SPREADING SPEEDS AND TRAVELING WAVES FOR SPACE-TIME PERIODIC NONLOCAL DISPERSAL COOPERATIVE SYSTEMS

. The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habi- tats with nonlocal dispersal. It is assumed that the trivial solution u = 0 of such a system is unstable and the system has a stable space-time periodic pos- itive solution u ∗ ( t,x ). We ﬁrst show that in any direction ξ ∈ S N − 1 , such a system has a ﬁnite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed c ∗ ( ξ ) in the direction of ξ . Next, we show that for any c > c ∗ ( ξ ), there are space-time periodic traveling wave solutions of the form u ( t,x ) = Φ ( x − ctξ,t,ctξ ) connecting u ∗ and 0 , and propagating in the direction of ξ with speed c , where Φ( x,t,y ) is periodic in t and y , and there is no such solution for c < c ∗ ( ξ ). We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.

1. Introduction. The present paper is concerned with the spatial spreading speeds and traveling wave solutions of the following nonlocal dispersal cooperative system in space-time periodic habitats, ∂u ∂t (t, x) = R N k(y − x)u(t, y)dy − u(t, x) + F(t, x, u(t, x)), x ∈ R N , (1.1) where k : R N → R is a C 1 nonnegative convolution kernel with compact support, and satisfies k(0) > 0 and R N k(z)dz = 1; the vector-valued function u(t, x) = (u 1 (t, x), . . . , u K (t, x)) represents the densities at the point (t, x) ∈ R × R N ; and F(t, x, u) = (F 1 (t, x, u), . . . , F K (t, x, u)) is the reaction term. The following hypotheses are standard.
(H1): F(t, x, 0) = 0 for any (t, x) ∈ R × R N . For each i = 1, . . . , K, F i (t, x, u) is C 1 in (t, x) ∈ R × R N and C 2 in u ∈ R K , and is periodic in (t, x) with period (T, P ) := (T, p 1 , p 2 , . . . , p N ), that is,  (H3): F(t, x, u) is cooperative in the sense that for any 1 ≤ i = j ≤ K, Here are some biological interpretations of the hypotheses (H1)-(H3). The space-time periodicity of F(·, ·, u) in (H1) indicates that the underlying environment of (1.1) is subject to space-time periodic variations. Note that a (T, P )periodic solution u * : R × R N → (0, ∞) K of (1.1) is referred to as a coexistence state in literature. The hypothesis (H2) means that (1.1) has a unique coexistence state which is globally stable with respect to strictly positive perturbations. The hypothesis (H3) indicates that the K species described by system (1.1) are cooperative.
System (1.1) is a nonlocal dispersal counterpart of the following system with random dispersal, Systems (1.1) and (1.2) model the population dynamics of a family of species with internal interaction or dispersal between individuals. Note that (1.2) is often used to model the evolution of population densities of cooperative species in which the internal interaction or movement of the individuals occurs randomly between adjacent spatial locations and is described by the differential operator u → ∆u, referred to as the random dispersal operator. System (1.1) arises in modelling the evolution of population densities of cooperative species in which the internal interaction or movement of the individuals occurs between non-adjacent spatial locations and is described by the integral operator u → R N k(y − x)u i (t, y)dy − u i (t, x), referred to as the nonlocal dispersal operator.
Among central dynamical issues in (1.1) and (1.2) are spatial spreading speeds and traveling wave solutions. A huge amount research has been carried out toward the spatial spreading speeds and traveling wave solutions of (1.2). See, for example, [4,8,11,12,13,16,17,19,20,23,24,25,27,28,29,46] for the study of (1.2) in space-time independent habitats, and [1,9,30,31,32,43,48,49] for the study of (1.2) in time periodic or space-time periodic habitats. We point out that, very recently, Fang, Yu and Zhao in [10] established the existence of spreading speeds and traveling wave solutions for abstract space-time periodic monotone semiflows with monostable structures. The abstract results in [10] can be applied to two species competitive reaction-advection-diffusion system with space-time periodic coefficients.
There are also several studies on the spatial spreading speeds and traveling wave solutions of some special cases of (1.1). See, for example, [14,18,26,33,34,35] and references therein for the study of the existence of traveling wave solutions of (1.1) in the space-time independent case, and [5,6,7,15,37,38,40,41,42] and references therein for the study of spectral theory of nonlocal dispersal operators and traveling wave solutions of nonlocal dispersal equations in space periodic habitats. Very recently, in [2], the authors established the existence, uniqueness and stability of periodic traveling wave solutions to nonlocal dispersal two species competitive systems with space periodic coefficients. Kong et al. [22] studied the spreading speeds of two species competitive systems with nonlocal dispersal and space-time periodic coefficients. However, these results can not be applied to the coupled system (1.1) with multiple variables directly. Due to the lack of compactness of solutions of (1.1), the abstract results on spreading speeds and traveling wave solutions established in [10] also can not be applied to (1.1). It is the objective of the present paper to carry out a study on spreading speeds and space-time periodic traveling wave solutions of (1.1).

