Spreading Speeds for Monostable Equations with Nonlocal Dispersal in Space Periodic Habitats

Transcription

1 Spreading Speeds for Monostable Equations with Nonlocal Dispersal in Space Periodic Habitats Wenxian Shen and Aijun Zhang Department of Mathematics and Statistics Auburn University Auburn University, AL U.S.A. Partially supported by NSF grant DMS

2 Abstract. The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established. Key words. Monostable equation; nonlocal dispersal; random dispersal; periodic habitat; spreading speed; spreading speed interval; traveling wave solution; principal eigenvalue; principal eigenfunction; variational principle. Mathematics subject classification. 45C05, 45G10, 45M20, 47G10, 92D25. 2

3 1 Introduction The current paper is devoted to the study of spatial spreading dynamics of the following monostable equation with nonlocal dispersal, u t = k δ (y x)u(t, y)dy u(t, x) + u(t, x)f(x, u(t, x)), x, (1.1) where δ > 0 represents the nonlocal dispersal distance and k δ ( ) is a symmetric convolution kernel supported on a ball of radius δ. More precisely, k δ ( ) is given by k δ (z) = 1 k( z ) δ N δ for all z and some C 1 function k( ) : R + which satisfies k(z) = k( z) for z, k(z) > 0 if z < 1, k(z) = 0 if z 1 ( denotes the norm in ), and k(z)dz = 1. A standard example is { ( ) C exp 1 for z < 1 z k(z) = 2 1 (1.2) 0 for z 1, where C > 0 is chosen such that k(z)dz = 1. The function f(x, u) is smooth in (x, u) R, periodic in x i with period p i (p i > 0, i = 1, 2,, N) (i.e. f( + p i e i, ) = f(, ), e i = (δ i1, δ i2,, δ in ), δ ij = 1 if i = j and 0 if i j, i, j = 1, 2,, N), and satisfies the following monostablility assumptions: (H1) f(x,u) u < 0 for x and u R and f(x, u) < 0 for x and u 1. (H2) There exists a continuous function u + : R + \ {0} satisfying that u + (x) is periodic in x i with period p i (i = 1, 2,, N) and u = u + (x) is an equilibrium solution of (1.1). Throughout this paper, a function h : R R is said to be smooth if h(x, u) is C N in x and C 1 in u R. To see the monostable feature of the assumptions (H1)-(H2), let X p = {u C(, R) u( + p i e i ) = u( ), i = 1,, N} (1.3) with norm u Xp = sup x u(x), and X + p = {u X p u(x) 0 x }. (1.4) Let I be the identity map on X p, and K δ, a 0 ( )I : X p X p be defined by ( Kδ u ) (x) = k δ (y x)u(y)dy, (1.5) (a 0 ( )Iu)(x) = a 0 (x)u(x), (1.6) where a 0 (x) = f(x, 0). For convenience, we introduce the following assumption. 3

4 (H3) u 0 is linearly unstable in X p, that is, λ 0 := sup{reλ λ σ(k δ I + a 0 ( )I)} is positive, where σ(k δ I + a 0 ( )I) is the spectrum of the operator K δ I + a 0 ( )I on X p. If (H1) and (H2) hold, then u u + (x) is an asymptotically stable solution of (1.1) with respect to perturbations in X p ; u 0 is linearly unstable in X p (hence (H3) holds); and there are no other equilibrium solutions of (1.1) in X p + except for u = 0 and u = u + (see Proposition 2.4 (1)). Therefore (H1)-(H2) imply that (1.1) has exactly two equilibrium solutions in X p +, i.e. u = 0 and u = u +, and u = 0 is linearly unstable and u = u + is asymptotically stable in X p, which reflects the monostable feature of the assumptions (H1) and (H2). We remark that if λ 0 in (H3) is an isolated algebraically simple eigenvalue of K δ I + a 0 ( )I, then (H1) and (H3) imply (H2) (see Proposition 2.4 (2)). Equation (1.1) is a nonlocal dispersal counterpart of the random dispersal equation u t = u + uf(x, u), x RN. (1.7) Random dispersal is essentially a local behavior which describes the movement of species between adjacent spatial locations and has been widely used to model the population spreading dynamics of species (see [1], [2], [5], [6], [7], [14], [19], [20], [37], [41], [54], [55], and references therein). In contrast, nonlocal dispersal characterizes the movements and interactions of species between non-adjacent spatial locations. As the movements and interactions of many species in biology and ecology can occur between non-adjacent spatial locations, nonlocal dispersal has also been used to model the population spreading dynamics of species by many people (see [3], [15], [16], [17], [18], [31], [32], [35], and references therein). One of the central problems for (1.1) and (1.7) is to understand how fast the population spreads as time evolves. E.g., letting ξ S N 1 := {ξ ξ = 1} and a given initial population u 0 satisfy for some δ 0 > 0 that u 0 (x) δ 0 for x with x ξ 1 and u 0 (x) = 0 for x with x ξ 1 (x ξ is the inner product of x and ξ), how fast does the population invade into the region with no population initially? The spatial spreading dynamics of (1.7) has been extensively studied and is well understood. For example, consider the following special case of (1.7) u t = 2 u + u(1 u), x R, (1.8) x2 which was introduced in papers by Fisher [20] and Kolmogorov, Petrowsky, Piscunov [37] for the evolutionary take-over of a habitat by a fitter genotype. Here u is the frequency of one of two forms of a gene. Fisher in [20] found traveling wave solutions u(t, x) = φ(x ct), (φ( ) = 1, φ( ) = 0) of all speeds c 2 and showed that there are no such traveling wave solutions of slower speed. He conjectured that the take-over occurs at the asymptotic speed 2. This conjecture was proved in [37] by Kolmogorov, Petrowsky, and Piscunov, 4

