t, z) of (3.3) locally uniformly such that W ≤ V s,∗ ≤ W . So does V s (t, z; W (0, ·)) provided that W (0, ·) ≤ W (ω, ·).

Proof. We only prove (ii), i.e., Qs [φ](z) := V s (ω, z; φ) is nondecreasing in z ∈ R provided φ is nondecreasing on R. For any η > 0, let W (t, z) = V s (t, z + η; φ). Since r(t, z) is nondecreasing in z, we have ⎛



Wt =d ⎝

R

⎛



≥d ⎝

⎞ J (y)W (t, z − y)dy − W ⎠ + cWz + W (r(t, z + η + s) − W ) ⎞ J (y)W (t, z − y)dy − W ⎠ + cWz + W (r(t, z + s) − W ),

t > 0,

R

which means that W (t, z) is a super-solution of (3.3). Since φ is nondecreasing on R, one has W (0, z) = V s (0, z + η; φ) = φ(z + η) ≥ φ(z) for any z ∈ R. Then by the comparison principle, we obtain W (t, z) ≥ V s (t, z; φ) for any t ≥ 0 and z ∈ R, which implies V s (ω, z + η; φ) ≥ V s (ω, z; φ) for z ∈ R, and hence, Qs [φ](z) is nondecreasing in z ∈ R. 2 Lemma 3.4. Assume that V ≡ 0 is a nonnegative and bounded solution of (3.1) with V (t + ω, z) ≡ V (t, z). Then 0 < V (t, z) < p(t) for (t, z) ∈ [0, ω] × R and V (t, −∞) = 0 uniformly for t ∈ [0, ω]. Proof. We first claim that V (t, z) > 0, ∀(t, z) ∈ R2 . Assume, by contradiction, that V (t0 , z0 ) = 0 for some (t0 , z0 ) ∈ R2 . By the argument similar to that in the proof of Theorem 1.2, it follows that V (t0 , ·) ≡ 0, and hence, V (t, ·) ≡ 0 for all t ≥ t0 . Now the time periodicity of V (t, z) implies

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that V ≡ 0 on R2 , which contradicts our assumption on V . For any M ≥ 1, it is easy to verify that Mp(t) is a super-solution of (3.1), where p(t) is the unique positive ω-periodic solution of (2.4). Since V is bounded, we can choose M large enough such that Mp(t) ≥ V (t, z) for (t, z) ∈ R2 . From Lemma 3.3 (iv), we obtain Qn0 [Mp(0)] ≥ Qn+1 0 [Mp(0)] ≥ V (0, ·),

∀n ≥ 0.

(3.5)

Thus, Qn0 [Mp(0)] converges to some ϕ locally uniformly as n → ∞. By Lemma 3.3 (ii)-(iii), we see that ϕ is nondecreasing and ϕ = Q0 [ϕ]. Thus, letting t = ω and z = 0 in (3.4), we obtain ϕ(−∞) = lim ϕ(s) = lim Q0 [ϕ](s) = lim Qs [ϕ(· + s)](0). s→−∞

s→−∞

s→−∞

(3.6)

Since ϕ(z + s) and r(t, z + s) converge to ϕ(−∞) and r(t, −∞) locally uniformly in(t, z) as s → −∞, respectively, we can obtain that V s (t, z; ϕ(· + s)) converges to V −∞ (t, z; ϕ(−∞)) locally uniformly in (t, z) as s → −∞. Then we have lim Qs [ϕ(· + s)](0) = lim V s (ω, 0; ϕ(· + s))

s→−∞

=V

−∞

s→−∞

(ω, 0; ϕ(−∞)) = Q−∞ [ϕ(−∞)].

(3.7)

Combining (3.6) and (3.7), we have ϕ(−∞) = Q−∞ [ϕ(−∞)]. By a similar argument, we can obtain ϕ(+∞) = Q+∞ [ϕ(+∞)]. Hence, ϕ(±∞) are spatially homogeneous fixed points of Q±∞ , respectively. It is easy to see that ϕ(−∞) = 0 and ϕ(+∞) = 0 or p(0). By (3.5), ϕ(·) ≥ V (0, ·). It then follows from ϕ(−∞) = 0 and (3.4) that V (t, −∞) = 0 uniformly for t ∈ [0, ω]. Since ϕ(·) ≥ V (0, ·), we obtain that ϕ(+∞) = p(0) due to V (t, z) > 0, which implies that V (t, z) ≤ p(t), ∀(t, z) ∈ R2 . It remains to prove that V (t, z) < p(t), ∀(t, z) ∈ R2 . Assume, by contradiction, that V (t0 , z0 ) = p(t0 ) for some (t0 , z0 ) ∈ R2 . Then w(t, z) := p(t) − V (t, z) ≥ 0, ∀(t, z) ∈ R2 and w(t0 , z0 ) = 0. By the argument similar to that in the proof of Theorem 1.2, we obtain w(t0 , ·) ≡ 0, and hence, V (t0 , z) = p(t0 ), ∀z ∈ R. This gives rise to 0 = V (t0 , −∞) = p(t0 ), a contradiction. 2 In order to prove the existence of ω-periodic solutions of (3.1), we make a change of variable W (t, z) = V (t, −z) for any z ∈ R. In view of (3.1) and (3.2), W (t, z) satisfies ⎛



Wt = d ⎝

⎞ J (y)W (t, z − y)dy − W ⎠ − cWz + W (r(t, −z) − W ),

(3.8)

R

with the periodic constraints W (t + ω, z) = W (t, z) for (t, z) ∈ R2 , and boundary conditions lim W (t, z) = p(t)

z→−∞

and

lim W (t, z) = 0 uniformly in t ∈ [0, ω].

z→+∞

(3.9)

Clearly, the existence of ω-periodic forced waves of (1.1) is equivalent to that of (3.8). We shall use the monotone iteration method together with the generalized super- and sub-solutions to prove the existence of ω-periodic solutions of (3.8) satisfying (3.9).

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11

At first, we construct a pair of ordered generalized super- and sub-solutions. In order to give a sub-solution, we introduce an auxiliary time-periodic nonlocal dispersal equation of ignition type: ut (t, x) = d(J ∗ u − u)(t, x) + f (t, u(t, x)),

(3.10)

where  is small enough such that r(t, +∞) −  > 0 for t ∈ R and f (t, u) is an ignition nonlinearity  f (t, u) =

u(r(t, +∞) −  − u), 0,

if u ≥ 0, if −  ≤ u < 0.

The following result on periodic traveling waves of (3.10) comes from [33–35]. Lemma 3.5. There exist a constant c > 0 and a spatially decreasing function W  (t, z) ∈ C 1,1 (R2 , R), z = x − c t solving ⎧      2 ⎪ ⎨Wt = d(J ∗ W − W ) + c Wz + f (t, W ), (t, z) ∈ R , W  (t, −∞) = p (t), W  (t, +∞) = −, t ∈ R, ⎪ ⎩  W (t, z) = W  (t + ω, z), (t, z) ∈ R2 ,

(3.11)

where p (t) is the unique positive ω-periodic solution of p (t) = p(t)(r(t, +∞) −  − p(t)).

(3.12)

Moreover, W  (t, z) is unique up to translation with respect to z. It follows from (3.11) that there exists a function z0 (t) ∈ R such that W  (t, z0 (t)) = 0. Similar to (2.5), the positive ω-periodic solution p (t) of (3.12) can also be given explicitly. It is easy to see that p (0) is monotone decreasing in  > 0, and so is p (t) for t > 0 by the comparison principle. Lemma 3.6. Let c∗ be the minimal wave speed of periodic traveling waves of (2.9) defined in (2.7). Then lim→0+ c = c∗ . Proof. We first prove that c is nonincreasing in  > 0. Let 2 > 1 > 0 and u1 (t, x) = W 1 (t, x − c1 t) and u2 (t, x) = W 2 (t, x − c2 t) be spatially decreasing solutions of (3.11) with  being replaced by 1 and 2 , respectively. We normalize W 1 (t, z) and W 2 (t, z) such that W 1 (0, 0) = W 2 (0, 0) =

1 min p(t) > 0. 2 t∈[0,ω]

(3.13)

By the definition of f , one has f1 ≥ f2 . In addition, W 1 (0, −∞) = p1 (0) > p2 (0) = W 2 (0, −∞) and W 1 (0, +∞) = −1 > −2 = W 2 (0, +∞). It is easy to see that any translation of a wave profile of (3.11) in space variable is also a wave profile. Hence, we can assume that W 1 (0, x) > W 2 (0, x) for any x ∈ R, i.e., u1 (0, x) > u2 (0, x) for x ∈ R. By the comparison

