Dynamics of a nonlocal multi-type SIS epidemic model with seasonality

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Dynamics of a nonlocal multi-type SIS epidemic model with seasonality Shi-Liang Wu ∗,1 , Panxiao Li, Huarong Cao School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 28 October 2017 Available online xxxx Submitted by J. Shi Keywords: Periodic and nonlocal epidemic model Spreading speed Time-periodic traveling wave front

a b s t r a c t In this paper, we propose and study a multi-type SIS nonlocal epidemic model with seasonality. We first establish the existence of the spreading speed c∗ and the non-existence of periodic traveling waves with wave speed c < c∗ . Then, we prove the existence and asymptotic behavior of the periodic traveling fronts with speed c > c∗ . Finally, the existence of the critical periodic traveling wave fronts is established by using a limiting argument. To overcome the difficulty of lack of compactness of solution maps of the nonlocal system with respect to compact open topology, we show that the solution sequence is pre-compact in Lloc (R2 , Rm ). © 2018 Elsevier Inc. All rights reserved.

1. Introduction To model the spatial spread of a deterministic epidemic in multi-types of population, Rass and Radcliffe [12] proposed and studied a multi-type SIS epidemic model. Consider m populations, each consisting of susceptible and infectious individuals. Let the numbers of susceptible and infectious individuals in the ith population at location x and time t be Si (x, t) and Ii (x, t), respectively. Suppose that the infection rate of a type i susceptible by a type k infectious individual is μi,k ≥ 0 and the corresponding contact distribution is pi,k (·). Assume that the infectious individuals in population i return to the susceptible state at rate νi . Then the model is described as follows: ⎧  m ∂Si (x,t) ⎪ = −Si (x, t) k=1 μi,k R Ik (x − y, t)pi,k (y)dy + νi Ii (x, t), ⎪ ∂t ⎪ ⎨  m ∂Ii (x,t) (1.1) = Si (x, t) k=1 μi,k R Ik (x − y, t)pi,k (y)dy − νi Ii (x, t), ∂t ⎪ ⎪ ⎪ ⎩ i = 1, · · · , m, x ∈ R, t > 0. * Corresponding author. E-mail address: [email protected] (S.-L. Wu). Partially supported by the NSF of China (11671315), the NSF of Shaanxi Province of China (2017JM1003) and the Science and Technology Activities Funding of Shaanxi Province of China. 1

https://doi.org/10.1016/j.jmaa.2018.03.011 0022-247X/© 2018 Elsevier Inc. All rights reserved.

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Assume further that the population size of the ith population is σi , i.e. σi = Si (x, t) + Ii (x, t). Then, system (1.1) can be re-written as

∂Ii (x, t)  = σi − Ii (x, t) μi,k ∂t m

k=1

 Ik (x − y, t)pi,k (y)dy − νi Ii (x, t), R

i = 1, · · · , m, x ∈ R, t > 0.

(1.2)

Denote yi (x, t) = Ii (x, t)/σi . Then system (1.2) reduces to

∂yi (x, t)  = 1 − yi (x, t) σk μi,k ∂t m

k=1

 yk (x − z, t)pi,k (z)dz − νi yi (x, t), R

i = 1, · · · , m, x ∈ R, t > 0.

(1.3)

Rass and Radcliffe [12, Chapter 8] made a complete analysis on the global dynamics of the spatially homogeneous m-dimensional system of (1.3). The speed of first spread of infection was also obtained by using the saddle point method. Weng and Zhao [19] further established the existence of the asymptotic speed of propagation of infection and showed that it coincides with the critical wave speed for traveling wave fronts. Their results also gave an affirmative answer to an open problem presented by Rass and Radcliffe [12]. Zhang and Zhao [23] considered the spreading speed and traveling wave fronts of a spatially discrete version of (1.3). Recently, Wu and Chen [20] studied the uniqueness and stability of the traveling wave fronts of (1.3). Note that the seasonal variation is not considered in the epidemic models (1.1) and (1.2). As pointed out by Altizer et al. [2], seasonal variations in temperature, rainfall, and resource availability are ubiquitous and can exert strong pressures on population dynamics. Therefore, it is more realistic to consider the time-periodic versions of (1.1) and (1.2). In this paper, we consider the following time-periodic and nonlocal epidemic model: m



k=1

R



∂ui (x, t)  = σi (t) − ui (x, t) μi,k (t) ∂t

uk (x − y, t)pi,k (y)dy − νi (t)ui (x, t),

i = 1, · · · , m, x ∈ R, t > 0,

(1.4)

where σi (t), μi,k (t), νi (t) are all T -periodic and nonnegative functions, T > 0 is a constant. It is clear that (1.4) is a time-periodic version of (1.2). The purpose of this paper is to study the dynamics of system (1.4), including the global attractivity of positive T −periodic solution, spreading speeds, and time-periodic traveling wave fronts. Since (1.4) is a cooperative system, we can apply the monotone semi-flow theory developed in [10,11,18] to study the spreading speeds of it, say c∗ . However, due to the lack of compactness of solution maps of (1.4), the abstract theory in [10,11,18] are difficult to be applied to study the existence of the periodic traveling wave fronts of (1.4). In this paper, we shall extend the monotone iteration technique and the limiting argument (cf. [3,15,26]) to the periodic and nonlocal system. More precisely, by constructing a pair of explicit super- and sub-solution (see Lemma 3.4), we shall prove existence of the periodic traveling fronts Φ(x + ct, t) with speed c > c∗ . The asymptotic behavior of the periodic traveling fronts is also obtained. To establish the existence of the periodic traveling fronts with speed c = c∗ (critical periodic traveling fronts for short), we consider a sequence of periodic traveling wave fronts {Φ(n) (x +cn t, t)}n∈N with {cn } ⊂ (c∗ , +∞) and limn→∞ cn = c∗ . Since {Φ(n) (x, t)}n∈N is not compact with respect to compact open topology, we can not obtain a convergence subsequence of it. To overcome

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the difficulty, by careful analysis, we shall prove that {Φ(n) (x, t)}n∈N is pre-compact in Lloc (R2 , Rm ) (see Lemma 4.2). Then, the existence of the critical periodic traveling fronts is established by taking n → ∞. It should be mentioned that, in the recent years, there have been many interesting studies on spreading speeds and traveling wave solutions for various evolution equations in space and/or time periodic media. Hamel [5] and Hamel and Roques [6] presented a complete analysis on the qualitative behavior, uniqueness, and stability of pulsating fronts for KPP equations in periodic media. Zhao and Ruan [26] established the existence, uniqueness and asymptotic stability of time-periodic traveling waves for periodic advection– reaction–diffusion systems. Liang and Zhao [10] extended the theory of spreading speeds and traveling waves for monotone autonomous semiflows to periodic semiflows. Rawal, Shen and Zhang [13] considered the spreading speeds and traveling waves of a single nonlocal monostable equation in time and space periodic habitats. Other related results on the spreading speed and traveling waves for spatially periodic or time periodic equations, we refer to [1,13,15,17,25]. Throughout this paper, we always use the usual notations for the standard ordering in Rm and denote by  ·  the Euclidean norm in Rm . We make also the following basic assumptions:   (C1 ) pi,k (y) ≥ 0, pi,k (y) = pi,k (−y), ∀y ∈ R, R pi,k (y)dy = 1, and for any λ > 0, R eλy pi,k (y)dy < ∞, 1 ≤ i, k ≤ m.