5 that is, they proved that for any nonnegative solution u(t, x) of (1.8), if at time t = 0, u is 1 near and 0 near, then lim t u(t, ct) is 0 if c > 2 and 1 if c < 2 (i.e. the population invades into the region with no initial population with speed 2). Since the pioneering works by Fisher ([20]) and Kolmogorov, Petrowsky, Piscunov ([37]), the spatial spreading dynamics of (1.7) has been widely studied (see [1], [2], [5], [6], [7], [19], [22], [25], [29], [34], [39], [40], [41], [44], [45], [47], [48], [49], [53], [54], [55], and references therein). The spatial spreading dynamics of (1.8) has been well extended to (1.7). For example, assume (H1) and that u 0 is a linearly unstable solution of (1.7). Then it is known that (1.7) has a unique positive equilibrium u + ( ) X p which is asymptotically stable with respect to perturbations in X p and it has been proved that for every ξ S N 1, there is a c (ξ) R such that for every c c (ξ), there is a traveling wave solution connecting u + and u 0 and propagating in the direction of ξ with speed c, and there is no such traveling wave solution of slower speed in the direction of ξ. The minimal wave speed c (ξ) is of some important spreading properties. For example, let X = {u C(, R) u is uniformly continuous on and sup x u(x) < } (1.9) with norm u X = sup x u(x), and Then and lim inf t X + = {u X u(x) 0 x }. (1.10) inf (u(t, x; u 0) u + (x)) = 0 x ξ ct lim sup t sup x ξ ct u(t, x; u 0 ) = 0 c < c (ξ) c > c (ξ), for all u 0 X + which satisfy for some δ 0 > 0 that u 0 (x) δ 0 for x with x ξ 1 and u 0 (x) = 0 for x with x ξ 1. Here u(t, x; u 0 ) denotes the solution of (1.7) with u(0, x; u 0 ) = u 0 (x) and inf x ξ ct (sup x ξ ct ) denotes the infimum (supremum) taken over all the x satisfying that x ξ ct (x ξ ct) for given ξ S N 1 and c, t R. Hence c (ξ) is also called the spreading speed of (1.7) in the direction of ξ. Moreover, it has the following variational characterization. Let λ(ξ, µ) be the eigenvalue of { u 2µ N i=1 ξ i u x i + (a 0 (x) + µ 2 )u = λu, x (1.11) u(x + p i e i ) = u(x), i = 1, 2,, N x with largest real part, where a 0 (x) = f(x, 0) (it is well known that λ(ξ, µ) is real and algebraically simple. λ(ξ, µ) is called the principal eigenvalue of (1.11) in literature). Then c λ(ξ, µ) (ξ) = inf. (1.12) µ>0 µ 5