Let
X p (d) = u ∈ C(R N , R d ) : u i (· + p l e l ) = u i (·), l = 1, . . . , N, i = 1, . . . , d , When d = K, we write X p = X p (K), X + p = X + p (K) and X ++ p = X ++ p (K). Consider the linearization of (1.1) at the zero solution 0, namely, Note that, for given µ ∈ R and ξ ∈ S N −1 := ξ ∈ R N : |ξ| = 1 , solutions of (1.5) of the form v(t, x) = e −µ(x·ξ−ct) φ(t, x) with φ ∈ X ++ p (if exist) play an important role in the study of spreading speeds and traveling wave solutions of (1.1) in the direction of ξ. Note also that, for such solutions (if exist), φ(t, x) and λ = µc satisfy x), x ∈ R N φ(· + T, ·) = φ(·, · + p l e l ) = φ(·, ·), l = 1, 2, · · · , N. (1.7) Let I be the identity map. Define the map K ξ,µ by setting where the kernel k is as in (1.1). The existence of φ ∈ X ++ p and λ ∈ R satisfying (1.7) is then related to the existence of the principal eigenvalue of −∂ t +K ξ,µ −I+A 0 in X p (see Definition 2.2 for the definition of the principal eigenvalue). Note that we do not specify the spaces on which the operators I and K ξ,µ are defined, but this should not cause any trouble.
Throughout this paper, we also assume that (H4): Let A 0 (t, x) be as in (1.6).
(a) For any (t, x) ∈ R × R N , the matrix A 0 (t, x) is quasi-positive (i.e., offdiagonal entries are nonnegative), and is in a block lower triangular form, namely, there is at least one nonzero entry to the left of each diagonal block of A 0 (t, x) other than the first block.
The strong irreducibility of B(t, x) implies that any limiting matrix of B(t, x) is irreducible, that is, if B * = lim n→∞ B(t n , x n ), then B * is irreducible. In [3], under the assumption that A 01 (t, x) is cooperative and strongly irreducible on R × R N , some criteria for the existence of the principle eigenvalue of −∂ t +K ξ,µ −I+A 01 were established (also see Proposition 2.3). The strong irreducibility of A 01 is implicitly used in the assumption (H4)(2) since if A 01 is not strongly irreducible, it may not make sense to assume λ 1 (ξ, µ, A 01 ) is the principal eigenvalue of the operator −∂ t + K ξ,µ − I + A 01 . (3) When A 0 (t, x) is independent of t and x, it is called the Frobenius form, in which all the diagonal blocks are irreducible, see [46]. Note that Weinberger et al. studied in [46] the spreading speeds and linear determinacy of the discrete-time recursion system u n+1 = Q[u n ], while Hu et al. studied in [18] the spreading speeds and traveling wave solutions of the cooperative system (1.1) when A 0 (t, x) is independent of t and x. It should be pointed out that (H4) is much more general than that in [ Regarding the existence of the principal eigenvalue of −∂ t + K ξ,µ − I + A 0 , we prove • (H4) implies that λ 1 (ξ, µ, A 01 ) is the principal eigenvalue of −∂ t +K ξ,µ −I+A 0 acting on X p , that is, −∂ t +K ξ,µ −I+A 0 has an eigenfunction φ(t, x; µ, ξ) 0 corresponding to λ 1 (ξ, µ, A 01 ) (see Proposition 2.4 for more details). Spatial spreading speeds from u * to 0 and traveling wave solutions connecting u * and 0 are among the most interesting dynamics of (1.1). Roughly, for any given is called the spreading speed interval of (1.1) from u * to 0 in the direction of ξ if for any u 0 ∈ X + satisfying 0 ≤ u 0 u * (0, ·), u 0 (x) = 0 for x · ξ 1 and lim inf x·ξ→−∞ u 0 (x) 0, there holds where φ * (t, x) = φ(t, x; µ * , ξ) with µ * satisfying Then, µ is the spreading speed of (1.1) in the direction of ξ (see Theorem 3.2 for details). Let ξ ∈ S N −1 . Roughly, an entire positive solution u(t, x) of (1.1) is called a traveling wave solution of (1.1) connecting u * and 0 propagating in the direction of ξ with speed c if there is bounded measurable function Φ : R N × R × R N → R K such that, for any i = 1, . . . , K, uniformly in (t, z) ∈ R × R N (see Definition 4.1 for details). Among others, assume (H1)-(H4), then for given ξ ∈ S N −1 , we prove the following results about traveling wave solutions of (1.1) (see Theorem 4.1 for details).
We point out the followings. First, for (1.2) in the case F(t, x, u) ≡ F(u), Weinberger et al. studied in [46] the weakly coupled reaction-diffusion system and provided conditions ensuring that the reaction-diffusion system has a spreading speed and is linearly determined, see [46,Theorem 4.2]. Our assumptions for the linear determinacy of spreading speeds of (1.1) in the case that the coupled term F(t, x, u) is independent of t and x are the same as those in [46,18]. Following from the assumptions in [22], it is easy to verify that our assumptions (H1)-(H4) and (1.9) hold for two species competitive system with nonlocal dispersal and the results on spreading speeds and linear determinacy in our work can also be applied to two species competitive system with nonlocal dispersal in space periodic habitats (see Subsection 5.2 for more details).
Second, in this work we establish the existence of space-time periodic traveling wave solutions of (1.1) for c > c * (ξ). It remains open, which remains open even for scalar nonlocal dispersal equations in space-time periodic habitats, whether there are traveling wave solutions propagating in the direction of ξ ∈ S N −1 with speed c = c * (ξ) for (1.1).
Third, we prove the uniqueness and continuity of traveling wave solutions in the case that F(t, x, u) is strictly sub-homogeneous, that is, F(t, x, αu) > αF(t, x, u) for u ∈ (0, u * ], (t, x) ∈ R × R N and α ∈ (0, 1). It remains open whether spacetime periodic traveling wave solutions of (1.1) without the strictly sub-homogeneous condition are continuous and unique. We will further study the spreading speeds and traveling wave solutions of some epidemic models by the results obtained in this paper somewhere else.
The rest of this paper is organized as follows. In Section 2, we state the definition of the principal eigenvalue for space-time periodic nonlocal dispersal operators, establish some useful properties for the principal eigenvalue, and present a comparison principle for (1.1) and some related linear cooperative systems with nonlocal dispersal. In Section 3, we investigate the existence of spreading speed intervals and linear determinacy of spreading speeds. The existence, nonexistence and uniqueness of space-time periodic traveling wave solutions of (1.1) is established in Section 4. In Section 5, we discuss the applications of the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.
2. Preliminary. In this section, we introduce some principal eigenvalue theory for space-time periodic linear cooperative systems with nonlocal dispersal, and present a comparison principle for (1.1) and some related linear cooperative systems with nonlocal dispersal.