(C2 ) The matrix Λ = μ ¯i,k m×m := min{σi (t)μi,k (t)} m×m ≥ 0 is irreducible in the sense that for every t≥0

i = k, there exists a distinct sequence i1 , i2 , · · · , ir with i1 = i, ir = k such that μ ¯is ,is+1 > 0, 1 ≤ s ≤ r − 1. The rest of the paper is organized as follows. Section 2 is devoted to the spreading speed c∗ and the non-existence of periodic traveling fronts with speed c ∈ (0, c∗ ). In Section 3, we prove the existence and asymptotic behavior of the time-periodic traveling wave fronts with speed c > c∗ . Section 4 is devoted to the existence of the time-periodic traveling wave fronts with speed c = c∗ . 2. Spreading speed This section is devoted to the spreading speed of (1.4). We first consider the threshold dynamics of the spatially homogeneous system of (1.4). Then, we shall give the well-posedness of initial value problem of (1.4), and establish a comparison theorem for super-solutions and sub-solutions. Further, we prove the existence of the spreading speed c∗ . As a consequence, we obtain the non-existence of the traveling waves with speed c < c∗ . Finally, we give the formula of c∗ . 2.1. Dynamics of the spatially homogeneous system In this section, we consider the existence and uniqueness of positive periodic solution and the threshold dynamics of the spatially homogeneous system of (1.4):

dwi (t)  = σi (t) − wi (t) μi,k (t)wk (t) − νi (t)wi (t), i = 1, · · · , m. dt m

(2.1)

k=1

Clearly, system (2.1) can be rewritten as dw(t) = F (t, w(t)), dt where w(t) = (w1 (t), · · · , wm (t))T and F (t, w(t)) = (F1 (t, w(t)), · · · , Fm (t, w(t)))T with

(2.2)

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m 

Fi (t, w(t)) = σi (t) − wi (t) μi,k (t)wk (t) − νi (t)wi (t). k=1

Denote

σM = max σ1 (t), · · · , maxσm (t) , D0 = [0, σ1 (0)] × · · · × [0, σm (0)], t≥0

t≥0

Dt = [0, σ1 (t)] × · · · × [0, σm (t)] and D = {(t, w) : 0 ≤ wi ≤ σi (t), t ≥ 0, i = 1, · · · , m}. Note that F (t, y) is continuous and Lipschitzian in y in any bounded set. From [16, Remark 5.2.1], it is easy to see that the following result holds. Lemma 2.1. For any w0 ∈ [0, l] with l ∈ Rm and l ≥ σM , system (2.2) has a unique solution w(t; w0 ) with w(0; w0 ) = w0 and w(t; w0 ) ∈ [0, l], ∀t > 0. Furthermore, if w0 ∈ D0 , then w(t; w0 ) ∈ Dt , ∀t > 0. Linearizing (2.1) at w = 0, we have dw(t) = Dw F (t, 0)w(t). dt

(2.3)

It is easy to verify that

Dw F (t, 0) = σi (t)μi,k (t) m×m − diag(ν1 (t), · · · , νm (t)). By assumption (C2), (2.3) is a cooperative and irreducible system. Let r0 be the spectral radius of the Poincare map associated with (2.3). We have the following result on the threshold dynamics of (2.1). Lemma 2.2. Assume (C1) and (C2). Then the following results hold. (i) If r0 ≤ 1, then zero solution is globally asymptotically stable for (2.1) in D0 . (ii) If r0 > 1, then (2.1) has a unique positive T −periodic solution β(t) which is globally asymptotically stable in D0 \ {0}. Proof. Since Fi (t, w) ≥ 0 for any (t, w) ∈ D with wi = 0, and the Jacobian matrix of F (t, w) is cooperative D, the solution semiflow {St }t≥0 of (2.1) is monotone in the sense that St (u) ≥ St (v) whenever u ≥ v in D0 . Next, we show that St is strongly monotone for all t ≥ mT , i.e. St (u)  St (v) whenever t ≥ mT and u > v in D0 . Let z(t) = (z1 (t), · · · , zm (t))T := w(t; u0 ) − w(t; v0 ) with u0 , v0 ∈ D0 and u0 > v0 . Then z(t) satisfies



dzi (t) = −zi (t) μi,k (t)wk (t; u0 ) + σi (t) − wi (t; v0 ) μi,k (t)zk (t) − νi (t)zi (t). dt m

m

k=1

k=1

Note that m 

 dzi (t) ≥− σ ¯ μi,k (t) + max νi (t) zi (t). t≥0 dt k=1

One can see that if there exists t0 ≥ 0 such that zi (t0 ) > 0, then zi (t) > 0 for all t ≥ t0 . In view of z(0) > 0, we suppose z1 (0) > 0. Thus, z1 (t) > 0 for all t ≥ 0. Fix any i∗ ∈ {2, · · · , m}. We now prove that there exists t¯ ∈ [(m − 1)T, mT ] such that zi∗ (t¯) > 0.



Since μ ¯i,k m×m = min{σi (t)μi,k (t)} m×m is irreducible, there exists a distinct sequence i1 , i2 , · · · , ir with t≥0

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i1 = i∗ and ir = 1 such that μ ¯is ,is+1 > 0, 1 ≤ s ≤ r − 1. It is clear that μis ,is+1 (t) > 0 for any t ≥ 0 and 1 ≤ s ≤ r − 1. We first prove the following claim: Claim. There exists t1 ∈ [0, T ] such that zir−1 (t1 ) > 0. In fact, if the claim is false, then zir−1 (t) = 0 for all t ∈ [0, T ]. From the equation for zir−1 (t), we get

dzir−1 (t)  = σir−1 (t) − wir−1 (t; v0 ) 0= μir−1 ,k (t)zk (t), t ∈ [0, T ]. dt m

(2.4)

k=1

Since wi (t; v0 ) ≤ σi (t) and zk (t) ≥ 0, ∀t ∈ [0, T ], we deduce from (2.4) that  σ1 (t) − w1 (t; v0 ) μir−1 ,1 (t)z1 (t) = 0.

(2.5)

If σ1 (t) = w1 (t; v0 ) for all t ∈ [0, T ], then it follows from w1 (t; v0 ) ≤ w1 (t; u0 ) ≤ σ1 (t) that z1 (t) = w1 (t; u0 ) − w1 (t; v0 ) = 0, ∀t ∈ [0, T ], which contradicts to the positivity of z1 (·). If there exists t0 ∈ [0, T ] such that σ1 (t0 ) > w1 (t0 ; v0 ), then it follows from (2.5) and the positivity of μir−1 ,1 (·) that z1 (t0 ) = 0, which is also a contradiction. As a conclusion, the claim holds. Similarly, we can show that there exists tk ∈ [(k − 1)T, kT ] such that zir−k (tk ) > 0, k = 2, · · · , r − 1. Take ¯ t = tr−1 . Thus, we see that t¯ ∈ [0, mT ] and zi∗ (t¯) > 0. Thus, zi∗ (t) > 0 for all t ≥ mT . From the arbitrary of i∗ , we conclude that z(t)  0 for all t ≥ mT . Therefore, St is strongly monotone for all t ≥ mT . Moreover, it is easy to verify that F (t, w) is strictly subhomogeneous in the sense that F (t, αw) > αF (t, w) for all α ∈ (0, 1) and w ∈ Dt with w  0. By [24, Theorem 2.3.4], as applied to SmT , we see that the assertion (i) holds. Further, when r0 > 1, there exists a unique mT −periodic solution β(t) which is globally asymptotically stable for all solutions of (2.1) with initial values in D0 \ {0}. Clearly, β(0) is the unique fixed point of SmT . Then, we have SmT (ST (β(0))) = ST (SmT (β(0))) = ST (β(0)). Hence, ST (β(0)) is also a fixed point of SmT . The uniqueness of the fixed point of SmT implies that ST (β(0)) = β(0). Thus, β(t) is a T −periodic solution of (2.1). Therefore, we deduce that the assertion (ii) holds. This completes the proof. 2 2.2. Initial value problem of (1.4) In this subsection, we shall give the well-posedness of initial value problem of (1.4), and establish a comparison theorem for super-solutions and sub-solutions. Let C be the space of all bounded and continuous functions from R to Rm . For any φ1 , φ2 ∈ C, we write 1 φ ≤ φ2 (φ1 φ2 ) if φ1 (x) ≤ φ2 (x) (φ1 (x) φ2 (x)), ∀x ∈ R and φ1 < φ2 if φ1 ≤ φ2 and φ1 = φ2 . We further equip C with the compact open topology, that is, a sequence φn converges to φ in C if and only if φn (x) converges to φ(x) in Rm uniformly for x in any bounded subset of R. The following norm on C can induce such topology:   max ϕ(x) ∞

x∈R,|x|≤l ϕC := , ∀ϕ ∈ C. 2l l=1

Clearly, the topology generated by  ·C and the compact open topology on C are equivalent on any uniformly bounded subset of C. Given q ∈ Rm with q  0. We denote Cq := {φ ∈ C : 0 ≤ φ(·) ≤ q}.