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principle, we have u1 (t, x) = W 1 (t, x − c1 t) > u2 (t, x) = W 2 (t, x − c2 t) for t > 0 and x ∈ R. We infer that c1 ≥ c2 . Otherwise, we let c1 < c2 . Since W i (t, z) is ω- periodic in t , i = 1, 2, for every positive integer k, we obtain W 1 (0, x − c1 kω) = W 1 (kω, x − c1 kω) > W 2 (kω, x − c2 kω) = W 2 (0, x − c2 kω). Taking x = c2 kω in above inequality, we get W 2 (0, 0) < W 1 (0, (c2 − c1 )kω) → −1 , as k → ∞, which leads a contradiction to the normalization condition (3.13). Hence, we have c1 ≥ c2 , which implies that c is nonincreasing in  > 0. It is easy to see that for any small  > 0, c ≤ c∗ . Thus, lim→0+ c exists. We set c˜ = lim→0+ c . Clearly, c˜ ≤ c∗ . For each n , n = 1, 2, · · · , there exists a unique solution (cn , W n ) satisfying (3.11) with  being replaced by n . Choosing a decreasing sequence {n } such that n → 0 as n → ∞ and normalizing W n (0, 0) = 12 mint∈[0,ω] p(t). By some a priori estimates, there is a subsequence of n , still denoted by n for simplicity, such that (cn , W n ) converges to a solution (c, ˜ W˜ ) satisfying ⎧ ⎪ ⎨W˜ t = d(J ∗ W˜ − W˜ ) + c˜W˜ z + f (t, W˜ ), W˜ (0, 0) = 12 mint∈[0,ω] p(t), ⎪ ⎩˜ W (t, z) = W˜ (t + ω, z), (t, z) ∈ R2 ,

(t, z) ∈ R2

where f (t, u) = u(r(t, +∞) − u). Since Wzn (t, z) ≤ 0 and 0 ≤ W n (t, z) ≤ maxt∈[0,ω] p(t) for (t, z) ∈ R2 , we have W˜ z ≤ 0 and 0 ≤ W˜ ≤ maxt∈[0,ω] p(t), which implies that W˜ (t, ±∞) exist and are periodic solutions of u (t) = f (t, u). From Section 2, we know that p(t) is the unique positive periodic solution. Thus, we obtain W˜ (t, ±∞) ∈ {0, p(t)}. By the monotonicity of W˜ (t, z) in z, one has 1 min p(t) ≤ lim W˜ (0, z) = W˜ (0, −∞), W˜ (0, +∞) = lim W˜ (0, z) ≤ W˜ (0, 0) = z→+∞ z→−∞ 2 t∈[0,ω] which implies W˜ (t, +∞) = 0 and W˜ (t, −∞) = p(t). Therefore, W˜ (t, x − ct) ˜ is a periodic traveling wave of (2.3) connecting p(t) and 0 with speed c. ˜ By Lemma 2.6, we have c˜ ≥ c∗ . Therefore, c˜ = c∗ . 2 Lemma 3.7. For any c > −c∗ , W (t, z) := max{W  (t, z), 0} is a generalized sub-solution of (3.8). Proof. Since c > −c∗ and lim→0+ c = c∗ , we obtain c > −c for sufficiently small . Without loss of generality, we assume that r(t, −z0 (t)) ≥ r(t, +∞) −  for t ∈ R. If z ≤ z0 (t), then W  (t, z) ≥ 0, and hence, W (t, z) = W  (t, z) for t ∈ R. Thus, one has ⎛ Wt − d ⎝



⎞ J (y)W (t, z − y)dy − W ⎠ + cW z − W (r(t, −z) − W )

R

⎛



≤ Wt − d ⎝

R

⎞ J (y)W  (t, z − y)dy − W  ⎠ + cWz − W  (r(t, −z0 (t)) − W  )

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⎛ ≤ Wt − d ⎝

13

⎞



J (y)W  (t, z − y)dy − W  ⎠ − c Wz − W  (r(t, +∞) −  − W  )

R

= 0. If z ≥ z0 (t), then W (t, z) = 0 for t ∈ R. Thus, we obtain ⎛ Wt − d ⎝  = −d



⎞ J (y)W (t, z − y)dy − W ⎠ + cW z − W (r(t, −z) − W )

R

J (y)W (t, z − y)dy ≤ 0. R

This completes the proof.

2

By the assumption (H1), we can choose z1 > 0 sufficiently large such that r(t, −z1 ) < 0 for t ∈ R. Then it is easy to see that for any c ∈ R, the following equation ⎛



d⎝

⎞ J (y)eλy dy − 1⎠ + cλ + r¯ (−z1 ) = 0

R

has a positive real root γ1 > 0. Define a continuous function φ(t) = e

t

r (−z1 )]ds 0 [r(s,−z1 )−¯

.

Since r(t, z) is ω-periodic in t , by the definition of r¯ (−z1 ), it can be easily verified that φ(t) is a ω-periodic function. Let   W (t, z) := min p(t), ρφ(t)e−γ1 (z−z1 ) , where ρ > 0 is chosen such that max p(t) ≤ ρ min φ(t), t∈[0,ω]

t∈[0,ω]

which implies that when z ≤ z1 , p(t) ≤ ρφ(t)e−γ1 (z−z1 ) , and hence, W (t, z) = p(t). Lemma 3.8. W (t, z) is a generalized super-solution of (3.8). Proof. If W (t, z) = ρφ(t)e−γ1 (z−z1 ) , then z ≥ z1 , and by (3.2), we have ⎛



Wt − d ⎝

R

⎞ J (y)W (t, z − y)dy − W ⎠ + cW z − W (r(t, −z) − W )

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14

⎡ ⎛



≥ ρφ (t)e−γ1 (z−z1 ) − ρφ(t)e−γ1 (z−z1 ) ⎣d ⎝

⎞

⎤

J (y)eγ1 y dy − 1⎠ + cγ1 + r(t, −z)⎦

R

≥ φ (t)e

−γ1 (z−z1 )

− φ(t)e

−γ1 (z−z1 )

(−¯r (−z1 ) + r(t, −z1 )) = 0.

If W (t, z) = p(t), then one has ⎛ ⎞  W t − d ⎝ J (y)W (t, z − y)dy − W ⎠ + cW z − W (r(t, −z) − W ) R

≥ p (t) − p(t)(r(t, −z) − p(t)) ≥ p (t) − p(t)(r(t, +∞) − p(t)) = 0. The last equality holds since p(t) is a solution of (2.4).

2

By the construction of W and W , we can see that W (t, z) ≥ W (t, z) for (t, z) ∈ R2 . Hence, we define a set   W (t, z) ≤ W (t, z) ≤ W (t, z),  = W ∈ C(R × R, R) : , W (t + ω, z) = W (t, z), and an operator H :  → C(R × R, R) by ⎛ ⎞  H(W )(t, z) :=βW (t, z) + d ⎝ J (y)W (t, z − y)dy − W (t, z)⎠ R

+ W (t, z)(r(t, −z) − W (t, z)), where β = d − mint∈[0,ω] r(t, −∞) + 2 supt∈[0,ω],z∈R W (t, z). Then (3.8) can be rewritten as follows Wt (t, z) + cWz (t, z) = −βW (t, z) + H(W )(t, z).

(3.14)

It is easy to verify that equation (3.14) is equivalent to the following integral equation t W (t, z) =

e−β(t−s) H(W )(s, z + c(s − t))ds.

(3.15)

0

Thus, the existence of solutions of (3.8) reduces to that of the fixed point of the operator F :  → C(R × R, R) defined by t F(W )(t, z) = 0

e−β(t−s) H(W )(s, z + c(s − t))ds, t ∈ (0, ω], z ∈ R.

(3.16)

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15

Lemma 3.9. F is a nondecreasing operator and maps  into . Moreover, if W ∈  is nonincreasing, then F(W )(t, z) is nonincreasing with respect to z. Proof. For any W1 , W2 ∈  with W1 (t, z) ≤ W2 (t, z) for (t, z) ∈ [0, ω] × R, we obtain H(W1 )(t, z) − H(W2 )(t, z)  = d J (y)(W1 (t, z − y) − W2 (t, z − y))dy R

+ [β − d + r(t, −z) − (W1 (t, z) + W2 (t, z))] (W1 (t, z) − W2 (t, z))   ≤ − min r(t, −∞) + 2 sup W (t, z) + r(t, −z) − (W1 (t, z) + W2 (t, z)) t∈[0,ω]

t∈[0,ω],z∈R

× (W1 (t, z) − W2 (t, z)) ≤ 0, where we used β = d − mint∈[0,ω] r(t, −∞) + 2 supt∈[0,ω],z∈R W (t, z). The above inequality means H(W )(t, z) is monotone non-decreasing in W ∈ . By the definition of F in (3.16), we have F(W1 )(t, z) ≤ F(W2 )(t, z) for all (t, z) ∈ [0, ω] × R. Thus, F(W )(t, z) ≤ F(W )(t, z) ≤ F(W )(t, z), for all W ∈ . Furthermore, we obtain t F(W )(t, z) =

e−β(t−s) H(W )(s, z + c(s − t))ds

0

t ≤

e−β(t−s) (W s (s, z + c(s − t)) + cW z (s, z + c(s − t)) + βW (s, z + c(s − t)))ds

0

t =

e−β(t−s) [(W (s, z + c(s − t)))s + βW (s, z + c(s − t))]ds

0

≤ W (t, z). Similarly, F(W )(t, z) ≥ W (t, z). Therefore, F() ⊆ . If W ∈  is nonincreasing in z ∈ R, then for any t ∈ [0, ω] and z1 , z2 ∈ R with z1 < z2 , one has H(W )(t, z1 ) − H(W )(t, z2 ) = (β − d − W (t, z1 ) − W (t, z2 ))(W (t, z1 ) − W (t, z2 )) + r(t, −z1 )W (t, z1 )  − r(t, −z2 )W (t, z2 ) + d J (y)(W (t, z1 − y) − W (t, z2 − y))dy R

≥ (β − d − W (t, z1 ) − W (t, z2 ) + r(t, −z2 ))(W (t, z1 ) − W (t, z2 ))

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16

 = − min r(t, −∞) + 2 t∈[0,ω]

 W (t, z) − W (t, z1 ) − W (t, z2 ) + r(t, −z2 )

sup t∈[0,ω],z∈R

× (W (t, z1 ) − W (t, z2 )) ≥ 0. Thus, we have t F(W )(t, z1 ) =

e−β(t−s) H(W )(s, z1 + c(s − t))ds

0

t ≥

e−β(t−s) H(W )(s, z2 + c(s − t))ds = F(W )(t, z2 ).