6 (See [5], [6], [7], [39], [44], [45], [55] and references therein for the above mentioned properties). Recently, the nonlocal dispersal equation of form (1.1) has also been studied by many authors. See, for example, [3], [17], [23], [31], [32], [35], [52], for the study of spectral theory for nonlocal dispersal operators and the existence, uniqueness, and stability of nontrivial positive stationary solutions. See, for example, [15], [16], [18], [54], [55] for the study of the existence of spreading speeds and traveling wave solutions connecting the trivial solution u = 0 and a nontrivial positive stationary solution. However, most existing works on spatial spreading dynamics of monostable equations with nonlocal dispersal deal with spatially homogeneous equations (i.e. f(x, u) in (1.1) is independent of x). There is little understanding of the spatial spreading dynamics of general nonlocal dispersal monostable equations. One major difference between (1.7) and (1.1) is that the solution operator of (1.7) in the phase space X p or X is compact with respect to the uniform convergence on bounded subsets of (i.e., is compact with respect to open compact topology), whereas the solution operator of (1.1) in X p or X does not exhibit such compactness features. It should be pointed out that Weinberger deals in [54] with spatial spreading dynamics for biological models in spatially homogeneous habitats. His existence and characterization results for spreading speeds do not require the usual compactness properties of the evolution operator, and they therefore apply to (1.1) in case that f(x, u) f(u). On the other hand, his results in [55] for very general population dispersal models in periodic habitats do not cover our situation since Hypotheses 2.1.vi in [55], which is used in the construction of the spreading speed in [55], see [55, page 25, line 19]), is not satisfied by the solutions of (1.1). In fact, Hypotheses 2.1.vi in [55] is a compactness assumption for the evolution operator under consideration with respect to the open compact topology. The objective of the current paper is to carry out a study on the spatial spreading dynamics of (1.1) and to explore the extent to which the existing theories on the spatial spreading dynamics of (1.7) can be extended to (1.1). In particular, we investigate the existence and characterization of spreading speeds of (1.1). It appears to be difficult to adopt the construction method for spreading speeds from [41], [54], [55], etc. in dealing with (1.1) in general due to the lack of compactness of the solution operator and the spatial inhomogeneity of the nonlinearity. Therefore we will employ the natural properties of spreading speeds to give a definition of this concept following an idea from [28], [50], [51] for time almost periodic KPP models and then investigate the existence and characterization of spreading speeds of (1.1). To be more precise, assume (H1) and (H2). In the following, inf x ξ r (sup x ξ r ) represents the infimum (supremum) taken over all the x satisfying that x ξ r for given ξ S N 1 and r R. Similarly, the notations inf x ξ ct, inf x ξ ct, inf x ct (sup x ξ ct, sup x ξ ct, sup x ct ) represent the infima (suprema) taken over all the x satisfying 6

7 the inequalities in the notations for given ξ S N 1 and c, t R. Let For a given ξ S N 1, let X + (ξ) = {u X + sup x u(x) < u + inf u + inf = inf x u + (x). (1.13) lim inf r inf u(x) > 0, x ξ r u(x) = 0 for x with x ξ 1}. (1.14) It follows from the general semigroup approach (see [26] or [46]) that (1.1) has a unique (local) solution u(t, x; u 0 ) with u(0, x; u 0 ) = u 0 (x) for every u 0 X. Moreover, a comparison principle in usual sense holds for solutions of (1.1), and u(t, x; u 0 ) exists for all t 0 if u 0 X + (see Proposition 2.1). Definition 1.1. Assume that (H1) and (H2) are fulfilled and that ξ S N 1. We call a number c (ξ) R the spreading speed of (1.1) in the direction of ξ if the following properties are satisfied: lim inf t inf (u(t, x; u 0) u + (x)) = 0 x ξ ct c < c (ξ) and lim sup t sup x ξ ct u(t, x; u 0 ) = 0 c > c (ξ) for every u 0 X + (ξ). Observe that our definition coincides with the notion of c (ξ) in [54] provided that f(x, u) is independent of x. The construction based definition used in [41], [54], [55] is different in the sense that our definition does not guarantee the existence of c (ξ). In fact, we focus in this paper on investigating the existence and characterization of c (ξ) for ξ S N 1. Among the main techniques employed in the existence proof and for the characterization of spreading speeds of (1.1) are the comparison principle, sub- and super-solutions, and the principal eigenvalue theory of the eigenvalue problem, ( Kδ,ξ,µ I + a( )I ) v = λv, v X p, (1.15) where ξ S N 1, µ R, a(x) is a smooth function periodic in x i with period p i > 0 (i.e. a(x + p i e i ) = a(x)) for i = 1, 2,, N. The operator a( )I has the same meaning as in (1.6) with a 0 ( ) being replaced by a( ), and K δ,ξ,µ : X p X p is defined by (K δ,ξ,µ v)(x) = e µ(y x) ξ k δ (y x)v(y)dy. (1.16) 7