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Consider the initial value problem of (1.4): 

 m  = σi (t) − ui (x, t) k=1 μi,k (t) R uk (x − y, t)pi,k (y)dy − νi (t)ui (x, t), u(x, 0) = ϕ(x) = (ϕ1 (x), · · · , ϕm (x)), i = 1, · · · , m, ∂ui (x,t) ∂t

(2.6)

where x ∈ R, t > 0. In the following, we denote the solution of (2.6) by u(x, t; ϕ). Definition 2.3. A function u ∈ CσM is called a super-solution of (1.4) if

∂ui (x, t)  ≥ σi (t) − ui (x, t) μi,k (t) ∂t m

k=1

 uk (x − y, t)pi,k (y)dy − νi (t)ui (x, t), i = 1, · · · , m, R

for any x ∈ R and t > 0. A sub-solution of (1.4) is defined by reversing the inequality. We now state the following result on the well-posedness of initial value problem of (1.4) and the comparison principle. Since its proof is similar to those of [19, Theorems 2.1 and 2.2], we omit it here. Lemma 2.4. Assume (C1) and (C2). The following results hold. (1) For any ϕ ∈ CσM , (1.4) admits a unique solution u(x, t; ϕ) satisfying u(·, 0; ϕ) = ϕ(·) and u(·, ·; ϕ) ∈ [0, σM ]. (2) Let u+ (x, t) and u− (x, t) be a super-solution and a sub-solution of (1.4), respectively. If u+ (·, 0) ≥ u− (·, 0), then u+ (·, t) ≥ u− (·, t) for all t ≥ 0. Remark 2.5. We would like to mention that one can easily prove that β(t) is globally stable for (1.4) provided that r0 > 1, which yields that 0 is unstable. In fact, based on Lemma 2.2 and using the comparison method (see e.g. [21, Theorem 3.2], one can show that for any δ ∈ Rm with δ > 0, lim u(·, t; ϕ) − β(t) = 0 uniformly for ϕ ∈ Xδ+ ,

t→∞

where X := BUC(R, Rm ) is the Banach space of all bounded and uniformly continuous functions from R into Rm with the supremum norm  ·  and   Xδ+ := ϕ ∈ X : ϕ(x) ∈ D0 , ∀x ∈ R and inf ϕ(x) ≥ δ . x∈R

We leave the details to the readers. 2.3. Existence of spreading speed According to Lemma 2.2, to consider the propagation phenomenon of (1.4), we always make the following assumption: (C3) r0 > 1. In the rest of this section, we always assume (C1)–(C3). Let R[ϕ](x) := ϕ(−x) be the reflection operator. Given h ∈ R, we define the translation operator Th [·] by Th [ϕ](x) = Th [ϕ](x − h). Let Q : Cγ → Cγ be a map, where γ  0 in Rm . In order to apply [10, Theorem 2.1], we need the following assumptions on Q: (A1) Q[R[u]] = R[Q[u]], Th [Q[u]] = Q[Th [u]] for h ∈ R; (A2) Q : Cγ → Cγ is continuous with respect to the compact open topology;

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(A3) {Q[u](x) : u ∈ Cγ , x ∈ R} is a bounded subset of Rm ; (A4) Q : Cγ → Cγ is monotone in the sense that Q[ϕ] ≥ Q[ψ] whenever ϕ ≥ ψ in Cγ ; (A5) Q : [0, γ] → [0, γ] admits exactly two fixed points 0 and γ, and limn→∞ Qn [u] = γ for any u ∈ [0, γ]\{0}. We recall the definition of periodic semiflow and spreading speed as follows. Definition 2.6 (See [10]). A family of operators {Qt }t≥0 is said to be a T -periodic semiflow on a metric ¯ provided that the following properties hold: space (X , d) (i) Q0 [ϕ] = ϕ, ∀ϕ ∈ X ; (ii) Qt [QT [ϕ]] = Qt+T [ϕ], ∀t ≥ 0 and ∀ϕ ∈ X ; (iii) Q(t, ϕ) := Qt [ϕ] is continuous in (t, ϕ) on [0, ∞) × X . The map QT is called the Poincaré map associated with this periodic semiflow. Definition 2.7 (See [8–11,18]). A function w : R × R+ → R+ is said to have an asymptotic speed of spread (spreading speed for short) c∗ > 0 if there exists ε > 0 such that lim

t→∞,|x|≥ct

and

lim inf

w(x, t) = 0 for any c > c∗

t→∞,|x|≤ct

w(x, t) ≥ ε for any c ∈ (0, c∗ ).

If all solutions of a system with initial functions having compact supports share the same c∗ and ε, then we call such c∗ the spreading speed of the system. From Lemma 2.4, we see that if ϕ ∈ Cβ(0) , then u(·, t; ϕ) ∈ Cβ(t) , ∀t > 0. To study the spreading speeds of (1.4), we define a family of maps {Qt }t≥0 from Cβ(0) to Cβ(t) by Qt [ϕ](x) = u(x, t; ϕ), ∀x ∈ R, t ≥ 0. We then have the following result. Lemma 2.8. {Qt }t≥0 is a monotone and sub-homogeneous periodic semiflow on Cβ(0) . Proof. We first prove that {Qt }t≥0 is a periodic semiflow on Cβ(0) . It is clear that Qt [·] satisfies the property (i). The semigroup property (ii) follows directly from the existence and uniqueness of solutions of (1.4). Now, we prove the property (iii). Given ϕ ∈ Cβ(0) . From (1.4), we have m ∂   

  max μi,k (t)βk (t) + νi (t)βi (t) , i = 1, · · · , m σi (t) + βi (t)  ui (x, t; ϕ) ≤ i=1,··· ,m,t∈R ∂t k=1

˜ > 0 such that for all (x, t) ∈ R × [0, ∞). Hence, there exists a constant L ˜ 1 − t2 |, ∀x ∈ R, t1 , t2 ∈ [0, ∞). u(x, t1 ; ϕ) − u(x, t2 ; ϕ) ≤ L|t Thus, for each ϕ ∈ Cβ(0) , Qt [ϕ] is continuous in t ∈ [0, ∞) with respect to the compact open topology. Note that for any (t0 , ϕ0 ) ∈ R+ × Cβ(0) , we have

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Qt [ϕ] − Qt0 [ϕ0 ]C ≤ Qt [ϕ] − Qt [ϕ0 ]C + Qt [ϕ0 ] − Qt0 [ϕ0 ]C . Thus, to prove the property (iii), it suffices to prove that for any t0 > 0, Qt [ϕ] = u(·, t; ϕ) is continuous in ϕ with respect to the compact open topology uniformly for t ∈ [0, t0 ]. ¯ φ˜ ∈ Cβ . For any given  > 0 and t0 > 0, we define Given any φ, ¯ − ui (x, t; φ)|, ˜ w(x, wi (x, t) :=|ui (x, t; φ) ¯ t) =

max wi (x, t), k0 :=

i=1,··· ,m

sup

w(x, ¯ t),

(x,t)∈R×[0,t0 ]

Ωh (z) :=[z − h, z + h], ∀h ∈ R+ , z ∈ R, and |φ|Ωh (z) :=

sup φ(x), ∀φ ∈ Cβ . x∈Ωh (z)

Moreover, we take 0 :=

 , 2(1 + L3 t0 )e(L1 +L2 +L3 )t0

where L1 =

max

i=1,··· ,m,t∈R

and L3 = m

νi (t), L2 = m

max

i=1,··· ,m,t∈R

max

i,k=1,··· ,m,t∈R

[σi (t) + βi (t)]

{μi,k (t)βi (t)},

max

i,k=1,··· ,m,t∈R

μi,k (t).

By the definition of k0 , there exist (x∗ , t∗ ) ∈ R × [0, t0 ] and i∗ ∈ {1, · · · , m} such that 0 ≤ wi (x, t) ≤ w(x, ¯ t) ≤ k0 ≤ wi∗ (x∗ , t∗ ) + 0 , ∀(x, t) ∈ R × [0, t0 ], i = 1, · · · , m. Since

 R

pi,k (y)dy = 1, ∀i, k = 1, · · · , m, we can choose an integer M > 1 such that  2

pi,k (y)dy

max

i=1,··· ,m,t∈R

βi (t) < 0 , ∀i, k = 1, · · · , m.

|y|≥M

Then, for any t ∈ [0, t0 ], it follows that  wk (x∗ − y, s)pi∗ ,k (y)dy ≤ |wk (·, s)|ΩM (x∗ ) + 0 . R

Let η =

 2e(L1 +L2 +L3 )t0

> 0. Next, we consider the following two sub-cases: Case (i). t∗ = 0; and Case (ii). t∗ ∈ (0, t0 ].