0

2

This completes the proof. Proof of Theorem 1.1. Set

W0 = W,

W n+1 = F(W n ), n = 0, 1, · · · .

It can be shown that W (t, z) ≤ W n+1 (t, z) ≤ W n (t, z) ≤ W (t, z) for all (t, z) ∈ [0, ω] × R. Therefore, the sequence {W n } converges to a function W . We further show that W is a fixed point of F . Since     |H(W n )(t, z)| =βW n (t, z) + d J (y)W n (t, z − y)dy − W n (t, z) R

  + W n (t, z)(r(t, −z) − W n (t, z)) ≤C0

sup

W (t, z),

t∈[0,ω],z∈R

where C0 := β + 2d +

sup

W (t, z) + max max{r(t, +∞), −r(t, −∞)}, t∈[0,ω]

t∈[0,ω],z∈R

by Lebesgue’s dominated convergence theorem, we have W (t, z) = lim W n+1 (t, z) n→∞

t = lim F(W )(t, z) = n

n→∞

e−β(t−s) lim H(W n )(s, z + c(s − t))ds n→∞

0

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t =

17

e−β(t−s) H(W )(s, z + c(s − t))ds = F(W )(t, z).

0

Note that W and W are periodic in t , and W is nonincreasing with respect to z. We can obtain that W n ∈  and W n is nonincreasing in z. Hence, W (t, z) is ω-periodic in t and nonincreasing in z. It remains to show the asymptotic behavior of W (t, z) as z → ±∞. Notice that for each t ≥ 0, limz→±∞ W (t, z) exist since Wz ≤ 0 and W is bounded. By the regularity of W with respect to t and the compactness of [0, ω], we have limz→±∞ W (t, z) = W (t, ±∞) uniformly in t ∈ R. In view of lim W (t, z) = lim W (t, z) = 0,

z→+∞

z→+∞

it follows that limz→+∞ W (t, z) = 0. Since W (t, z) is a nonnegative and bounded solution of (3.8), by Lemma 3.4, 0 < W (t, z) < p(t). In addition, W (t, z) ≥ W (t, z) implies that W (t, −∞) ≥ p (t) > 0 for t ∈ [0, ω]. Note that W (t, z) satisfies the integral equation (3.15). By taking the limit to (3.15) as z → −∞ and then differentiating it with respect to t , we obtain W (t, −∞) is a positive periodic solution of (2.4). Due to the uniqueness of positive periodic solution of (2.4), we have W (t, −∞) = p(t) uniformly for t ∈ R. 2 4. The nonexistence and uniqueness In this section, we discuss the nonexistence and uniqueness of periodic forced waves. Let c∗ be defined in (2.7). Since r¯ (+∞) > 0, [25, Lemma 2.3] implies that the eigenvalue equation ⎛



d⎝

⎞ J (y)eλy dy − 1⎠ − cλ + r¯ (+∞) = 0

(4.1)

R

admits two negative real roots λ2 ≤ λ1 < 0 if and only if c ≤ −c∗ . For every c ∈ R, since r¯ (−∞) < 0, the eigenvalue equation ⎛



d⎝

⎞ J (y)eλy dy − 1⎠ − cλ + r¯ (−∞) = 0

(4.2)

R

has a unique negative root μ1 < 0. Proof of Theorem 1.2. Assume, by contradiction, that (3.1) admits a positive solution V (t, z) connecting 0 to p(t) with speed c ≤ −c∗ . We claim that V (t, z) = o(e−(μ1 +η)z ) for any small η > 0 as z → −∞, where μ1 is the unique negative root of (4.2). Let 0 := −¯r (−∞) > 0. For any given  ∈ (0, 0 ), there exists η ∈ (0, −μ1 ) such that ⎛



d⎝

R

⎞ J (y)e(μ1 +η)y dy − 1⎠ − c(μ1 + η) + r¯ (−∞) +  = 0,

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18

and there exists −z > 0 large enough such that r(t, z) ≤ r(t, −∞) +  t



for

z ≤ z .

(4.3)

(μ +η)y

Define ϕ (t) := e 0 [d( R J (y)e 1 dy−1)−c(μ1 +η)+r(s,−∞)+]ds . Clearly, ϕ (t) is an ω-periodic function. It is easy to see that for above z , there exists ρ > 0 such that ρ e−(μ1 +η)z ϕ (t) ≥ max p(s), s∈[0,ω]

z ≥ z , t ∈ R.

(4.4)

Define   v(t, ˜ z) := min p(t), ρ e−(μ1 +η)z ϕ (t) . It is easy to verify that v(t, ˜ z) is a generalized super-solution of (3.1). Note that if v(t, ˜ z) ≥ V (t, z) for (t, z) ∈ R2 , then the claim is proved. Define D := {(t, z) ∈ ˜ z) ≥ p(t)}. Let ∂D be the boundary of D . By the definition of v(t, ˜ z), we see that R2 : v(t, there exists zˆ (t) such that ∂D := {(t, zˆ (t)), t ∈ R}. It then follows from (4.4) that zˆ (t) ≤ z , ˜ z) ≥ V (t, z) for (t, z) ∈ D . ∀t ∈ R. By Lemma 3.4, V (t, z) < p(t) for (t, z) ∈ R2 , and hence, v(t, So it suffices to prove that v(t, ˜ z) ≥ V (t, z) for (t, z) ∈ / D . For this purpose, we try to find a positive periodic function ψ(t) such that for any δ > 0, v(t, ˜ z) + δψ(t), (t, z) ∈ / D , is also a super-solution of (3.1). If this is true, then by the fact that v(t, ˜ z) + δψ(t) ≥ v(t, ˜ z) ≥ V (t, z) for (t, z) ∈ D , v(0, ˜ z) + δψ(0) ≥ V (0, z) for any z ≤ z˜ (0) and the comparison principle, we obtain that v(t, ˜ z) + δψ(t) > V (t, z) for δ > 0 and (t, z) ∈ / D , and hence, letting δ → 0, we get v(t, ˜ z) ≥ V (t, z) for (t, z) ∈ / D . For (t, z) ∈ / D , by (4.4), we have z < z , and hence, (4.3) holds. t Choose ψ(t) := e 0 [r(s,−∞)+0 ]ds . Clearly, ψ(t) is ω-periodic due to 0 = −¯r (−∞). A simple calculation yields that ⎛ (v˜ + δψ)t − d ⎝



⎞ J (y)(v˜ + δψ)dy − (v˜ + δψ)⎠ − c(v˜ + δψ)z

R

− (v˜ + δψ)(r(t, z) − (v˜ + δψ))   ≥ δψ r(t, −∞) + 0 − (r(t, z) − (v˜ + δψ)) ≥ δψ [r(t, −∞) + 0 − r(t, −∞) − ] ≥ 0, due to  < 0 . Hence, for any δ > 0, v(t, ˜ z) + δψ(t), (t, z) ∈ / D , is a super-solution of (3.1). t [r(s,+∞)−¯r (+∞)]ds ρ −λ z 1 0 Define v (t, z) = ρϕ(t)e , where ρ > 0, ϕ(t) = e and λ1 is the negative root of the eigenvalue equation (4.1). It is easy to verify that v ρ (t, z) is a super-solution of (3.1). In view of (4.1), one has ⎛



d⎝

R

⎞ J (y)eλ1 y dy − 1⎠ − cλ1 + r¯ (−∞) = −¯r (+∞) + r¯ (−∞) < 0,

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which implies μ1 < λ1 due to (4.2). Thus, there exist ρ ∗ > 0 and (t0 , z0 ) ∈ R2 such that ∗

v ρ (t, z) ≥ V (t, z)

∗

and v ρ (t0 , z0 ) = V (t0 , z0 ).

∗

Define w := v ρ − V . Then we have w(t, z) ≥ 0, ∀(t, z) ∈ R2 , and w(t0 , z0 ) = 0. Since ⎛ ⎞  ∗ ∗ ∗ ∗ ∗ ρ vt − d ⎝ J (y)v ρ (t, z − y)dy − v ρ ⎠ − cvzρ − v ρ r(t, +∞) = 0, R

we see from (3.1) that ⎛



wt − d ⎝

⎞ J (y)w(t, z − y)dy − w ⎠ − cwz − wr(t, +∞)

R

= −V (r(t, z) − V ) + V r(t, +∞) ≥ 0.