8 We point out the following relation between (1.1) and (1.15): if u(t, x) = e µ(x ξ λ µ t) φ(x) with φ X p \ {0} is a solution of the linearization of (1.1) at u = 0, u t = k δ (y x)u(t, y)dy u(t, x) + a 0 (x)u(t, x), x, (1.17) where a 0 (x) = f(x, 0), then λ is an eigenvalue of (1.15) with a( ) = a 0 ( ) or K δ,ξ,µ I + a 0 ( )I and v = φ(x) is a corresponding eigenfunction. Definition 1.2. Let σ(k δ,ξ,µ I + a( )I) be the spectrum of K δ,ξ,µ I + a( )I on X p. A number λ(δ, ξ, µ, a) R is called the principal eigenvalue of (1.15) or K δ,ξ,µ I + a( )I if it is an algebraically simple eigenvalue of K δ,ξ,µ I + a( )I with an eigenfunction v X + p, and for every λ σ(k δ,ξ,µ I + a( )I) \ {λ(δ, ξ, µ, a)}, Reλ < λ(δ, ξ, µ, a). Similarly, due to the lack of compactness of the semigroup generated by K δ,ξ,µ I + a( )I on X p and the inhomogeneity of a( ), the existence of a principal eigenvalue and eigenfunction of (1.15) cannot be obtained from standard theory (e.g. the Krein-Rutman theorem). We hence first develop a principal eigenvalue theory for (1.15) or K δ,ξ,µ I + a( )I. It should be pointed out that recently the principal eigenvalue problem for nonlocal dispersal has been studied by several authors (see [33], [35], [52], etc.). However, the existing results cannot be applied directly to (1.15). In this paper, we prove Theorem A. There is δ 0 > 0 such that the following properties hold for every 0 < δ δ 0 : (1) the principal eigenvalue λ(δ, ξ, µ, a) of (1.15) exists for all ξ S N 1 and µ R; (2) for each ξ S N 1, λ(δ, ξ, µ, a) is convex in µ; (3) there are m δ > 0 and µ 0 > 0 such that λ(δ, ξ, µ, a) m δ µ 2 for all µ µ 0 and ξ S N 1 ; (4) if λ(δ, ξ, 0, a) > 0, then for every ξ S N 1, there is a µ (ξ) (0, ) such that λ(δ, ξ, µ (ξ), a) µ (ξ) λ(δ, ξ, µ, a) = inf µ>0 µ (1.18) and λ(δ, ξ, µ, a) µ > λ(δ, ξ, µ (ξ), a) µ (ξ) for 0 < µ < µ (ξ). (1.19) Theorem B. (1) If a(x) satisfies that max x a(x) min x a(x) < 1, then for every δ > 0, the conclusions in Theorem A hold. 8