˜ Ω (x ) < η, then using (2.7), we have For Case (i), if |φ¯ − φ| M ∗ |w(·, ¯ t)|ΩM (x∗ ) ≤ 0 + wi∗ (x∗ , t∗ ) ¯ ∗ ) − φ(x ˜ ∗ )| ≤ 0 + η < , for any t ∈ [0, t0 ]. = 0 + wi∗ (x∗ , 0) = 0 + |φ(x ˜ Ω (x ) < η and t ∈ [0, t0 ], we have For Case (ii), if |φ¯ − φ| M ∗ |w(·, ¯ t)|ΩM (x∗ ) ≤ 0 + wi∗ (x∗ , t∗ )

(2.7)

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−L1 t∗

≤ 0 + wi∗ (x∗ , 0)e

 t∗   ¯ +  e−L1 (t∗ −s) [L1 − νi (s)]ui∗ (x∗ , s; φ) 0

m 

¯ − σi∗ (s) − ui∗ (x∗ , s; φ) μi∗ ,k (s) k=1

t∗ −

9



 ¯ i ,k (y)dy ds uk (x∗ − y, s; φ)p ∗

R

 ˜ e−L1 (t∗ −s) [L1 − νi (s)]ui∗ (x∗ , s; φ)

0 m 

˜ − σi∗ (s) − ui∗ (x∗ , s; φ) μi∗ ,k (s) k=1

t∗ ≤ 0 + η +

 R

 L1 wi∗ (x∗ , s)

0 m  

¯ + σi∗ (s) − ui∗ (x∗ , s; φ) μi∗ ,k (s) k=1

+ wi∗ (x∗ , s)

m

 μi∗ ,k (s)

k=1

t∗ ≤ 0 + η +

  ˜ i ,k (y)dy ds uk (x∗ − y, s; φ)p ∗

 wk (x∗ − y, s)pi∗ ,k (y)dy R

 ˜ i ,k (y)dy ds uk (x∗ − y, s; φ)p ∗

R

 [L1 + L2 ]wi∗ (x∗ , s) + L3 [|wk (·, s)|ΩM (x∗ ) + 0 ] ds



0

t∗ ≤ 0 + η + 0 L3 t0 + [L1 + L2 + L3 ]|w(·, ¯ s)|ΩM (x∗ ) ds. 0

It then follows from Gronwall’s inequality that |w(·, ¯ t)|ΩM (x∗ ) ≤ (0 + η + 0 L3 t0 )e(L1 +L2 +L3 )t0 = , ∀t ∈ [0, t0 ]. Summarizing the above two cases, we conclude that for any  > 0, t0 > 0 and compact set A ⊆ R, there exist η > 0 and compact set ΩM (x∗ ) such that A ⊆ ΩM (x∗ ) and ˜ Ω (x ) < η. |w(·, t)|A ≤ |w(·, ¯ t)|ΩM (x∗ ) < , ∀t ∈ [0, t0 ] whenever |φ¯ − φ| M ∗ Using this conclusion, one can easily show that Qt [ϕ] = u(·, t; ϕ) is continuous in ϕ with respect to the compact open topology, uniformly for t ∈ [0, t0 ]. Consequently, Qt [ϕ] = u(·, ·; ϕ) is continuous in (t, ϕ) with respect to the compact open topology. Hence, {Qt [·]}t≥0 is a periodic semiflow on Cβ(0) . The monotonicity of Qt [·] follows from Lemma 2.4 (2). Finally, we prove that Qt [·] is sub-homogeneous on Cβ(0) , i.e. Qt [γϕ] ≥ γQt [ϕ] for all γ ∈ [0, 1] and ϕ ∈ Cβ(0) . In fact, we can show that the functions u ¯(x, t) := u(x, t; γϕ) and u(x, t) := γu(x, t; ϕ) constitute a pair of super- and subsolution of (1.4) with u ¯(·, 0) = u(·, 0). By comparison principle, we have u(x, t; γϕ) ≥ γu(x, t; ϕ) for any x ∈ R, t ≥ 0, which implies that Qt [·] is sub-homogeneous on Cβ(0) . This completes the proof. 2 Lemma 2.9. The Poincaré map Q = QT satisfies (A1)–(A5) with r = β(0), and Qt satisfies (A1) for any t > 0.

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Proof. It is easy to see that Q = QT satisfies (A1) and (A3), and Qt satisfies (A1) for any t > 0. Moreover, (A2), (A4) and (A5) follow from Lemma 2.8, Lemma 2.4 (2) and Lemma 2.2, respectively. This completes the proof. 2 By [10, Theorem A], it then follows that the Poincaré map QT has a spreading speed c∗T > 0. Moreover, [10, Theorem 2.1] implies that c∗ =: c∗T /T is the spreading speed for solutions of (1.4), that is, the following result holds. Theorem 2.10. Let c∗T be the spreading speed of QT and u(x, t; ϕ) be the solution of (1.4) with u(·, 0; ϕ) = ϕ. Then the following statements hold true: (1) For any c > c∗ , if ϕ ∈ Cβ(0) with 0 ≤ ϕ β(0), and ϕ(x) = 0 for x outside a bounded interval, then lim u(x, t; ϕ) = 0. t→∞,|x|≥ct

(2) For any c < c∗ , if ϕ ∈ Cβ(0) with ϕ(x) > 0, then

lim

t→∞,|x|≤ct

u(x, t; ϕ) − β(t) = 0.

As a straightforward consequence of [10, Theorem 2.2], we have the following result on the non-existence of the T −periodic traveling waves. Theorem 2.11. For any c ∈ (0, c∗ ), system (1.4) has no T −periodic traveling wave Φ(x + ct, t) connecting 0 to β(t). 2.4. Formula of spreading speed In this section, we give the formula of the spreading speed c∗ . Since c∗ = c∗T /T , we only need to compute In order to compute c∗T , we consider the linearized system of (1.4) at the zero solution:

c∗T .

m



k=1

R

∂ui (x, t) = σi (t) μi,k (t) ∂t

uk (x − y, t)pi,k (y)dy − νi (t)ui (x, t), i = 1, · · · , m.

(2.8)

For any λ ∈ R+ , let u(x, t) = e−λx w(t). Then w(t) = (w1 (t), · · · , wm (t)) satisfies the following ordinary differential system

dwi (t) = σi (t) μi,k (t)wk (t) dt m

k=1

 eλy pi,k (y)dy − νi (t)wi (t), i = 1, · · · , m.

(2.9)

R

Let r(λ) be the spectral radius of the Poincare map of (2.9). Then, we have the following computation formulae for c∗T . Lemma 2.12. c∗T = inf λ>0

ln r(λ) λ .

Proof. Let w(t) be the solution of (2.9). Then, one can see that u(x, t) = e−λx w(t) is a solution of (2.8). Let Mt be the solution map of (2.8). Define Bλt : Rm → Rm by Bλt (z) := Mt (e−λx z)(0) = w(t). Therefore, Bλt (z) is the solution map of the linear system (2.9) on Rm , and hence, r(λ) be the spectral radius of the Poincare map BλT . Similar to the proof of Lemma 2.2, we can show that Bλt is strongly

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11

positive for all t ≥ mT . Thus, one can easily verify that BλmT = (BλT )m is compact and strongly positive operator. It follows from [11, Lemma 3.1] that r(λ) > 0 which is a simple eigenvalue of BλT with a strongly positive eigenvector νλ . Moreover, using an argument similar to that in the proof of [22, Lemma 2.1], we see that there exists a T −periodic function ωλ (t) = (ω1,λ (t), · · · , ωm,λ (t)) such that eM (λ)t ωλ (t) is a solution of (2.9), where M (λ) = T1 ln r(λ) and ωλ (0) = νλ . Thus, Bλt (ωλ (0)) = eM (λ)t ωλ (t). In particular, BλT (ωλ (0)) = eM (λ)T ωλ (0), which implies that eM (λ)T is the principle eigenvalue of BλT with strongly positive eigenvector ωλ (0). Define the function ΠT (λ) =

M (λ)T ln r(λ) ln eM (λ)T = = . λ λ λ

Since r(0) = r0 > 1, the condition (C7) in [11] is satisfied.