(4.5)

It then follows from (4.5) and min(t,z)∈R2 w(t, z) = w(t0 , z0 ) = 0 that ⎛

⎞

0 ≤wt (t0 , z0 ) − d ⎝

J (y)w(t0 , z0 − y)dy − w(t0 , z0 )⎠



R

− cwz (t0 , z0 ) − w(t0 , z0 )r(t0 , +∞)  = − d J (y)w(t0 , z0 − y)dy ≤ 0. R



This implies that R J (y)w(t0 , z0 − y)dy = 0. In view of (J1), there exists y0 > 0 such that J (y0 ) = 0. By the continuity of J , there exists δ0 > 0 such that J (y) = 0, ∀y ∈ [y0 − δ0 , y0 + δ0 ]. Since J (y) = J (−y), it follows that   J (y)w(t0 , z0 − y)dy = J (y)w(t0 , z0 + y)dy = 0. R

R

Thus, w(t0 , z0 ± y) = 0, ∀y ∈ [y0 − δ0 , y0 + δ0 ], and hence, w(t0 , y) = 0, ∀y ∈ z0 + [−y0 − δ0 , −y0 + δ0 ] ∪ [y0 − δ0 , y0 + δ0 ]. Let s = z0 + y0 + δ0 and observe that w(t0 , s) = 0. Then we can argue as above to obtain w(t0 , y) = 0, ∀y ∈ s + [−y0 − δ0 , −y0 + δ0 ] ∪ [y0 − δ0 , y0 + δ0 ]. In particular, w(t0 , y) = 0, ∀y ∈ z0 + [0, 2δ0 ]. Repeating the argument with s = z0 + y0 − δ0 , we have w(t0 , y) = 0, ∀y ∈ z0 + [−2δ0 , 0]. It then follows that w(t0 , y) = 0, ∀y ∈ z0 + [−2δ0 , 2δ0 ]. ∗ Working inductively, we see that w(t0 , z) ≡ 0, ∀z ∈ R, and hence, v ρ (t0 , z) ≡ V (t0 , z), ∀z ∈ R, ∗ which contradicts the fact that V (t0 , +∞) = p(t0 ) and v ρ (t0 , +∞) = +∞. 2

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Proof of Theorem 1.3. Let L > 0 be a sufficiently large number such that suppJ ⊆ [−L, L]. Let V1 (t, z) and V2 (t, z) be two wave profiles with the same force speed c > −c∗ satisfying Vi (t, −∞) = 0 and Vi (t, +∞) = p(t), i = 1, 2. For any given  > 0, we define K := {k ≥ 1 : kV1 (t, z) ≥ V2 (t, z) − q(t, z), ∀(t, z) ∈ R2 }, where q(t, z) is a positive function satisfying q(t, z) = q(t + ω, z),

q(t, ±∞) > σ for some σ > 0

and will be specified later. It is easy to see that V2 (t, z) − q(t, z) p(t) − q(t, +∞) = z→+∞ V1 (t, z) p(t) lim

and

V2 (t, z) − q(t, z) = −∞, z→−∞ V1 (t, z) lim

uniformly in t ∈ R. Thus, there exists some positive constant L such that V2 (t,z)−q(t,z) ≤ L for V1 (t,z) 2 (t, z) ∈ R , and hence, K = ∅. Set k := inf K . It is clear that k V1 (t, z) ≥ V2 (t, z) −q(t, z) for (t, z) ∈ R2 . For any 0 < 1 < 2 , one can see that k1 V1 (t, z) ≥ V2 (t, z) − 1 q(t, z) > V2 (t, z) − 2 q(t, z) for any (t, z) ∈ R2 . Thus, k1 ≥ k2 , which means k is nonincreasing in  > 0. Define k ∗ = lim→0+ k . It is easy to see that k ∗ ∈ [1, +∞]. If k = 1 for all  > 0, then k ∗ = 1. If there exists a 0 > 0 such that k0 > 1, then by the monotonicity of k in , we obtain that k ≥ k0 > 1 for  ∈ (0, 0 ], and hence k ∗ > 1. We shall show that this case does not hold. For any given  ∈ (0, 0 ], we define w (t, z) := k V1 (t, z) − V2 (t, z) + q(t, z),

(t, z) ∈ R2 .

Then w (t, z) ≥ 0 for (t, z) ∈ R2 . It is easy to see that w (t, z) w (t, z) (k − 1)p(t) + q(t, +∞) = +∞ and lim = > 0, z→−∞ V1 (t, z) z→+∞ V1 (t, z) p(t) lim

since k > 1. Then by the definition of k , we can find (t , z ) such that w (t , z ) = 0 and w (t, z) ≡ 0 in any neighborhood of (t , z ).

(4.6)

Otherwise, there exists  such that w (t, z) > 0 for any (t, z) ∈ R2 . Since k > 1, by choosing  (t,z) η > 0 sufficiently small, we can get k − η > 1 and w V1 (t,z) ≥ η, and hence, (k − η)V1 (t, z) ≥ V2 (t, z) − q(t) for (t, z) ∈ R2 , which leads a contradiction to the definition of k . Since w is periodic in t, we may assume that t ∈ [0, ω]. There are three different possibilities for {z }: (i) {z } is bounded; (ii) z → +∞ (up to subsequence) and (iii) z → −∞ (up to subsequence). We first consider the case (i). In this case, there exists a sequence {n }∞ n=1 ⊂ (0, 0 ] with limn→∞ n = 0 such that limn→∞ (tn , zn ) = (t ∗ , z∗ ). Then we have V2 (tn , zn ) − n p(tn , zn ) V2 (t ∗ , z∗ ) = < +∞. n→∞ V1 (tn , zn ) V1 (t ∗ , z∗ )

k ∗ = lim kn = lim n→∞

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Hence, we have k ∗ ∈ (1, +∞). Define w(t, z) = k ∗ V1 (t, z) − V2 (t, z) for (t, z) ∈ R2 . It is easy to see that w(t, −∞) = 0 and w(t, +∞) = (k ∗ − 1)p(t) > 0. Since w(t, z) = limn→∞ wn (t, z), we have w(t, z) ≥ 0 for (t, z) ∈ R2 and w(t ∗ , z∗ ) = 0. In addition, w(t, z) satisfies wt =k ∗ (V1 )t − (V2 )t ⎛ ⎞  =d ⎝ J (y)w(t, z − y)dy − w ⎠ + cwz + k ∗ V1 (r(t, z) − V1 ) − V2 (r(t, z) − V2 ) R

⎛ =d ⎝



⎞ J (y)w(t, z − y)dy − w ⎠ + cwz + wr(t, z) − k ∗ V12 + V22

R

⎛ ≥d ⎝



⎞ J (y)w(t, z − y)dy − w ⎠ + cwz + w[r(t, z) − (k ∗ V1 + V2 )].

R

By an argument similar to that in the proof of Theorem 1.2, we can show that w(t ∗ , z) ≡ 0, ∀z ∈ R, which contradicts the fact that w(t ∗ , +∞) = (k ∗ − 1)p(t ∗ ) > 0. For the case (ii), there exists δ > 0 such that when (t, z) ∈  := [0, ω] × (z − δ , z + δ ), it follows (w )t =k (V1 )t − (V2 )t + qt ⎛ ⎞  =d ⎝ J (y)w (t, z − y)dy − w ⎠ + c(w )z + k V1 (r(t, z) − V1 ) − V2 (r(t, z) − V2 ) R

⎛

+  ⎝qt − d



⎞ J (y)q(t, z − y)dy + dq − cqz ⎠

R

⎞ ⎛  ≥d ⎝ J (y)w (t, z − y)dy − w ⎠ + c(w )z + [r(t, z) − (k V1 + V2 )](k V1 − V2 ) R

⎛

+  ⎝qt − d



⎞ J (y)q(t, z − y)dy + dq − cqz ⎠

R

⎞ ⎛  ≥d ⎝ J (y)w (t, z − y)dy − w ⎠ + c(w )z + [r(t, z) − (k V1 + V2 )]w , R

provided that  qt − d

J (y)q(t, z − y)dy + dq − cqz − [r(t, z) − (k V1 + V2 )]q ≥ 0. R

(4.7)

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Since w (t, z) ≥ 0, ∀(t, z) ∈ R2 and w (t , z ) = 0 due to (4.6), we can easily obtain 

L J (y)w (t , z − y)dy =

J (y)w (t , z − y)dy = 0.