9 (2) If the partial derivatives of a(x) up to order N 1 at some x 0 are zero, where x 0 is such that a(x 0 ) = max x a(x), then for every δ > 0, the conclusions in Theorem A hold. We remark that the proof of Theorem A (1) and of the existence part of the principal eigenvalue in Theorem B rely on techniques from the perturbation theory of Burger [8] (see [8, Proposition 2.1 and Theorem 2.2]) and on the arguments in [35, Theorem 2.6]. However, special care is required in view of the dependence of K δ,ξ,µ on ξ S N 1 and µ R. Note that the conclusions are independent of ξ S N 1 and µ R (i.e. for proper δ > 0 and a, λ(δ, ξ, µ, a) exists for every ξ S N 1 and µ R). Theorem A (1) is proved in [35] for µ = 0, all other results in Theorems A and B are new. As it is well known, the principal eigenvalue of a random or local dispersal operator always exists. By Theorem A, if the nonlocal dispersal operator K δ,ξ,µ I +a( )I is nearly local in the sense that the dispersal distance δ is sufficiently small, then we obtain a similar principal eigenvalue theory as for random dispersal operators. Observe that K δ,ξ,µ : X p X p is a compact and positive operator and Kδ,ξ,µ n is strictly positive for n sufficiently large. If a(x) a is independent of x, then it is not difficult to see that λ(δ, ξ, µ, a) := a 1+ e µδz ξ k(z)dz is the principal eigenvalue of K δ,ξ,µ I +a( )I for all δ > 0, ξ S N 1, and µ R. By Theorem B, if a( ) has certain homogeneity features, then the nonlocal dispersal operator K δ,ξ,µ I + a( )I also possesses a principal eigenvalue for all δ > 0. More precisely, Theorem B (1) shows that if a(x) is nearly globally homogeneous or globally flat in the sense that max x a(x) min x a(x) < 1, then the principal eigenvalue λ(δ, ξ, µ, a) of the nonlocal dispersal operator K δ,ξ,µ I + a( )I exists for all δ > 0, ξ S N 1, and µ R. Note that if K δ,ξ,µ I + a( )I in (1.15) is replaced by ν[k δ,ξ,µ I] + a( )I with a general positive dispersal rate ν > 0, Theorem B (1) holds provided that max x a(x) min x a(x) < ν. Hence biologically the condition max x a(x) min x a(x) < 1 means that the variation in the habitat is less than the dispersal rate of the nonlocal dispersal operator K δ I. We say a( ) is nearly homogeneous or flat in some region where it is most conducive to population growth in the zero-limit population (which will be referred to as nearly locally homogeneous in the following) if all partial derivatives of a(x) up to order N 1 are zero at some x 0 with a(x 0 ) = max x a(x). Theorem B (2) shows that if a( ) is nearly locally homogenous, then for any δ > 0, ξ S N 1, and µ R, the principal eigenvalue λ(δ, ξ, µ, a) of K δ,ξ,µ I + a( )I exists, too. It should be pointed out that a(x) is nearly globally homogeneous may not imply that it is nearly locally homogeneous. Clearly, the flatness condition for a(x) in Theorem B (2) is always satisfied for N = 1 or 2. Hence when N = 1 or 2, the principal eigenvalue of K δ,ξ,µ I + a( )I exists for all ξ S N 1, µ R, and δ > 0. In general, if N 3 and δ is not small, the principal eigenvalue of (1.15) may not exist (see example in section 6). This reveals an 9

10 essential difference between nonlocal dispersal operators and random dispersal operators. As mentioned in [33], it is not clear what the biological implications of the above dimension issue are, and further investigation is warranted. Theorems A and B will play an important role in the investigation of spreading speeds of (1.1) and are also of independent interest, for example, they provide some basic tools for the study of persistence and invasion in two species competition systems with nonlocal dispersal (see [32], [35]). Note that if µ = 0, (1.15) is independent of ξ and hence we put λ(δ, a) := λ(δ, ξ, 0, a) ξ S N 1 (1.20) (if λ(δ, ξ, 0, a) exists). In terms of the principal eigenvalue, we make the following assumption: (H4) The principal eigenvalue λ(δ, ξ, µ, a 0 ) of (1.15) exists for every ξ S N 1 and µ 0, where a 0 (x) = f(x, 0). To state our main results on the spreading speed of (1.1), consider the space shifted equation of (1.1) u t = k δ (y x)u(t, y)dy u(t, x) + u(t, x)f(x + z, u(t, x)), x (1.21) where z. Let u(t, x; u 0, z) be the solution of (1.21) with u(0, x; u 0, z) = u 0 (x) for u 0 X. Applying the principal eigenvalue theory for (1.15) and the comparison principle for solutions of (1.1) together with some sub- and super-solutions of (1.1), we obtain the following existence theorem and variation principle for spreading speeds of (1.1). Theorem C. (Existence and symmetry of spreading speeds) Assume (H1) and (H3). For a given δ > 0, assume also (H4). Then (H2) is satisfied and the following properties hold: (1) the spreading speed c (ξ) of (1.1) in the direction of ξ S N 1 exists for every ξ S N 1 and c λ(δ, ξ, µ) (ξ) = inf, µ>0 µ where λ(δ, ξ, µ) is the principal eigenvalue of (1.15) with a(x) = f(x, 0); (2) c (ξ) = c ( ξ) for every ξ S N 1 ; (3) for every u 0 X + (ξ) and c < c (ξ), lim inf t inf (u(t, x; u 0, z) u + (x + z)) = 0 uniformly in z ; x ξ ct 10