Next, we prove that ΠT (∞) = ∞. Since min{σi (t)μi,k (t)} m×m ≥ 0 is irreducible, there exist i0 ∈ t≥0

{1, · · · , m} such that min{σ1 (t)μ1,i0 (t)} > 0. In view of eM (λ)t ωλ (t) is a solution of (2.9), we see that t≥0

 ω1,λ (t) = σ1 (t)

m

 eλy p1,k (y)dy − [ν1 (t) + M (λ)]ω1,λ (t)

μ1,k (t)ωk,λ (t)

k=1

R



≥ ωi0 ,λ (t)

eλy p1,i0 (y)dy min{σ1 (t)μ1,i0 (t)} − [max ν1 (t) + M (λ)]ω1,λ (t), t≥0

t≥0

R

which implies that T 0=

 ω1,λ (t) dt ω1,λ (t)

0

T ≥ min{σ1 (t)μ1,i0 (t)} t≥0

ωi0 ,λ (t) dt ω1,λ (t)

0

 eλy p1,i0 (y)dy − [max ν1 (t) + M (λ)]T. t≥0

R

Hence, we deduce that M (λ)T ≥ min{σ1 (t)μ1,i0 (t)} t≥0 λ

T

ωi0 ,λ (t) dt ω1,λ (t)

 R

eλy p1,i0 (y)dy maxt≥0 ν1 (t) − . λ λ

0

Since have

 R

p1,i0 (y)dy = 2  R

∞ 0

p1,i0 (y)dy = 1, we can choose X0 > 0 such that

eλy p1,i0 (y)dy ≥ λ

∞ X0

eλy p1,i0 (y)dy λ

≥

eλX0

∞ X0

p1,i0 (y)dy λ

∞ X0

p1,i0 (y)dy > 0. Thus, we

→ ∞ as λ → ∞.

Then, we conclude that ΠT (∞) = ∞. Therefore, ΠT (λ) attains its minimum at some finite value λ∗ . Note that the solution u(x, t; φ) is a sub-solution of the linear system (2.9), we have Qt [φ] ≤ Mt [φ], ∀φ ∈ Cβ(0) , t ≥ 0. It then follows from [11, Theorem 3.10 (i)] that c∗T ≤ inf λ≥0 ΠT (λ). For any λ > 0, let r (λ) be the spectral radius of the Poincare map of the following ordinary differential system

dwi (t) = (1 − )σi (t) μi,k (t)wk (t) ∂t m

k=1

 eλy pi,k (y)dy − νi (t)wi (t), i = 1, · · · , m. R

(2.10)

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Similarly, we can show that c∗T ≥ inf λ≥0

ln r  (λ) . λ

Letting  → 0, we obtain

ln r(λ) = inf ΠT (λ). λ≥0 λ≥0 λ

c∗T ≥ inf

Therefore, c∗T = inf λ≥0 ΠT (λ). The proof is complete.

2

3. Existence of periodic traveling wave fronts In this section, we prove the existence and asymptotic behavior of the periodic traveling fronts with speed c > c∗ . We first give the definition of the periodic traveling wave solution, see [10,13,15]. Definition 3.1. (i) A measurable function u(x, t) : R2 → Rm is called an entire solution of (1.4) if u(x, t) is differentiable in t ∈ R and it satisfies (1.4) for all x, t ∈ R. (ii) An entire solution u(x, t) is called a periodic traveling wave solution connecting 0 and β(t) if there exists a bounded measurable function Φ(z, t) : R2 → Rm such that u(x, t) = Φ(x + ct, t), Φ(·, · + T ) = Φ(·, ·), and Φ(−∞, t) = 0 and Φ(+∞, t) = β(t) uniformly in t ∈ R. Moreover, if Φ(z, t) is non-decreasing in z ∈ R, then we call it a periodic traveling wave front. (iii) A periodic traveling wave solution is said to be continuous if it is continuous in x. Let Π(λ) :=

M (λ) ΠT (λ) = . T λ

From [11, Lemma 3.8], there following results hold. Lemma 3.2. (i) (ii) (iii) (iv)

There exists λ∗ ∈ (0, ∞) such that c∗ := Π(λ∗ ) = inf λ>0 Π(λ). limλ→0+ Π(λ) → ∞ and limλ→∞ Π(λ) = ∞. Π(λ) is decreasing near 0. Π (λ) changes sign at most once on (0, ∞).

Given any c > c∗ . It follows from Lemma 3.2 that there exists 0 < λ1 := λ1 (c) < λ∗ < λ2 := λ2 (c) < ∞ such that c = Π(λ1 ) and c > Π(λ) for any λ ∈ (λ1 , λ2 ). Take  ∈ (0, λ2 − λ1 ) and denote λ = λ1 + . Let eM (λ1 )t ωλ1 (t) be the solution of (2.9) corresponding to λ = λ1 . For simplicity, we denote ωλ1 (t) by ω1 (t). The main results of this section are stated as follows. Theorem 3.3. [Existence of non-critical periodic traveling fronts] Assume (C1 )–(C3 ). Then, for each c > c∗ , (1.4) admit a periodic traveling front Φ(x + ct, t) = (Φ1 (x + ct, t), · · · , Φm (x + ct, t)) connecting 0 and β(t) satisfying lim

ξ→−∞

Φi (ξ, t) = 1, i = 1, · · · , m. eλ1 ξ ω1,i (t)

(3.1)

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3.1. Existence of non-critical periodic traveling fronts To prove the existence of non-critical periodic traveling fronts, i.e. Theorem 3.3, we first give some lemmas. Take  ∈ (0, λ2 − λ1 ) and denote λ = λ1 + . Let eM (λ )t ωλ (t) be the solution of (2.9) corresponding to λ = λ . For simplicity, we denote ωλ (t) by ω (t). For q > 1, denote z∗ =

1 1 1  1 ω1,i (t)  ω1,i (t)  ln max and zi := zi (t) = ln , i = 1, · · · , m.  q i=1,··· ,m,t≥0 ω,i (t)  q ω,i (t)

We can choose q > 1 sufficiently large such that zi (t) ≤ z∗ ≤ 0 and eλ1 z∗

max

i=1,··· ,m,t≥0

ω1,i (t) ≤

min

i=1,··· ,m,t≥0

βi (t).

Define two functions v¯(z, t) = (¯ v1 (z, t), · · · , v¯m (z, t)) and v(z, t) = (v 1 (z, t), · · · , v m (z, t)) as follows: v¯(z, t) = min{β(t), eλ1 z ω1 (t)}, (z, t) ∈ R2 , and   v(z, t) = max 0, eλ1 z ω1 (t) − qeλ z ω (t) , (z, t) ∈ R2 . Lemma 3.4. The following results hold. (i) u(x, t) := v(x + ct, t) ≤ u ¯(x, t) := v¯(x + ct, t), ∀(x, t) ∈ R2 ; (ii) u(x, t) is a sub-solutions of (1.4) provided that q > 1 is sufficiently large; (iii) u ¯(x, t) is a super-solution of (1.4). Proof. For convenience, we denote Q[u](z, t) = (Q1 [u](z, t), · · · , Qm [u](z, t)), where

∂ui (z, t)  ∂ui (z, t) +c − σi (t) − ui (z, t) μi,k (t) ∂t ∂z m

Qi [u](z, t) :=

k=1

 uk (z − y, t)pi,k (y)dy R

+ νi (t)ui (z, t), i = 1, · · · , m. To prove this lemma, it suffices to prove that (1) v(z, t) ≤ v¯(z, t), ∀(z, t) ∈ R2 ; (2) for each z ∈ R, Q[v](z, t) ≤ 0 for a.e. t ∈ R, provided that q is sufficiently large; (3) for each z ∈ R, Q[¯ v ](z, t) ≥ 0 for a.e. t ∈ R. Next, we prove the statements (1)–(3). (1) It is clear that if z ≥ z∗ , v(z, t) = 0. If z < z∗ , then v(z, t) ≤ eλ1 z∗ ω1 (t) ≤ β(t). Then, we see that v(z, t) ≤ v¯(z, t), ∀(z, t) ∈ R2 . + 2 2 (2) Denote A− i = {(z, t) ∈ R : z > zi (t)} and Ai = {(z, t) ∈ R : z < zi (t)}, i = 1, · · · , m. Note that − λ1 z 2 0 ≤ v i (z, t) ≤ e ω1,i (t), ∀(z, t) ∈ R , i = 1, · · · , m. If (z, t) ∈ Ai , then v i (z, t) = 0, and hence Qi [v](z, t) ≤

∂v i (z, t) ∂v i (z, t) + − νi (t)v i (z, t) ≤ 0. ∂t ∂z

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λ1 z If (z, t) ∈ A+ ω1,i (t) − qeλ z ω,i (t) ≥ 0, and i , then v i (z, t) = e



zk (t)



v k (y, t)pi,k (z − y))dy =

eλ1 y ω1,k (t) − qeλ y ω,k (t) pi,k (z − y)dy

−∞

R

∞ ≥



eλ1 y ω1,k (t) − qeλ y ω,k (t) pi,k (z − y)dy, k = 1, · · · , m.