(4.8)

−L

R

If k ∗ = +∞, then by the definition of w , we have lim→0+ w (t, z) = +∞ for any (t, z) ∈ R2 . Taking the limit  → 0+ to (4.8), we get a contradiction. Thus, we assume k ∗ ∈ (1, +∞). Denote w(t, z) := lim→0+ w (t, z) = k ∗ V1 (t, z) − V2 (t, z). Since t ∈ [0, ω], there exists a sequence {n } such that n → 0+ and tn → t˜ as n → ∞. Meanwhile, zn → +∞ as n → ∞. It then follows from (4.8) that  0 = lim

L J (y)wn (tn , zn − y)dy = lim

n→∞

n→∞ −L

R

J (y)wn (tn , zn − y)dy = w(t˜, +∞),

which contradicts w(t˜, +∞) = (k ∗ − 1)p(t˜) > 0. For the case (iii), we also obtain ⎛



(w )t ≥ d ⎝

⎞ J (y)w (t, z − y)dy − w ⎠ + c(w )z + [r(t, z) − (k V1 + V2 )]w

R

for (t, z) ∈  . Similar to the case (ii), we can derive a contradiction. Therefore, we have k ∗ = 1. It then follows that V1 (t, z) ≥ V2 (t, z) for (t, z) ∈ R2 . By interchanging the role of V1 and V2 , we also have V2 (t, z) ≥ V1 (t, z) for (t, z) ∈ R2 . Thus, we obtain V1 ≡ V2 , which completes the proof of this theorem. It remains to construct q(t, z) such that (4.7) holds for t ∈ R and all sufficiently large |z|. By limz→−∞ Vi (t, z) = 0 and limz→+∞ Vi (t, z) = p(t), i = 1, 2, we see lim [r(t, z) − (k V1 (t, z) + V2 (t, z))] = r(t, −∞)

(4.9)

lim [r(t, z) − (k V1 (t, z) + V2 (t, z))] = r(t, +∞) − (k + 1)p(t).

(4.10)

z→−∞

and z→+∞

In view of (4.9) and the fact that r(t, −∞) < 0 in (H1), there exists z1 < −L such that r(t, z) − (k V1 (t, z) + V2 (t, z)) ≤ σ1 (t),

z ≤ z1 ,

where 1 σ1 (t) := r(t, −∞) − ω

ω r(t, −∞)dt. 0

(4.11)

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Similarly, by (4.10), there exists z2 > L such that r(t, z) − (k V1 (t, z) + V2 (t, z)) ≤ σ2 (t),

z ≥ z2 ,

where

σ2 (t) := r(t, +∞) − (k + 1)p(t) + k

1 ω

ω p(t)dt. 0

It is easy to see that

ω 0

σ1 (t)dt =

ω 0

σ2 (t)dt = 0, since

ω

ω [r(t, +∞) − p(t)]dt =

0

p (t) dt = 0. p(t)

0 t

We choose q ∈ C 1,1 (R × R) such that q(t, z) is ω-periodic in t , and satisfies q(t, z) = e 0 σ1 (s)ds t t for z < z1 , q(t, z) = e 0 σ2 (s)ds for z > z2 , 0 ≤ q(t, z) ≤ e 0 σ1 (s)ds for z1 ≤ z < 0, and 0 ≤ t q(t, z) ≤ e 0 σ2 (s)ds for 0 < z ≤ z2 . We can easily verify that (4.7) holds for all t ∈ [0, ω] and z < z1 or z > z2 . In fact, for z < z1 , we obtain  qt − d

J (y)q(t, z − y)dy + dq − cqz − [r(t, z) − (k V1 + V2 )]q R

L = σ1 (t)q(t, z) + d

J (y)(q(t, z) − q(t, z − y))dy − [r(t, z) − (k V1 + V2 )]q

−L

0 =d

J (y)(q(t, z) − q(t, z − y))dy + [σ1 (t) − r(t, z) + (k V1 + V2 )]q

−L

≥ 0, due to (4.11) and z − y < z1 − y ≤ z1 + L < 0 for y ∈ [−L, 0]. The verification for z > z2 is similar, so we omit it here. 2 5. The global exponential stability In view of Theorems 1.1 and 1.3, it follows that for any c > −c∗ , system (1.1) admits a unique periodic forced wave V (t, x − ct) connecting 0 to p(t). In this section, we will show that the periodic forced wave V (t, x − ct) is globally exponentially stable. Let u(t, x; φ) be the unique solution of (1.1) with the initial value u(0, ·; φ) = φ ∈ X+ \ {0} and lim infx→+∞ φ(x) > 0.

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We first consider the following ω-periodic nonlocal dispersal equation ut = d(J ∗ u − u) + ug(t, u),

(5.1)

where g(t, u) is nonincreasing in u and g(t + ω, u) = g(t, u). Lemma 5.1. If of (5.1).

ω 0

g(t, 0)dt < 0, then u = 0 is the only nonnegative and bounded entire solution

Proof. Let T (t) be the semigroup of the nonlocal dispersal equation ut = d(J ∗ u − u). It then follows that the linear nonlocal dispersal equation ut = d(J ∗ u − u) + g(t, 0)u admits an evolution operator (t, s), t ≥ s, which is given by (t, s)ϕ = e

t s

g(τ,0)dτ

T (t − s)ϕ, ∀t ≥ s.

t ω Since 0 g(t, 0)dt < 0, it easily follows that there exist k > 0 and a > 0 such that e s g(τ,0)dτ ≤ ¯ x) be a nonnegative and bounded entire solution of (5.1). Since ke−a(t−s) for any t ≥ s. Let u(t, g(t, u) ¯ ≤ g(t, 0), we obtain

u¯ t ≤ d(J ∗ u¯ − u) ¯ + g(t, 0)u, ¯ ∀(t, x) ∈ R2 . By the comparison principle, we have 0 ≤ u(t, ¯ ·) ≤ (t, s)u(s, ¯ ·) = e

t s

g(τ,0)dτ

T (t − s)u(s, ¯ ·), ∀t ≥ s.

(5.2)

Since u(t, ¯ x) is bounded for (t, x) ∈ R2 , it is easy to see that T (t − s)u(t, ¯ ·) is also bounded. Letting s → −∞ in (5.2), we then obtain 0 ≤ u(t, ¯ x) ≤ 0, ∀(t, x) ∈ R2 . Thus, u(t, ¯ x) ≡ 0 for (t, x) ∈ R2 .

2

Next we show that the solutions of (1.1) with “large” initial data converge to the time-periodic forced wave uniformly. The proof of Lemma 3.4 implies that for any M ≥ p(0), the solution u(t, x; M) converges to V (t, x − ct) connecting 0 to p(t) locally uniformly in the variable z = x − ct as t → +∞. The following lemma shows a stronger convergence result. Lemma 5.2. Assume that (J1)-(J2) and (H1)-(H2) hold. For each fixed c > −c∗ , let V (t, x − ct) be the unique periodic forced wave of (1.1) connecting 0 to p(t). Then for any M ≥ p(0), we have limt→+∞ |u(t, x; M) − V (t, x − ct)| = 0 uniformly in x ∈ R.

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Proof. Since V (t, x) is non-decreasing in x ∈ R, we have V (0, x) ≤ V (0, +∞) = p(0) ≤ M for any x ∈ R. By the comparison principle, we have V (t, x − ct) ≤ u(t, x; M),

(t, x) ∈ [0, +∞) × R.

(5.3)

Let w(t) be the solution of (2.4) with initial value w(0) = M, i.e., w (t) = w(r(t, +∞) − w). Since r(t, z) ≤ r(t, +∞) for z ∈ R, the comparison principle yields that u(t, x; M) ≤ w(t). Note that p(t) is globally asymptotically stable for (2.4) with positive initial value. Then we have limt→+∞ |w(t) − p(t)| = 0. Hence, lim sup[u(t, x; M) − p(t)] ≤ lim sup |w(t) − p(t)| = 0. t→+∞

(5.4)

t→+∞

In view of V (t, +∞) = p(t), combining (5.3) and (5.4), and the established local convergence of u(t, x; M) to V (t, x − ct) in the variable x − ct, we see that for some x0  1, lim sup[u(t, x; M) − V (t, x − ct)] = 0

(5.5)

t→+∞

uniformly in x − ct ≥ x0 . It remains to prove that (5.5) holds uniformly in x − ct ≤ x0 . Since the local uniform convergence of u(t, x; M) to V (t, x − ct) in the variable z = x − ct as t → +∞, it suffices to show that (5.5) holds uniformly when z = x − ct is negatively sufficiently large. If this is not true, by considering the fact that V (t, −∞) = 0 uniformly for t ∈ R, we can assume that there exists m0 > 0 and tn , xn such that xn − ctn → −∞ and u(tn , xn ; M) = m0 . Let [tn /ω] be the greatest integer less than or equal to tn /ω. Then tn − [tn /ω]ω ∈ [0, ω). Without loss of generality, we assume that limn→∞ (tn − [tn /ω]ω) = t ∗ . Define wn (t, x) := u(t + tn , x + xn ; M). It is easy to verify that wn (t, x) satisfies (wn )t (t, x) =d[(J ∗ wn )(t, x) − wn (t, x)] + wn (t, x)[r(t + tn , x − ct + xn − ctn ) − wn (t, x)].

(5.6)

Now we give a priori estimates on wn (t, x). By Lemma 2.4, u(t, x; M) is bounded. Hence, there exists K > 0 such that 0 ≤ u(t, x; M) ≤ K for (t, x) ∈ [0, +∞) × R, which implies 0 ≤ wn (t, x) ≤ K for (t, x) ∈ [−tn , +∞) × R. It then easily follows that  |(wn )t | ≤ 2dK + K

 max r(t, +∞) + K =: C1 .

t∈[0,ω]

For any η ∈ R, let w˜ n (t, x) = wn (t, x + η) − wn (t, x). Without loss of generality, we assume that w˜ n (t, x) ≥ 0. It then follows from (5.6) that  (w˜ n )t ≤d |J (x + η − y) − J (x − y)|wn (t, y)dy − d w˜ n R

+ r(t + tn , x + η − ct + xn − ctn )w˜ n + wn (x)[r(t + tn , x + η − ct + xn − ctn ) − r(t + tn , x − ct + xn − ctn )]. (5.7)

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Since J ∈ L1 (R) by (J1) and (J2), there exists L > 0 such that 

1  |J (x + η − y) − J (x − y)|dy ≤ |η|

|J (x − y + θ η)|dθ dy ≤ L|η|.

0 R

R

 Hence, for any  > 0, there exists δ1 = L > 0 such that R |J (x + η − y) − J (x − y)|dy ≤  provided that |η| ≤ δ1 . Since r(t, x) is continuous in (t, x) ∈ R2 , there exists δ2 > 0 such that |r(t + tn , x + η − ct + xn − ctn ) − r(t + tn , x − ct + xn − ctn )| ≤  provided that |η| ≤ δ2 . Choose δ = min{δ1 , δ2 }. Then for |η| ≤ δ, we obtain from (5.7) that   (w˜ n )t ≤ (d + 1)K − d − max r(t, +∞) w˜ n . t∈[0,ω]

In view of w˜ n (−tn , x) = 0 for x ∈ R, by using the argument similar to that in the proof of [25, := C2 , i.e., |wn (x + η, t) − wn (x, t)| ≤ C2  Lemma 3.9], we see that w˜ n ≤ d−max(d+1)K t∈[0,ω] r(t,+∞) provided |η| ≤ δ. Furthermore, we get |(wn )t (t, x + η) − (wn )t (t, x)| ≤ C3  for some positive constant C3 , provided |η| ≤ δ. Similarly, we obtain |(wn )t (t + η, x) − (wn )t (t, x)| ≤ C4  for some positive constant C4 , as η → 0. By the above a priori estimates, wn (t, x) converges to some w(t, x) ≥ 0 locally uniformly in (t, x) as n → ∞. Thus, by (5.6) and considering r(t + tn , ·) = r(t + tn − [tn /ω]ω, ·), we obtain that w(t, x) solves the following equation wt (t, x) =d[(J ∗ w)(t, x) − w(t, x)] + w(t, x)[r(t + t ∗ , −∞) − w(t, x)], (t, x) ∈ R2 . (5.8) ω Since 0 r(t + t ∗ , −∞)dt < 0, it follows from Lemma 5.1 that w = 0 is the only nonnegative and bounded entire solution of (5.8). As such, we have w(t, x) ≡ 0, which contradicts w(0, 0) = limn→∞ wn (0, 0) = m0 > 0. Consequently, (5.5) holds uniformly in x ∈ R. 2 Lemma 5.3. Assume that (J1)-(J2) and (H1)-(H2) hold. When c ∈ (−c∗ , c∗ ), we have limt→+∞ supx≤μt |u(t, x; φ) − V (t, x − ct)| = 0 for any μ ∈ (c, c∗ ). Proof. Assume that σ ∈ (0, (c∗ − c)) is fixed. By definition, it suffices to show that for any  > 0, there exists T0 > 0 such that sup

x≤(c∗ −σ )t

|u(t, x; φ) − V (t, x − ct)| < ,

∀t ≥ T0 .

(5.9)

Since V (t, +∞) = p(t) uniformly in t ∈ R, there exists x0 > 0 such that V (t, x) > p(t) − /2 for any t ∈ R and x ≥ x0 , and c0 > c∗ − σ/2, where c0 is the spreading speed of solutions of wt (t, x) = d[(J ∗ w)(t, x) − w(t, x)] + w(t, x)[r(t, x0 ) − w(t, x)],

(5.10)

with initial value w(0, x) = φ(x). By [46, Theorem 5.2.1], the ordinary differential equation

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w (t) = w(t)[r(t, x0 ) − w(t)] admits a unique positive periodic solution p0 (t) with p0 (t) > p(t) − /4 for t ∈ R. We prove (5.9) by splitting the region x ≤ (c∗ − σ )t into two regions: (i) x ≤ x0 + ct; (ii) x ∈ [x0 + ct, (c∗ − σ )t]. In case (i), by using the argument similar to that in the proof of Lemma 5.2, we obtain u(t, x; φ) converges to V (t, x − ct) uniformly in the variable x − ct ≤ x0 . Hence, T0 can be found. In case (ii), we have x − ct ≥ x0 , and hence, V (t, x − ct) ≥ V (t, x0 ) ≥ p(t) − /2 for all t ≥ 0. It then suffices to find T0 > 0 such that sup

x∈[x0 +ct,(c∗ −σ )t]

|u(t, x; φ) − p(t)| < /2,

∀t ≥ T0 .

(5.11)

It is clear that when x ≥ x0 + ct, r(t, x − ct) ≥ r(t, x0 ). Let w(t, x; φ) be the solution of (5.10) with w(0, x) = φ(x). It is easy to see that u(t, x0 + ct) and w(t, x0 + ct) solve the same equation dv(t, x) = d[(J ∗ v)(t, x) − v(t, x)] + cvx (t, x) + v(t, x)[r(t, x0 ) − v(t, x)], dt

(5.12)

where x = x0 + ct. Moreover, u(0, x0 ) = φ(x0 ) = w(0, x0 ). Hence, by the uniqueness of solutions of (5.12), we have u(t, x0 + ct) = w(t, x0 + ct), ∀t ≥ 0. It then follows from the comparison principle that u(t, x; φ) ≥ w(t, x; φ), ∀x ≥ x0 + ct. Since c0 is the spreading speed of w(t, x; φ) of (5.10), we have lim

sup |w(t, x; φ) − p0 (t)| = 0,

t→+∞ |x|≤μt

μ ∈ (0, c0 ).

Take μ := c∗ − σ . Then we have μ < c0 . Thus, there exists T1 > 0 such that u(t, x; φ) ≥ w(t, x; φ) ≥ p0 (t) − /4 ≥ p(t) − /2,

∀t ≥ T1 , x ∈ [x0 + ct, (c∗ − σ )t]. (5.13)

On the other hand, we choose ρ > 1 such that ρ supx∈R φ(x) > p(0). Then by Lemma 5.2, there exists T2 > 0 such that u(t, x; φ) ≤ u(t, x; ρ sup φ(x)) ≤ p(t) + /2,

∀t ≥ T2 , x ∈ R.

(5.14)

x∈R

Let T0 = max{T1 , T2 }. Combining (5.13) and (5.14), we see that (5.11) holds.

2

The following result shows that the periodic forced wave V (t, x −ct) with speed c ∈ (−c∗ , c∗ ) is globally asymptotically stable. Lemma 5.4. Assume that (J1)-(J2) and (H1)-(H2) hold. Then for each fixed c ∈ (−c∗ , c∗ ), there holds limt→+∞ supx∈R |u(t, x) − V (t, x − ct)| = 0. Proof. By Lemma 5.3, we only need to prove that lim sup |u(t, x) − V (t, x − ct)| = 0

t→+∞ x≥μt

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for some fixed positive number μ ∈ (c, c∗ ). Since V (t, z) is periodic in t ∈ R and nondecreasing in z ∈ R with V (t, +∞) = p(t), we have sup |p(t) − V (t, x − ct)| = p(t) − inf V (t, x − ct) x≥μt

x≥μt

= p(t) − inf V (t, y + (μ − c)t) ≤ p(t) − V (t, (μ − c)t). y≥0

In view of μ − c > 0 and V (t, +∞) = p(t), we obtain lim sup |p(t) − V (t, x − ct)| = 0.

t→+∞ x≥μt

By the triangular inequality, it then suffices to prove lim sup |u(t, x) − p(t)| = 0.

t→+∞ x≥μt

(5.15)

Fix δ ∈ (0, min{μ, μ − c}). Let x0 be some positive number and will be specified later. Then we obtain lim sup sup |u(t, x) − p(t)| ≤ lim sup t→+∞ x≥μt

sup

t→+∞ x≥(μ−δ)t+x0

|u(t, x) − p(t)|

= lim sup sup |u(t, x + (μ − δ)t) − p(t)|. t→+∞ x≥x0

Set w(t, x) := u(t, x + (μ − δ)t). Then it is easy to see that w(t, x) satisfies wt (t, x) =d[(J ∗ w)(t, x) − w(t, x)] + (μ − δ)wx (t, x) + w(t, x)[r(t, x + (μ − δ − c)t) − w(t, x)],

(5.16)

with lim infx→+∞ w(0, x) = lim infx→+∞ u(0, x) > 0. Choose M > supx∈R w(0, x). Then w(t), ˜ the solution of w˜ = w(r(t, ˜ +∞) − w) ˜ with w(0) ˜ = M, is a super-solution of (5.16). By the comparison principle, we have w(t, x) ≤ w(t), ˜ ∀(t, x) ∈ [0, +∞) × R, and hence, lim sup sup [w(t, x) − p(t)] ≤ lim sup[w(t) ˜ − p(t)] = 0. t→+∞ x≥x0

t→+∞

It then remains to show that lim supt→+∞ supx≥x0 [p(t) − w(t, x)] ≤ 0, which implies that (5.15) holds. Thus, we just need to prove that for any  > 0, there exists x0 > 0 and T0 > 0 such that inf w(t, x) ≥ p(t) − ,

x≥x0

∀t ≥ T0 .