Comparison principle.
In this subsection, we present a comparison principle for system (1.1) and the following linear cooperative system with nonlocal dispersal, holds for all t ∈ [0, T ). ( Proof. The proposition follows from the arguments in [41, Proposition 2.1].

2.2.
General principal eigenvalue theory. Let K ξ,µ be as in (1.8). Consider the following eigenvalue problem In the sequel, results in this subsection with different d will be used. We point is a solution of (2.1) with e −µ(y−x)·ξ k(y − x) being replaced by k(y − x) , then λ is an eigenvalue of (2.2) and w = φ(t, x; µ, ξ, A) is a corresponding eigenfunction.
Let σ(L ξ,µ,A ) be the spectrum of L ξ,µ,A on X p (d). Set

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which is spectral bound of the operator L ξ,µ,A . Observe that if µ = 0, then (2.2) is independent of ξ, and hence we set Throughout the rest of this subsection, we assume (A1) and (A2). We note that the principal spectrum point λ(ξ, µ, A) of L ξ,µ,A belongs to σ(L ξ,µ,A ). In general, λ(ξ, µ, A) may not be the principal eigenvalue of L ξ,µ,A and hence L ξ,µ,A may not have a principal eigenvalue. The reader is referred to [5] and [41] for examples in the case d = 1. In the recent paper [3], the first two authors of the current paper established some useful criteria for the existence of a principal eigenvalue of L ξ,µ,A . For example, for any fixed x ∈ R N , let λ(x, A) be the principal eigenvalue (i.e., the eigenvalue with largest real part and with a positive eigenfunction) of Consider the inhomogeneous linear system where B : R × R N → R d is a continuous and (T, P )-periodic vector-valued function.
We have the following proposition on the existence of bounded entire solutions. For given ρ ≥ 0, let . Define the solution operator of (2.1) by ) the spectrum radius of Φ p (T ; ξ, µ, A). It follows from arguments as in [37, Proposition 3.10] that For Then, the uniqueness of solutions of (2.1) with initial function in for all x ∈ R N , l = 1, . . . , N and i = 1, . . . , d. Then, for any i, j = 1, . . . , d, Hence m ij (x − p l e l ; y, dy) = m ij (x; y + p l e l , dy), ∀ i, j = 1, . . . , d.
The following corollary follows from the convexity of λ(ξ, µ, A) in µ and Proposition 2.3.
We remark that, by Proposition 2.4, results proven in Subsection 2.2 apply to the operator L ξ,µ,A0 in Proposition 2.4.

3.
Spreading speeds and linear determinacy. In this section, we investigate the spreading speeds of (1.1) and explore the linear determinacy for the spreading speeds. Recall that u(t, x; u 0 ) is the unique solution of (1.1) with u(0, ·; u 0 ) = u 0 ∈ X.
(1) Under the assumptions (H1)-(H5), we have shown that (1.1) has a spreading speed and it is linearly determined. In particular, our assumptions and results extend the results in Weinberger et al. [46] for discretetime recursion system u n+1 = Q[u n ] to more general nonlocal dispersal cooperative system (1.1) in space-time periodic habitats.
(2) We will apply the results of the present paper on spreading speeds, that is, Theorems 3.1 and 3.2 to two species competitive system with nonlocal dispersal and space-time periodic coefficients.
3.1. Proof of Theorem 3.1. In this subsection, we prove Theorem 3.1. We first present some lemmas. Let us consider the space shifted systems of (1.1), i.e., where z ∈ R N . Let u(t, x; u 0 , z) be the unique solution of (3.3) with u(0, ·; u 0 , z) = u 0 ∈ X. In particular, u(t, x; u 0 , 0) = u(t, x; u 0 ). By assumptions (H1)-(H3) and Proposition 2.1, we have the following lemma. Let K 1 be the dimension of the first diagonal block of A 0 , that is, then for any c < c, there holds lim sup x·ξ≤c t,t→∞ Proof. Due to the stability of u * , for each u 0 ∈ X + (ξ) there holds the convergence |u(t, x; u 0 , z) − u * (t, x + z)| → 0 as t → ∞ for x ∈ R N and z ∈ R N . By (H4), we know that an increase in the first K 1 components will increase all components as time elapses. Hence, the conclusion of the lemma can be shown using the comparison principle and arguments similar to those in [22,Lemma 3.4]. Here we omit the details.
We are now ready to prove Theorem 3.1.
First, we present the following lemma, which provides a lower bound for c * inf (ξ) and plays a crucial role in the proof of Theorem 3.2.
Lemma 3.2. Assume that there exists a space-time periodic K × K matrix A(t, x) satisfying (H4) with A 0 (t, x) being replaced by A(t, x) such that F(t, x, u) ≥ A(t, x)u for 0 ≤ u ≤ β with β ∈ R K and 0 < β i 1, i = 1, . . . , K. Let A 1 (t, x) be the first diagonal block of A(t, x). For any ξ ∈ S N −1 , let λ 1 (ξ, µ, A 1 ) be the principal eigenvalue of L ξ,µ,A1 for any µ > 0. Then Lemma 3.2 can be proven by the strategy which has been used in several papers (see [27,Proposition 3.9], [38,Lemma 4.4], [44, Lemma 9.1]). We provide the proof of Lemma 3.2 in Appendix A for interested readers.
Next, we present a lemma, which will be used in the proof of Theorem of 3.2 to get an upper bound for c * sup (ξ). To this end, for given M > 0, let where µ * is as in (3.1) and λ 1 (µ * ) and φ * (t, x) are as in (3.2). Let x, e −M t u), i = 1, . . . , K.
Remark 4.1. We refer to [47,Definition 2.3.1] for more information about strict sub-homogeneity. When F(t, x, u) = uf (t, x, u), if ∂f ∂u (t, x, u) < 0 for (t, x) ∈ R×R N and f (t, x, u) < 0 for t ∈ R, x ∈ R N and u 1, we know from [38] that the strictly sub-homogeneous condition holds for the following nonlocal monostable equation and our results can be applied to the above nonlocal dispersal equation (see Subsection 5.1).