−∞

Since eM (λ1 )t ω1 (t) and eM (λ )t ω (t) are solutions of (2.9) corresponding to λ = λ1 and λ = λ , respectively, we have  ω1,i (t) + M (λ1 )ω1,i (t) = σi (t)

m



k=1  (t) + M (λ )ω,i (t) = σi (t) ω,i

m

eλ1 y pi,k (y))dy − νi (t)ω1,i (t),

μi,k (t)ω1,k (t) R



eλ y pi,k (y))dy − νi (t)ω,i (t).

μi,k (t)ω,k (t)

k=1

R

Note that M (λ1 ) = cλ1 and M (λ ) < cλ , direct computations show that Qi [v](z, t)

∂v (z, t) ∂v (z, t) ≤ i +c i − σi (t) μi,k (t) ∂t ∂z m

k=1

+ eλ1 z ω1,i (t)

m

=e

 [ω1,i (t)

− σi (t)

m



eλ1 y ω1,k (t) − qeλ y ω,k (t) pi,k (z − y)dy

−∞



eλ1 (z−y) ω1,k (t)pi,k (y)dy + νi (t)v i (z, t)

μi,k (t)

k=1 λ1 z

∞

R

 + cλ1 ω1,i (t)] − qeλ z [ω,i (t) + cλ ω,i (t)]

∞ μi,k (t)

k=1

 λ1 y e ω1,k (t) − qeλ y ω,k (t) pi,k (z − y)dy

−∞

+ eλ1 z ω1,i (t)

m

 eλ1 (z−y) ω1,k (t)pi,k (y)dy

μi,k (t)

k=1

R

 + νi (t) eλ1 z ω1,i (t) − qeλ z ω,i (t) λ1 z

=e

 [ω1,i (t)

+ M (λ1 )ω1,i (t) − σi (t)

m



k=1  (t) + cλ ω,i (t) − σi (t) − qeλ z [ω,i

m

R



+ e2λ1 z ω1,i (t)

k=1

eλ y pi,k (y)dy + νi (t)ω,i (t)]

μi,k (t)ω,k (t)

k=1 m

eλ1 y pi,k (y)dy + νi (t)ω1,i (t)]

μi,k (t)ω1,k (t)

R

 eλ1 y ω1,k (t)pi,k (y)dy

μi,k (t) R

 m  

μi,k (t) eλ1 y ω1,k (t)pi,k (y)dy = eλ z − q[cλ − M (λ )] + e(λ1 −)z ω1,i (t) k=1

R

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 m  

≤ eλ z − q[cλ − M (λ )] + ω1,i (t) μi,k (t) eλ1 y ω1,k (t)pi,k (y)dy ≤ 0, k=1

R

provided that q > 1 is sufficiently large. Similarly, we can prove the statement (3). This completes the proof. 2 Let un (x, t) := u(x − cnT, t + nT ; u ¯(·, 0)). The following result shows that un (x, t) is non-increasing in n and non-decreasing in x. Lemma 3.5. For any given t ∈ R and n ∈ N with t + nT > 0, x ∈ R and h > 0, un+1 (x, t) ≤ un (x, t) and un (x + h, t) ≥ un (x, t). Proof. Since ω1 (· + T ) = ω1 (·) and β(· + T ) = β(·), it is easy to see that u ¯(· − cT, T ) = u ¯(·, 0) and u ¯(x, t) is non-decreasing in x ∈ R. By Lemmas 2.4 and 3.4, for any given t ∈ R and n ∈ N with t + nT > 0, x ∈ R and h > 0, we have un+1 (x, t) = u(x − c(n + 1)T, t + (n + 1)T ; u ¯(·, 0)) = u(x − cnT, t + nT ; u(· − cT, T ; u ¯(·, 0))) ≤ u(x − cnT, t + nT ; u ¯(· − cT, T )) = u(x − cnT, t + nT ; u ¯(·, 0)) = un (x, t) and un (x + h, t) = u(x − cnT + h, t + nT ; u ¯(·, 0)) = u(x − cnT, t + nT ; u ¯(· + h, 0)) ≥ u(x − cnT, t + nT ; u ¯(·, 0)) = un (x, t). This completes the proof. 2 The following result will be used to prove the upward convergence of periodic traveling waves. Lemma 3.6. For any q > 1, there exists L > 0 such that inf

x≥L,t≥0

u(x − ct, t, u ¯(·, 0)) > 0.

Proof. By comparison theorem, we have u(x, t) ≤ u(x, t, u ¯(·, 0)) ≤ u ¯(x, t). Noting that for x < 0 with |x| 1, there holds u(x − ct, t) = eλ1 x ω1 (t) − qeλ x ω (t) > 0.

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Thus, we can choose L < 0 with |L| 1 such that u(L − ct, t, u ¯(·, 0)) ≥ u(L − ct, t) ≥ κ := min[eλ1 L ω1 (t) − qeλ L ω (t)] > 0, ∀t ≥ 0. t≥0

Since u(x − ct, t, u ¯(·, 0)) is non-decreasing in x ∈ R, we see that u(x − ct, t, u ¯(·, 0)) ≥ u(L − ct, t, u ¯(·, 0)) ≥ κ > 0, ∀x ≥ L, t ≥ 0. Therefore, we conclude that inf

x≥L,t≥0

u(x − ct, t, u ¯(·, 0)) > 0.

This completes the proof. 2 By monotonicity, we can define U (x, t) := lim un (x, t) and U 0 (x) := lim un (x, 0). n→∞

n→∞

(3.2)

Thus, U (x, t) and U 0 (x) are upper-semi-continuous in x, t ∈ R. In the following lemma, we prove that U (x, t) is an entire solution of (1.4) with initial value U 0 (x). Lemma 3.7. Let U (x, t) and U 0 (x) be defined as in (3.2). Then we have U (x, t) = u(x, t; U 0 (·)) for all x, t ∈ R and U (x, t) is non-decreasing in x ∈ R. Proof. Notice that t uni (x, t)

= uni (x, 0)

∂uni (x, t) dt ∂t

+ 0

= uni (x, 0)

t   + σi (s) − uni (x, s) 0

×

m

 μi,k (s)

k=1

 unk (x − y, s)pi,k (y)dy − νi (s)uni (x, s) ds, i = 1, · · · , m.

R

It then follows from Lebesgue dominated convergence theorem that

Ui (x, t)

= Ui0 (x)

t  +

 σi (s) − Ui (x, s)

0

×

m

k=1

 μi,k (s)

 Uk (x − y, s)pi,k (y)dy − νi (s)Ui (x, s) ds, i = 1, · · · , m.

R

This yields that U (x, t) = u(x, t; U 0 (·)) for all x, t ∈ R. Since un (x, t) is non-decreasing in x ∈ R, it is clear that U (x, t) is also non-decreasing in x ∈ R. This completes the proof. 2

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Now, we are ready to prove that U (x, t) is a periodic traveling front of (1.4) connecting 0 and β(t) with speed c > c∗ . Proof of Theorem 3.3. Take Φ(x, t) := U (x − ct, t) = u(x − ct, t; U 0 (·)), i.e. u(x, t; U 0 (·)) = Φ(x + ct, t) = U (x, t). By comparison theorem, we have u ¯(x − cnT, t + nT ) ≥ u(x − cnT, t + nT ; u ¯(·, 0)) ≥ u(x − cnT, t + nT ). Note that u ¯(· − cnT, · + nT ) = u ¯(·, ·) and u(· − cnT, · + nT ) = u(·, ·). It follows that u ¯(x, t) ≥ U (x, t) = lim u(x − cnT, t + nT ; u ¯(·, 0)) ≥ u(x, t). n→∞

Thus, u ¯(x − ct, t) ≥ Φ(x, t) ≥ u(x − ct, t). By the definition of u ¯(x, t) and u(x, t), we have for x < 0 with |x|  1, u ¯(x − ct, t) = eλ1 x ω1 (t) and u(x − ct, t) = eλ1 x ω1 (t) − qeλ x ω (t) Noting that λ = λ1 +  > λ1 , the squeezing argument gives lim

ξ→−∞

Φi (ξ, t) = 1, i = 1, · · · , m. eλ1 ξ ω1,i (t)