(5.17)

For any  > 0, there exists γ > 0 such that r¯ (+∞) − γ > 0 and β(t) > p(t) − /2, where β(t) is the unique positive ω-periodic solution of β (t) = β(t)[r(t, +∞) − β(t) − γ ]. For the above γ and w(0, x), there exists x0 > 0 such that r(t, x0 ) ≥ r(t, +∞) − γ for t ∈ R and w(0, x) ≥ 1 2 lim infx→+∞ w(0, x) for x ≥ x0 . Since μ − δ − c > 0 and r(·, x) is nondecreasing in x, we obtain that r(t, x + (μ − δ − c)t) ≥ r(t, x0 ) ≥ r(t, +∞) − γ for any t ≥ nω and x > x0 − (μ −

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δ − c)nω. Let xn := x0 − (μ − δ − c)nω for any n ≥ 0, and νn and L be some positive numbers that will be specified later. Define  φ n (x) :=

π νn e(μ−δ)(x−xn ) cos 2L (x − xn − L), νn e(μ−δ)L ,

x ∈ [xn , xn + L], x > xn + L.

It is easy to see that φ n (x) is non-decreasing in x ≥ xn , since μ − δ > 0. Consider a family of nonlocal semi-line problems 

wtn = d[J ∗ w n − w n ] + (μ − δ)wxn + w n [r(t, +∞) − γ − w n ], ∀t > 0, x ≥ xn , (5.18) ∀t > 0, x < xn . w n (t, x) = 0,

By the comparison argument similar to that in the proof of [31, Theorem A], we see that for all sufficiently small νn , w n (mω + t, x; φ n ) converges to the unique positive time-periodic solution w n,∗ (t, x) of (5.18), as m → ∞. Moreover, w n,∗ (t, x) is increasing in n, non-decreasing in x, and w n,∗ (t, +∞) = β(t). Letting n → ∞, we obtain that w ∗ (t, x) := limn→∞ w n,∗ (t, x) solves the following equation wt∗ = d[J ∗ w ∗ − w ∗ ] + (μ − δ)wx∗ + w ∗ [r(t, +∞) − γ − w ∗ ], ∀(t, x) ∈ R2 . It is clear that w ∗ (t, x0 ) := limn→∞ w n,∗ (t, x0 ) ≥ w 1,∗ (t, x0 ) > 0. Since μ − δ ∈ (0, c∗ ), we obtain the entire solution w ∗ (t, x) ≡ β(t). For sufficiently small νn , we see w(nω, x) ≥ φ n (x), ∀x ≥ xn . It then follows from the comparison principle that lim inf inf w(nω + t, x) ≥ lim inf inf w n (nω + t, x) ≥ w ∗ (t, x0 ) = β(t) n→∞ x≥x0

n→∞ x≥x0

uniformly for t ∈ [0, ω]. Therefore, (5.17) holds.

2

In order to establish the exponential stability of the periodic forced wave V (t, x − ct) with speed c ∈ (−c∗ , c∗ ), we need to construct a pair of super- and sub-solutions of (1.1). Lemma 5.5. Assume that (J1) and (H1) hold. Let w ± (t, x) := V (t, x − ct) ± ρe−σ (t−ξ0 ) [1 + MV (t, x − ct)] for (t, x) ∈ [0, +∞) × R, where ξ0 ∈ R, M, ρ and σ are appropriate positive real numbers. Then w ± is a pair of super- and sub-solutions of (1.1). Proof. We first verify that w + is a super-solution of (1.1). By (3.1), we obtain wt+ − d[J ∗ w + − w + ] − w + [r(t, x − ct) − w + ]  = ρe−σ (t−ξ0 ) 2V (t, z) + MV 2 (t, z) − r(t, z)

 + (1 + MV (t, z))(−σ + ρe−σ (t−ξ0 ) (1 + MV (t, z))) .

Note that V (t, −∞) = 0, V (t, +∞) = p(t) and 0 < V (t, z) < p(t). By choosing M > 0 large enough, we can get 2V (t, z) + MV 2 (t, z) − r(t, z) > 0 for (t, z) ∈ [0, +∞) × R. Hence, taking ρ and σ > 0 sufficiently small, we obtain

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wt+ − d[J ∗ w + − w + ] − w + [r(t, x − ct) − w + ] ≥ 0, Similarly, we can show that w − is a sub-solution of (1.1).

∀(t, x) ∈ [0, +∞) × R.

2

Now we are in a position to prove the global exponential stability of the periodic forced wave V (t, x − ct) with speed c ∈ (−c∗ , c∗ ). Proof of Theorem 1.4. For the number ρ defined in Lemma 5.5, it follows from Lemma 5.4 that there exists some large T0 > 0 such that |u(T0 , x) − V (T0 , x − cT0 )| < ρ,

∀x ∈ R.

Taking ξ0 = T0 in Lemma 5.5, we see that w ± (t, x) are respectively super- and sub-solutions of (1.1) for all (t, x) ∈ [T0 , +∞) × R. By the comparison principle, we obtain w − (t, x) ≤ u(t, x; φ) ≤ w+ (t, x),

∀(t, x) ∈ [T0 , +∞) × R.

This implies that |u(t, x; φ) − V (t, x − ct)| ≤ ρe

σ T0

! 1 + M max p(t) e−σ t ,

Then we obtain (1.4) by choosing 0 < μ < σ .

t∈[0,ω]

∀(t, x) ∈ [T0 , +∞) × R.

2

6. Spreading properties In this section, we establish the spreading properties for solution u(t, x; φ) of system (1.1) with u(0, ·) = φ(·) ∈ X+ \{0} in the sense of Theorem 1.5, which is a straightforward consequence of Propositions 6.1, 6.2 and 6.3 below. Propositions 6.1 and 6.2 indicate that the species will eventually become extinct in space under certain conditions on the initial functions and the shifting speed of the habitat. Proposition 6.1. The following statements are valid: (i) If there exists a constant X1 such that φ(x) = 0 for all x ≤ X1 , then for c > −c∗ , limt→+∞ supx≤(c−σ )t u(t, x; φ) = 0 for any σ > 0. (ii) If φ(x) = 0 for x outside a bounded interval, then for c > c∗ , limt→+∞ u(t, x; φ) = 0 uniformly in x ∈ R. Proof. Let V (t, x − ct) be the positive periodic forced wave of (1.1) with speed c > −c∗ connecting 0 to p(t). By the monotonicity of V (t, x) in x ∈ R, we have 0 < V (0, x) ≤ V (0, +∞) = p(0). Since φ(x) is bounded for x ∈ R and φ(x) = 0 for all x ≤ X1 , we can choose a larger positive number ρ > 1 such that φ(x) ≤ ρV (0, x) for x ∈ R. Let w(t, x) := ρV (t, x − ct) and denote z = x − ct. Then w(t, x) satisfies

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wt (t, x) =ρ(Vt (t, z) − cVz (t, z)) ⎞ ⎛  =dρ ⎝ J (y)V (t, z − y)dy − V (t, z)⎠ + ρV (t, z)(r(t, z) − V (t, z)) R

≥d(J ∗ w − w)(t, x) + w(t, x)(r(t, x − ct) − w(t, x)), which indicates that w(t, x) is a super-solution of (1.1). Since w(0, x) = ρV (0, x) ≥ φ(x) for x ∈ R, the comparison principle yields u(t, x; φ) ≤ w(t, x) = ρV (t, x − ct) for t ≥ 0 and x ∈ R. For any  > 0, since V (t, −∞) = 0 uniformly for t ∈ R, there exists a large number M > 0 such that V (t, −M) < ρ for t ∈ R. Thus, we get u(t, x; φ) ≤ ρV (t, x − ct) ≤ ρV (t, −M) < , ∀t ≥ 0, x ≤ −M + ct.

(6.1)

It is clear that for any σ > 0, there exist T ∗ > 0 such that (c − σ )t < −M + ct for t ≥ T ∗ . Thus, for t ≥ T ∗ and x ≤ (c − σ )t , we obtain u(t, x; φ) < , which implies the statement (i) of the theorem. For the statement (ii), since φ(x) = 0 for x outside a bounded interval, without loss of generality, we assume that φ(x) = 0 for all x ≤ X1 . Hence, (6.1) holds. Note that c∗ is the spreading speed of solutions of periodic nonlocal dispersal Fisher-KPP equation (2.3) (see Lemma 2.5). We let v(t, x; φ) be the unique solution of (2.3) with initial value v(0, x) = φ(x). Fix c1 ∈ (c∗ , c). Then by Lemma 2.5, limt→+∞ supx≥c1 t v(t, x; φ) = 0. It is easy to see that v(t, x; φ) is a supersolution of (1.1). Hence, by the comparison principle, we get u(t, x; φ) ≤ v(t, x; φ) for t ≥ 0 and x ∈ R. Thus, we have limt→+∞ supx≥c1 t u(t, x; φ) = 0. It follows that there exists T0 > 0 such that u(t, x; φ) ≤ ,

t ≥ T0 , x ≥ c1 t.