11 (4) for every u 0 X + (ξ) and c > c (ξ), lim sup t sup x ξ ct u(t, x; u 0, z) = 0 uniformly in z. The following two theorems are about the spreading features of the spreading speeds. Theorem D. (Spreading features of spreading speeds) Suppose that the conditions in Theorem C hold. (1) If u 0 X + satisfies that sup x u 0 (x) < u + inf and u 0 (x) = 0 for x with x ξ 1, then for each c > c (ξ), lim sup t sup x ξ ct u(t, x; u 0, z) = 0 uniformly in z. (2) Assume that ξ S N 1 and 0 < c < c (ξ). Then for each σ > 0, there is r σ > 0 such that lim inf t inf (u(t, x; u 0, z) u + (x + z)) = 0 uniformly in z x ξ ct for every u 0 X + satisfying u 0 (x) σ for all x with x ξ r σ. Theorem E. (Spreading features of spreading speeds) Suppose that the conditions in Theorem C hold. (1) If u 0 X + satisfies that sup x u 0 (x) < u + inf and u 0 (x) = 0 for x with x 1, then lim sup t sup x ct for all c > sup ξ S N 1 c (ξ). u(t, x; u 0, z) = 0 uniformly in z. (2) Assume that 0 < c < inf ξ S N 1{c (ξ)}. Then for any σ > 0, there is r σ > 0 such that lim inf t inf (u(t, x; u 0, z) u + (x + z)) = 0 uniformly in z x ct for every u 0 X + satisfying u 0 (x) σ for x with x r σ. 11

12 We remark that (H1) and (H3) in Theorems C-E are quite natural assumptions. If the nonlocal dispersal in (1.1) is nearly local or the inhomogeneity in (1.1) is nearly globally homogeneous or nearly locally homogeneous, then by Theorems A and B, (H4) is satisfied, and hence by Theorems C, D, and E, (1.1) possesses similar spatial spreading properties as those of the random dispersal equation (1.7). It remains open whether in general the spreading speed c (ξ) of (1.1) exists for any δ > 0 and ξ S N 1. We show in section 4 that (1.1) has a so-called spreading speed interval [c inf (ξ), c sup(ξ)] in the direction of ξ for every δ > 0 and ξ S N 1 (see Definition 4.1 and Corollary 4.1). The spreading speed interval [c inf (ξ), c sup(ξ)] provides upper and lower bounds for the spreading speed c (ξ) if it exists and c (ξ) exists if and only if [c inf (ξ), c sup(ξ)] is a singleton (see the proof of Theorem C in section 5.) Theorems C-E extend the spreading speed theory for (1.7) to (1.1) and establish some fundamental theories for the further study of the spreading and propagating dynamics of (1.1). The next natural and important problems to address include the existence, uniqueness, and stability of traveling wave solutions of (1.1) in the direction of ξ connecting u + and u with speed c c (ξ) (in case that c (ξ) exists), which will be explored elsewhere, and the effects of spatial variations and dispersal strategies on the spreading speeds, which is investigated in [27]. Note that in [10], [11], [24], [30], [56], [61], etc., the authors studied the spreading and propagating dynamics for discrete monostable equations of the following form, u i,j = ν[u i 1,j + u i+1,j + u i,j 1 + u i,j+1 4u i,j ] + u i,j f((i, j), u i,j ). (1.22) Some results in [54] and [55] can also be applied to the above discrete monostable equation. The reader is referred to [4], [9], [12], [13], [19], [38], [42], [43], [57], [58], [59], [60], and references therein for the study of the front propagation dynamics of (1.1), (1.7), and (1.22) when the nonlinear term is of bistable type. The rest of the paper is organized as follows. In section 2, we establish some basic properties of solutions of equation (1.1) and some related nonlocal linear equations for the use in later sections. We investigate the eigenvalue problem (1.15) and prove Theorems A and B in section 3. We introduce in section 4 the notion of the spreading speed interval [c inf (ξ), c sup(ξ)] of (1.1) in the direction of ξ and explore some basic properties of the spreading speed intervals. Spreading speeds of (1.1) are investigated and Theorems C- E are proved in section 5. In section 6, we provide an example which shows that the principal eigenvalue of (1.15) may not exist if N 3 and δ is not small. Acknowledgement. The authors thank the referee for valuable comments and suggestions which improved the presentation considerably. 12