Now, we show that Φ(·, t + T ) = Φ(·, t), ∀t ∈ R. For any (x, t) ∈ R2 , we have Φ(x, t + T ) = U (x − c(t + T ), t + T ) ¯(·, 0)) = lim u(x − c(t + (n + 1)T ), t + (n + 1)T ; u n→∞

= lim u(x − ct − cnT, t + nT ; u ¯(·, 0)) n→∞

= U (x − ct, t) = Φ(x, t). It is clear that Φ(−∞, t) = 0. It remains to show that Φ(+∞, t) = β(t). Since Φ(x, t) = U (x − ct, t) and U (x, t) is non-decreasing in x ∈ R, Φ(x, t) is non-decreasing in x ∈ R. In view of β(t) ≥ u ¯(x − ct, t) ≥ Φ(x, t) ≥ u(x − ct, t) ≥ 0, we see that Φ(+∞, t) =: γ(t) ∈ [0, β(t)], ∀t ∈ R. Moreover, γ(t +T ) = γ(t). Noting that Φ(x +ct, t) = U (x, t), U (x, 0) = U 0 (x), and

Ui (x, t)

= Ui0 (x)

+

t   0

m

σi (s) − Ui (x, s)

 × R

μi,k (s)

k=1

 Uk (x − y, s)pi,k (y)dy − νi (s)Ui (x, s) ds, i = 1, · · · , m,

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we obtain t  Φi (ξ, t) = Φi (ξ, 0) + 0

m

 μi,k (s) σi (s) − Φi (ξ + c(s − t), s)

 ×

k=1

 Φk (ξ − y + c(s − t), s)pi,k (y)dy − νi (s)Φi (ξ + c(s − t), s) ds.

R

Thus, for t ∈ [0, T ], one can verify that t  γi (t) = γ(0) +

m

 μi,k (s) σi (s) − γi (s) k=1

0



 γk (s)pi,k (y)dy − νi (s)γi (s) ds.

R

Hence, γ(t) satisfies m 

γi (t) = σi (t) − γi (t) μi,k (t)γk (t) − νi (s)γi (t), k=1

that is, γ(t) is a nonnegative and periodic solution of (2.1). From Lemma 3.6, we have u(x − ct − cnT, t + nT ; u ¯(·, 0)) ≥ κ for all x ≥ L + cnT and t ≥ 0. Thus, Φ(x, t) = U (x − ct, t) = lim u(x − ct − cnT, t + nT ; u ¯(·, 0)) ≥ κ n→∞

for all x ≥ L + cnT and t ≥ 0, which implies that γ(t) ∈ (0, β(t)], t ∈ R. It follows from Lemma 2.2 that γ(t) = β(t). Therefore, we conclude that Φ(+∞, t) = β(t), ∀t ∈ R. This completes the proof of Theorem 3.3. 2 4. Existence of critical periodic traveling fronts In this section, we prove that the existence of the critical periodic traveling front connecting 0 and β(t). More precisely, we have the following result. Theorem 4.1. [Existence of critical periodic traveling front] Assume (C1 )–(C3 ). Then, (1.4) admit a periodic traveling front Φ(x + c∗ t, t) = (Φ1 (x + c∗ t, t), · · · , Φm (x + c∗ t, t)) with speed c = c∗ and connecting 0 and β(t). We choose a sequence {cn } ⊂ (c∗ , +∞) such that limn→∞ cn = c∗ . According to Theorem 3.3, there exists a periodic traveling wave front (Φ(n) (x + cn t, t), cn ) of (1.4) for each n connecting 0 and β(t). To obtain the convergence sequence of {Φ(n) (x, t)}n∈N , we prove the following result. Lemma 4.2. {Φ(n) (x, t)}n∈N is pre-compact in Lloc (R2 , Rm ). Proof. The idea of the proof comes from the works of [7, Lemma 5] and [14, Lemma 2.5]. It is clear that one (n) need to prove that {Φi (x, t)}n∈N is pre-compact in Lloc (R2 , R) for i = 1, · · · , m. Given any i ∈ {1, · · · , m}. (n) (n) (n) Since 0 ≤ Φi (x, t) ≤ max βi (t) and Φi (x, t) is monotone in x, we know that {Φi (·, t) : n ∈ N, t ∈ R} is t≥0

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19

pre-compact in Lloc (R, R). Then, for any r > 0, there exists a continuous and non-decreasing function zr (·) which satisfying zr (0) = 0 and 

(n)

(n)

|Φi (x + x, t) − Φi (x, t)|dx ≤ zr (|x|), ∀n ∈ N and t ∈ R.

(4.1)

|x|≤r

We may assume that zr1 (·) ≤ zr2 (·) for r1 ≤ r2 . Thus, to prove this claim, it suffices to prove that for any r > 0, there exists a continuous and nondecreasing function z¯r (·) which satisfies z¯r (0) = 0 and 

(n)

(n)

|Φi (x, t + t) − Φi (x, t)|dx ≤ z¯r (|t|), ∀n ∈ N and t ∈ R.

(4.2)

|x|≤r

Let δ(·) be an infinitely differentiable function on R with +∞  δ(·) ≥ 0, δ(s) = 0 for |s| ≥ 1 and δ(s)ds = 1. −∞

Given r > 0. For any h ∈ (0, r), we define  ϑ(x) :=

(n)

(n)

sign(Φi (x, t + t) − Φi (x, t)), if |x| ≤ r − h, 0, if |x| > r − h, +∞ 

ρ(x) = ϑh (x) := −∞

1 x−y δ( )ϑ(y)dy, x ∈ R. h h

 Clearly, suppρ ⊆ {x ∈ R|x| ≤ r}, |ρ(·)| ≤ 1 and |ρ (·)| ≤ C1 /h, where C1 > 0 is a constant independent of n and h. Note that Φ(n) (x, t) satisfies

∂Φi (x, t) ∂Φ (x, t)  (n) + cn i = σi (t) − Φi (x, t) μi,k (t) ∂t ∂x (n)

m

(n)

k=1

−

(n) νi (t)Φi (x, t),



(n)

Φk (x − y, t)pi,k (y)dy R

i = 1, · · · , m.

For any h ∈ (0, min{1, r/2}], we have 

(n)

(n)

ρ(x)[Φi (x, t + t) − Φi (x, t)]dx |x|≤r



t+ t 

(n)

ρ(x)(Φi )t (x, t)dtdx

= |x|≤r



t t+ t 

 (n) (n) ρ(x) − cn (Φi )x + νi (t)Φi

= |x|≤r

t m 

(n) + σi (t) − Φi (z, t) μi,k (t) k=1

 R

 (n) Φk (z − y, t)pi,k (y)dy dtdx

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20 t+ t 



 (n) (n) cn ρ (x)Φi + νi (t)Φi ρ(x)

= |x|≤r

t

m 

(n) + ρ(x) σi (t) − Φi (z, t) μi,k (t) k=1 t+ t 



≤ C2

 (n) Φk (z − y, t)pi,k (y)dy dtdx

R

  |ρ (x)| + |ρ(x)| dxdt

|x|≤r

t

≤ |t|



C2 (C1 + h) C2 (C1 + 1) ≤ |t| . 2 h h2

According to [14, Lemma 2.1] (see also [7]), for any h ∈ (0, min{1, r/2}], we obtain 

(n)

(n)

|Φi (x, t + t) − Φi (x, t)|dx |x|≤r



(n)

≤

(n)

|Φi (x, t + t) − Φi (x, t)|dx + C3 h |x|≤r−h



(n)

≤

(n)

ρ(x)[Φi (x, t + t) − Φi (x, t)]dx + C4 zr−h (h) + C3 h |x|≤r−h



(n)

(n)

ρ(x)[Φi (x, t + t) − Φi (x, t)]dx + C4 zr (h) + C3 h

≤ |x|≤r

≤ C5

 |t| + zr (h) + h , 2 h

where C3 and C4 are constants independent of h, n, t and |t|, and C5 := max{C2 (C1 + 1), C3 , C4 }. Then it follows that 

(n)

(n)

|Φi (x, t + t) − Φi (x, t)|dx ≤ C5 |x|≤r

min

h∈(0,min{1, r2 }]

 |t| + zr (h) + h =: z¯r (|t|). 2 h

This proves (4.2). The proof is completed. 2 Now, we give the proof of Theorem 4.1. (n)

(n)

Proof of Theorem 4.1. Since Φi (−∞, t) = 0 and Φi (+∞, 0) = βi (0), we can choose ξn such that (n) (n) Φi (ξn , 0) = βi2(0) . Define W (n) (·, t) = Φi (· + ξn , t). Then, W (n) (0, 0) =

βi (0) . 2

(4.3)

By Lemma 4.2, there exist a function W (·, ·) = (W1 (·, ·), · · · , Wm (·, ·))T ∈ Lloc (R2 , Rm ) and a subsequence of {W (n) (·, ·)}, still denoted by {W (n) (·, ·)}, such that W (n) (x, t) → W (x, t) for a.e. (x, t) ∈ R2 .