(6.2)

M Let T = max{T0 , c−c }. Then one has −M + ct ≥ c1 t for t ≥ T . Thus, by (6.1) and (6.2), we 1 obtain that u(t, x; φ) <  for t ≥ T and x ∈ R. Therefore, limt→+∞ u(t, x; φ) = 0 uniformly in x ∈ R. 2

Proposition 6.2. Assume that c > c∗ . If there exists a constant X2 such that φ(x) = 0 for all x ≥ X2 , then limt→+∞ supx≥ct u(t, x; φ) = 0. Proof. For any σ > 0, let λσ be the smaller positive root of the following equation ⎛



d⎝

⎞

 σ J (y)eλy dy − 1⎠ − c∗ + λ + r¯ (+∞) = 0. 2

R ∗

Define u(t, ¯ x) = Mϕ(t)e−λσ (x−(c +σ/2)t) , where M > 0 is a constant that will be specified later t and ϕ(t) = e 0 (r(s,+∞)−¯r (+∞))ds . It can be verified that u(t, ¯ x) solves ut = d(J ∗ u − u) + r(t, +∞)u,

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which implies that u¯ is a super-solution of (1.1). Since φ(x) is bounded for x ∈ R and φ(x) = 0 for all x ≥ X2 , we can choose M large enough such that φ(x) ≤ u(0, ¯ x) = Me−λσ x for x ∈ R. By the comparison principle, we then obtain 0 ≤ u(t, x; φ) ≤ u(t, ¯ x) ≤ Ce−λσ (x−(c

∗ +σ )t)

e−

λσ σ 2

t

∀t ≥ 0, x ∈ R,

,

where C := M maxt∈[0,ω] ϕ(t). It then follows that 0 ≤ u(t, x; φ) ≤ Ce−

λσ σ 2

t

,

∀t ≥ 0, x ≥ (c∗ + σ )t.

Hence, limt→+∞ supx≥(c∗ +σ )t u(t, x; φ) = 0, which implies that limt→+∞ supx≥ct u(t, x; φ) = 0 for any c > c∗ . 2 The following result shows that if the habitat shifts towards right with a moderate speed c ∈ (0, c∗ ), then the species will persist in space. Proposition 6.3. Assume c ∈ (0, c∗ ). Then for each σ ∈ (0, c lim

sup

t→+∞ (c+σ )t
∗ −c

2

), there holds

|u(t, x; φ) − p(t)| = 0.

Proof. By definition, it suffices to prove that for any  > 0, there exists T0 > 0 such that sup

x∈[(c+σ )t,(c∗ −σ )t]

|u(t, x; φ) − p(t)| < ,

∀t ≥ T0 .

(6.3)

Since x > (c + σ )t , we have x − ct > σ t. Thus, there exists T˜ > 0 such that x − ct > σ t > x0 for t > T˜ , where x0 is defined in the proof of Lemma 5.3. By using the argument similar to that for (5.11) in Lemma 5.3, we then obtain (6.3). 2 Remark 6.4. Our method also applies to the following general nonlocal dispersal evolution equation ut = d[J ∗ u − u] + ug(t, x − ct, u), provided that the function g satisfies the following assumptions: (G1) g ∈ C 1 (R × R × R+ , R) and g(t, z, u) is ω-periodic in t for some ω > 0; (G2) g(t, z, u) is non-decreasing in z ∈ R and non-increasing in u ∈ R+ , g(t, +∞, u) exists and is  ωstrictly decreasing in u ∈ R+ ; (G3) 0 g(t, +∞, 0)dt > 0 and  ω there exists M > 0 such that g(t, +∞, M) ≤ 0; (G4) g(t, −∞, u) exists and 0 g(t, −∞, 0)dt < 0.

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Acknowledgments We are very grateful to the anonymous referee for careful reading and helpful suggestions which led to an improvement of our original manuscript. G.-B. Zhang was supported by NSF of China (11861056), NSF of Gansu Province (18JR3RA093) and a postdoctoral fellowship at the Memorial University of Newfoundland, and X.-Q. Zhao was supported in part by the NSERC of Canada. References [1] M. Alfaro, H. Berestycki, G. Raoul, The effect of climate shift on a species submitted to dispersion, evolution, growth and nonlocal competition, SIAM J. Math. Anal. 49 (2017) 562–596. [2] N.D. Alikakos, P.W. Bates, X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Am. Math. Soc. 351 (1999) 2777–2805. [3] X.-X. Bao, Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable LotkaVolterra competition system, J. Differ. Equ. 255 (2013) 2402–2435. [4] W.-J. Bo, G. Lin, S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst. 38 (2018) 4329–4351. [5] W.-J. Bo, G. Lin, Y. Qi, Propagation dynamics of a time periodic diffusion equation with degenerate nonlinearity, Nonlinear Anal., Real World Appl. 45 (2018) 376–400. [6] H. Berestycki, O. Diekmann, C. Nagelkerke, P. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol. 71 (2009) 399–429. [7] H. Berestycki, L. Rossi, Reaction-diffusion equations for population dynamics with forced speed, I - the case of the whole space, Discrete Contin. Dyn. Syst. 21 (2008) 41–67. [8] H. Berestycki, L. Rossi, Reaction-diffusion equations for population dynamics with forced speed, II - cylindrical type domains, Discrete Contin. Dyn. Syst. 25 (2009) 19–61. [9] H. Berestycki, J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differ. Equ. 264 (2018) 2157–2183. [10] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. 185 (2006) 461–485. [11] J. Coville, Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Differ. Equ. 2007 (2007) 1–23. [12] J. Coville, J. Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differ. Equ. 244 (2008) 3080–3118. [13] Y. Du, L. Wei, L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dyn. Differ. Equ. 30 (2018) 1389–1426. [14] J. Fang, X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal. 46 (2014) 3678–3704. [15] J. Fang, Y. Lou, J. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math. 76 (2016) 1633–1657. [16] J. Fang, R. Peng, X.-Q. Zhao, Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment, preprint, 2018. [17] P. Fife, Some nonclassical trends in parabolic-like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, 2003, pp. 153–191. [18] C. Hu, B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differ. Equ. 259 (2015) 1967–1989. [19] H. Hu, X. Zou, Existence of an extinction wave in the Fisher equation with a shifting habitat, Proc. Am. Math. Soc. 145 (2017) 4763–4771. [20] L.I. Ignat, J.D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal. 251 (2007) 399–437. [21] Y. Jin, X.-Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity 22 (2009) 1167–1189. [22] C.T. Lee, et al., Non-local concepts and models in biology, J. Theor. Biol. 210 (2001) 201–219. [23] C. Lei, Y. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst., Ser. B 22 (2017) 895–911.

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[24] B. Li, S. Bewick, J. Shang, W.F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math. 5 (2014) 1397–1417. [25] W.-T. Li, L. Zhang, G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst. 35 (2015) 1531–1560. [26] W.-T. Li, J.-B. Wang, X.-Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci. 28 (2018) 1189–1219. [27] X. Liang, Y. Yi, X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution system, J. Differ. Equ. 231 (2006) 57–77. [28] X. Liang, X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monostable semiflows with application, Commun. Pure Appl. Math. 60 (2007) 1–40. [29] F. Lutscher, E. Pachepsky, M.A. Lewis, The effect of dispersal patterns on stream populations, SIAM J. Appl. Math. 65 (2005) 1305–1327. [30] J. Murray, Mathematical Biology, 3rd ed., Springer, Berlin-Heidelberg, New York, 1993. [31] T. Nguyen, N. Rawal, Coexistence and extinction in time-periodic Volterra-Lotka type systems with nonlocal dispersal, Discrete Contin. Dyn. Syst., Ser. B 23 (2018) 3799–3816. [32] N. Rodriguez, On an integro-differential model for pest control in a heterogeneous environment, J. Math. Biol. 70 (2015) 1177–1206. [33] W. Shen, Z. Shen, Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity, Discrete Contin. Dyn. Syst. 37 (2017) 1013–1037. [34] W. Shen, Z. Shen, Transition fronts in time heterogeneous and random media of ignition type, J. Differ. Equ. 262 (2017) 454–485. [35] W. Shen, Z. Shen, Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, Trans. Am. Math. Soc. 369 (2017) 2573–2613. [36] Z. Shen, H.-H. Vo, Nonlocal dispersal equations in time-periodic media: principal spectral theory, limiting properties and long-time dynamics, J. Differ. Equ. 267 (2019) 1423–1466. [37] H.-H. Vo, Persistence versus extinction under a climate change in mixed environments, J. Differ. Equ. 259 (2015) 4947–4988. [38] J.-B. Wang, X.-Q. Zhao, Uniqueness and global stability of forced waves in a shifting environment, Proc. Am. Math. Soc. 147 (2019) 1467–1481. [39] Z.-C. Wang, L. Zhang, X.-Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dyn. Differ. Equ. 30 (2018) 379–403. [40] P. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differ. Equ. 229 (2006) 270–296. [41] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., Kyoto Univ. 45 (2009) 925–953. [42] X. Yu, X.-Q. Zhao, A periodic reaction-advection-diffusion model for a stream population, J. Differ. Equ. 258 (2015) 3037–3062. [43] G.-B. Zhang, W.-T. Li, Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equ. 252 (2012) 5096–5124. [44] G. Zhao, S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl. 95 (2011) 627–671. [45] G. Zhao, S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differ. Equ. 257 (2014) 1078–1147. [46] X.-Q. Zhao, Dynamical Systems in Population Biology, second edition, Springer-Verlag, New York, 2017.