13 2 Comparison Principle, Convergence, and Positive Equilibrium In this section, we establish some basic properties of solutions of equation (1.1) and some related nonlocal linear evolution equations, including the comparison principle and monotonicity of solutions with respect to initial conditions, convergence of solutions on compact sets, and existence, uniqueness, and stability of a positive equilibrium of (1.1). Consider equation (1.1) and the following nonlocal linear evolution equation, u t = e µ(y x) ξ k δ (y x)u(t, y)dy u(t, x) + a(x)u(t, x), x (2.1) where µ R, ξ S N 1, and a is smooth and periodic in x i with period p i (i = 1, 2,, N). Note that if µ = 0 and a(x) = a 0 (x)(:= f(x, 0)), (2.1) reduces to (1.17), i.e., the linearization of (1.1) at u 0. Let X p and X be as in (1.3) and (1.9), respectively. For given ρ 0, let X(ρ) = {u C(, R) the function x e ρ x u(x) belongs to X} (2.2) equipped with the norm u X(ρ) = sup x e ρ x u(x). Note that X(0) = X. It follows from the general linear semigroup theory (see [26] or [46]) that for every u 0 X(ρ) (ρ 0), (2.1) has a unique solution u(t, ; u 0, ξ, µ) X(ρ) with u(0, x; u 0, ξ, µ) = u 0 (x). Put Φ(t; ξ, µ)u 0 = u(t, ; u 0, ξ, µ). (2.3) Note that for every µ R and ρ 0, there is ω(µ, ρ) > 0 such that Φ(t; ξ, µ)u 0 X(ρ) e ω(µ,ρ)t u 0 X(ρ) t 0, ξ S N 1, u 0 X(ρ). (2.4) Note also that if u 0 X p, then Φ(t; ξ, µ)u 0 X p for t 0. By general nonlinear semigroup theory (see [26] or [46]), (1.1) has a unique (local) solution u(t, x; u 0 ) with u(0, x; u 0 ) = u 0 (x) for every u 0 X. Also if u 0 X p, then u(t, x; u 0 ) X p for t in the existence interval of the solution u(t, x; u 0 ). Throughout this section, we assume that ξ S N 1 and µ R are fixed, unless otherwise specified. 2.1 Comparison principle and monotonicity Let X + p and X + be as in (1.4) and (1.10), respectively. Let Int(X + p ) = {v X p v(x) > 0, x }. (2.5) 13

14 For v 1, v 2 X p, we define and v 1 v 2 (v 1 v 2 ) if v 2 v 1 X + p (v 1 v 2 X + p ), v 1 v 2 (v 1 v 2 ) if v 2 v 1 Int(X + p ) (v 1 v 2 Int(X + p )). For u 1, u 2 X, we define u 1 u 2 (u 1 u 2 ) if u 2 u 1 X + (u 1 u 2 X + ). A continuous function u(t, x) on [0, T ) is called a super-solution or sub-solution of (2.1) if u exists and is continuous on [0, T ) t RN and satisfies u t e µ(y x) ξ k δ (y x)u(t, y)dy u(t, x) + a(x)u(t, x), x or u t e µ(y x) ξ) k δ (y x)u(t, y)dy u(t, x) + a(x)u(t, x), x for t (0, T ). Super-solutions and sub-solutions of (1.1) are defined in an analogous way. Proposition 2.1 (Comparison principle). (1) If u 1 (t, x) and u 2 (t, x) are sub-solution and super-solution of (2.1) on [0, T ), respectively, u 1 (0, ) u 2 (0, ), and u 2 (t, x) u 1 (t, x) β 0 for (t, x) [0, T ) and some β 0 > 0, then u 1 (t, ) u 2 (t, ) for t [0, T ). (2) If u 1 (t, x) and u 2 (t, x) are bounded sub- and super-solutions of (1.1) on [0, T ), respectively, and u 1 (0, ) u 2 (0, ), then u 1 (t, ) u 2 (t, ) for t [0, T ). (3) For every u 0 X +, u(t, x; u 0 ) exists for all t 0. Proof. (1) We prove the proposition by modifying the arguments of [33, Proposition 2.4]. First let v(t, x) = e ct( u 2 (t, x) u 1 (t, x) ). Then v(t, x) satisfies v t e µ(y x) ξ k δ (y x)v(t, y)dy + p(x)v(t, x), x (2.6) for t (0, T ), where p(x) = a(x) 1 + c. Take c > 0 such that p(x) > 0 for all x. We claim that v(t, x) 0 for t [0, T ) and x. Let p 0 = sup x p(x). It suffices to prove the claim for t (0, T 0 ) and x, where 1 T 0 = min{t, }. Assume that there are p 0 t (0, T +1 0 ) and x such that v( t, x) < 0. Then there is t 0 (0, T 0 ) such that v inf := inf v(t, x) < 0. (t,x) [0,t 0 ] 14