Sub-and super-solutions.
In this subsection, we construct sub-and supersolutions of some equations related to (3.3) that are used in the proof of Theorem 4.1. Throughout the rest of this section, we assume (H1)-(H5).
Proposition 4.4. The following statements hold.
(2) For any i = 1, . . . , K, there is a constant C such that Proof.
(1) It follows from similar arguments as in Lemma 3.3.

4.2.
Proof of Theorem 4.1. In this subsection, we investigate the existence and uniqueness of traveling wave solutions of (1.1), that is, we prove Theorem 4.1. For convenience, we set u − (t, x; z) = u − (t, x; z, d, ρ 1 ). Let u ± 0,z be given in Proposition 4.3 and Proposition 4.4.
Lemma 4.1. Let u n (t, x, z) = (u n 1 (t, x, z), . . . , u n K (t, x, z)) and u n (t, x, z) = (u n1 (t, x, z), . . . , u nK (t, x, z)) be defined by . . , K. Then, for any given bounded interval I ⊂ R, there is N 0 ∈ N such that u n (t, x, z) is non-increasing in n for n ≥ N 0 and u n (t, x, z) is non-decreasing in n for n ≥ N 0 , t ∈ I, x ∈ R N and z ∈ R N .
(1) follows directly from the definition of Φ ± and (2) follows from Propositions 4.1 and 4.4.
Next, we prove the main results of Theorem 4.1.
Proof of Theorem 4.1.
(1) Let Φ = Φ + . It suffices to prove that Φ generates a traveling wave solution of (1.1) with speed c in the direction of ξ. First, it follows from Lemma 4.2 that Φ i satisfies (4.1) for any i = 1, . . . , K. On the other hand, Φ(x, t, z) is periodic in space x and time t, that is, for any x, x ∈ R N with x · ξ = x · ξ (see also [38,Theorem 5.1]), which imply (4.2) and (4.3) hold true. Next, we prove that Note that there is N 0 ∈ N such that for t ∈ [0, T ] and n ≥ N 0 , x + z) for t ∈ R and x, z ∈ R N , i = 1, . . . , K. Then, by Lemma 3.1, for any > 0 and c < 0, there is N * ∈ N with N * ≥ N 0 such that T ] and x · ξ ≤ c (N * + 1)T . Then, (4.19) follows from (4.20) and (4.2), and hence, Φ generates a traveling wave solution of (1.1) in the direction of ξ with speed c.
Let c , c ∈ (c 1 , c * (ξ)) with c > c . From Lemma 3.1, we know that lim sup x·ξ≤c t,t→∞ Since u(t, x) is the solution of (1.1) with speed c 1 ∈ (0, c * (ξ)) connecting u * and 0, which leads to a contradiction. Hence, there is no traveling wave solution of (1.1) connecting u * and 0 in the direction of ξ with speed c < c * (ξ).
(3) It follows from standard arguments using strict sub-homogeneity and some trick using the decay rate of Φ as x → ∞ given in (1). We refer the read to the proof in [39, Theorem 2.2] for more details.

5.
Application. In this section, we discuss the applications of the results obtained in Sections 2-4 to a nonlocal KPP equation and a two-species competitive system with nonlocal dispersal.

A nonlocal KPP equation.
In this subsection, we consider the applications of the results obtained in Sections 2-4 to the following nonlocal monostable equation in space-time periodic habitats, where k(·) is the same as in (1.1).
Then, the following theorem, which recovers the results of [38] for spreading speeds and traveling wave solutions of (5.1), holds. is the spreading speed of (5.1) in the direction of ξ, and for any c > c * (ξ), (5.1) has a continuous periodic traveling wave solution u(t, x) = Φ(x − ctξ, t, ctξ) connecting u * (t, x) and 0 in the direction of ξ.
It is easy to see that v(s, z) is continuous in (s, z) and each of its component is nonincreasing in s and vanishes for s ≥ 0. Let η := min z∈R,1≤i≤K1 −k i (µ, z) + τ i .