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Thus, there is a set D0 ⊆ R with m(R \ D0 ) = 0 such that for any t ∈ D0 , W (n) (x, t) → W (x, t) for a.e. x ∈ R.

(4.4)

Using Helly’s theorem (see [4, P165]) and the monotonicity of W (n) (x, t) in x, we may assume that 0 ∈ D0 and for each t ∈ R \ D0 , there is {nl } ⊆ {n} such that W (nl ) (x, t) → W (x, t) for a.e. x ∈ R.

(4.5)

Due to the boundedness of W (n) (·, ·), we may assume that W (nl ) (0, 0) → W (0, 0) and hence W1 (0, 0) = β1 (0)/2. Moreover, we may assume that W (x, t) is non-decreasing in x. Let u(n) (x, t) := W (n) (x + c∗ t, t) ¯ (x, t) := W (x + c∗ t, t). It is clear that and W (n)

(n)

∂u (x, t) ∂ui (x, t) + (cn − c∗ ) i ∂t ∂x  m

 (n) (n) (n) = σi (t) − ui (x, t) μi,k (t) uk (x − y, t)pi,k (y)dy − νi (t)ui (x, t). k=1

(4.6)

R

From (4.4) and (4.5), for given 0 < t < ∞, there exists {nl } ⊆ {n} such that for any s ∈ (D0 ∩ [0, t]) ∪ {0, t}, ¯ (x, s) for a.e. x ∈ R. u(nl ) (x, s) → W

(4.7)

By (4.6) and (4.7), using the dominated convergence theorem and Fubini theorem for Lebesgue integrals, for any s(·) ∈ C ∞ (R) with compact support, we have  ¯ i (x, t)s(x)dx W R



= lim

(nl )

ui

l→∞

(x, t)s(x)dx

R

  t  (n ) ∂u l (x, s) (nl ) (n ) = lim + νi (s)ui l (x, s) ui (x, 0)s(x) + s(x) − (cnl − c∗ ) i l→∞ ∂x R

0

m 

(n ) + σi (s) − ui l (x, s) μi,k (s) k=1

 = lim

(nl )

ui

l→∞

t

 

l→∞ 0



 (x − y, s)pi,k (y)dy ds dx

R

   (n ) (n ) (cnl − c∗ )ui l (x, s)s (x) + s(x)νi (s)ui l dxds

0

R

l→∞

+ lim

(nl )

uk

t (x, 0)s(x)dx + lim

R



m

 (n ) μi,k (s) s(x) σi (s) − ui l (x, s) k=1

R



(nl )

uk

 (x − y, s)pi,k (y)dy dxds

R

¯ i (x, 0)s(x)dx W

= R

t  + 0

R

 m  

¯ ¯ k (x − y, s)pi,k (y)dy + νi (s)W ¯ i dxds. s(x) σi (s) − Wi (x, s) μi,k (s) W k=1

R

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Hence,

¯ i (x, t) = W ¯ i (x, 0) + W

t   ¯ i (x, s) + σi (s) − W ¯ i (x, s) νi (s)W 0

×

m



μi,k (s)

k=1

 ¯ k (x − y, s)pi,k (y)dy ds, for a.e. x ∈ R. W

(4.8)

R

Define ¯ (x + h, t) = lim W (x + c∗ t + h, t) W − (x, t) := lim− W − h→0

h→0

−

and Φ(x + c∗ t, t) := W (x, t) for all (x, t) ∈ R2 . ¯ (x, t) in x that W − (x, t) satisfies (4.8) for any x ∈ R, Then it follows from (4.8) and the monotonicity of W t > 0. Let w(x, t) be the solution of the following initial value problem: ⎧ ⎨

∂ui (x,t) ∂t

 m  = σi (t) − ui (x, t) k=1 μi,k (t) R uk (x − y, t)pi,k (y)dy − νi (t)ui (x, t),

⎩ u(x, 0) = W − (x, 0), i = 1, · · · , m.

(4.9)

By the uniqueness of the solution of (4.9), we see that w(x, t) = W − (x, t). This implies that W − (x, t) is an entire solution of (1.4). Since Φ(n) (·, t) = Φ(n) (·, t + T ), one can see that Φ(·, t) = Φ(·, t + T ), ∀t ∈ R. It remains to prove the boundary behaviors of Φ(·, t) at ±∞. It is easy to see that γ ± (t) := Φ(±∞, t) both exists, γ ± (t) ∈ [0, β(t)] and are periodic solutions of the system:

dvi (t)  = σi (t) − vi (t) μi,k (t)vk (t) − νi (t)vi (t), i = 1, · · · , m. dt m

(4.10)

k=1

Since W (x, t) is non-decreasing in x, it follows that Φ1 (0, 0) = W1− (0, 0) = lim− W1 (h, 0) ≤ W1 (0, 0) = β1 (0)/2, h→0

and Φ1 (1, 0) = W1− (1, 0) = lim− W1 (1 + h, 0) ≥ W1 (0, 0) = β1 (0)/2. h→0

Then, γ1− (0) ≤ β1 (0)/2 and γ1+ (0) ≥ β1 (0)/2. By Lemma 2.2, system (4.10) has exactly two periodic solutions 0 and β(t) on [0, β(t)]. Thus, Φ(−∞, t) = 0 and Φ(+∞, t) = β(t). This completes the proof of Theorem 4.1. 2 Remark 4.3. It should be mentioned that the periodic traveling wave solutions obtained in Theorems 3.3 and 4.1 are only semi-continuous in x. The continuity of such traveling waves remains open. Another interesting problem is the stability of the traveling wave solutions. We leave these problems for future research. References [1] N.D. Alikakos, P.W. Bates, X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc. 351 (1999) 2777–2805.

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[2] S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual, P. Rohani, Seasonality and the dynamics of infectious diseases, Ecol. Lett. 9 (2006) 467–684. [3] K.J. Brown, J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc. 81 (1977) 431–433. [4] J.L. Doob, Measure Theory, Springer-Verlag, New York, 1994. [5] F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl. 89 (2008) 355–399. [6] F. Hamel, L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. 13 (2011) 345–390. zkov, First order quasilinear equations in several independent variables, Math. USSR, Sb. 10 (1970) 217–243. [7] S.N. Kru˘ [8] B. Li, M.A. Lewis, H.F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol. 58 (2009) 323–338. [9] B. Li, L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity 24 (2011) 1759–1776. [10] X. Liang, Y. Yi, X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations 231 (2006) 57–77. [11] X. Liang, X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007) 1–40, Erratum: 61 (2008) 137–138. [12] L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, Math. Surveys Monogr., vol. 102, Amer. Math. Soc., Providence, RI, 2003. [13] Nar Rawal, W. Shen, A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst. 35 (2015) 1609–1640. [14] W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal. 54 (2003) 319–339. [15] W. Shen, A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal. 19 (2012) 73–101. [16] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., vol. 41, Amer. Math. Soc., Providence, RI, 1995. [17] F.-B. Wang, A periodic reaction–diffusion model with a quiescent stage, Discrete Contin. Dyn. Syst. Ser. B 17 (2012) 283–295. [18] H.F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982) 353–396. [19] P. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations 229 (2006) 270–296. [20] S.-L. Wu, G.-S. Chen, Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model, Nonlinear Anal. Real World Appl. 36 (2017) 267–277. [21] S.-L. Wu, C.-H. Hsu, Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction– diffusion systems, J. Differential Equations 258 (2015) 1058–1105. [22] F. Zhang, X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl. 325 (2007) 496–516. [23] F. Zhang, X.-Q. Zhao, Spreading speed and traveling waves for a spatially discrete epidemic model, Nonlinearity 21 (2008) 97–112. [24] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [25] 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. [26] G. Zhao, S. Ruan, Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems, J. Differential Equations 257 (2014) 1078–1147.