15 Observe that there are t n (0, t 0 ] and x n such that By (2.6), we have v(t n, x n ) v(0, x n ) tn 0 tn 0 v(t n, x n ) v inf as n. [ RN [ RN = t n (1 + p 0 )v inf t 0 (1 + p 0 )v inf e µ(y xn) ξ k δ (y x n )v(t, y)dy + p(x n )v(t, x n ) ] dt e µ(y xn) ξ k δ (y x n )v inf dy + p 0 v inf ] dt for n = 1, 2,. Note that v(0, x n ) 0 for n = 1, 2,. We then have for n = 1, 2,. Letting n, we get v(t n, x n ) t 0 (1 + p 0 )v inf v inf t 0 (1 + p 0 )v inf > v inf (since β 0 v inf < 0). This is a contradiction. Hence v(t, x) 0 for (t, x) [0, T ) and then u 1 (t, x) u 2 (t, x) for (t, x) [0, T ). (2) Let v(t, x) = e ct (u 2 (t, x) u 1 (t, x)). Then v(t, ) 0 and v(t, x) satisfies v t k δ (y x)v(t, y)dy + p(t, x)v(t, x), x for t (0, T ), where p(t, x) = c 1 + f(x, u 2 (t, x)) + [ u 1 (t, x) 1 0 f u (x, su 1 (t, x) + (1 s)u 2 (t, x))ds ] v(t, x) for t [0, T ), x. By the boundedness of u 1 (t, x) and u 2 (t, x), there is c > 0 such that inf t [0,T ),x p(t, x) > 0. (2) then follows from the arguments in (1) with p(x) and p 0 being replaced by p(t, x) and sup t [0,T ),x p(t, x), respectively. (3) By (H1), there is M > 0 such that u 0 (x) M and f(x, M) < 0 for x. Let u M (t, x) M for x and t R. Then u M is a super-solution of (1.1) on [0, ). Let I(u 0 ) R be the maximal interval of existence of the solution u(t, ; u 0 ) of (1.1). Then by (2), one obtains 0 u(t, x; u 0 ) M for x, t I(u 0 ) [0, ). It then follows easily that [0, ) I(u 0 ) and u(t, x; u 0 ) exists for all t 0. 15

16 Proposition 2.2 (Strong monotonicity). Suppose that u 1, u 2 X p and u 1 u 2, u 1 u 2. (1) Φ(t; ξ, µ)u 1 Φ(t; ξ, µ)u 2 for all t > 0. (2) u(t, ; u 1 ) u(t, ; u 2 ) for every t > 0 at which both u(t, ; u 1 ) and u(t, ; u 2 ) exist. Proof. (1) We apply the arguments in Theorem 2.1 of [35]. First, assume that u 0 X + p \ {0}. Then by Proposition 2.1 (1), Φ(t; ξ, µ)u 0 0 for t > 0. We claim that e K δ,ξ,µt u 0 0 for t > 0. In fact, note that e K δ,ξ,µt u 0 = u 0 + tk δ,ξ,µ u 0 + t2 (K δ,ξ,µ ) 2 u 0 2! tn (K δ,ξ,µ ) n u 0 n! Let x 0 be such that u 0 (x 0 ) > 0. Then there is r > 0 such that u 0 (x 0 ) > 0 for x B(x 0, r) := {y y x 0 < r}. This implies that (K δ,ξ,µ u 0 )(x) = e µ(y x) ξ k δ (y x)u 0 (y)dy > 0 for x B(x 0, r + δ). By induction, (K δ,ξ,µ u 0 ) n (x) > 0 for x B(x 0, r + nδ), n = 1, 2, This together with the periodicity of u 0 (x) implies that e K δ,ξ,µt u 0 0 for t > 0. Let m > 1 min x a(x). Note that Φ(t; ξ, µ)u 0 = u(t, ; u 0 ) = e (K δ,ξ,µ I+a( )I+mI mi)t u 0 = e mit e (K δ,ξ,µ I+a( )I+mI)t u 0 and (e mit v)(x) = e mt v(x) for every x. Note also that e (K δ,ξ,µ I+a( )I+mI)t u 0 = e K δ,ξ,µt u 0 + t 0 e K δ,ξ,µ(t s) ( I +a( )I +mi)u(s, ; u 0 )ds for t > 0. It then follows that Φ(t; ξ, µ)u 0 0 for all t > 0. Now let u 0 = u 2 u 1. Then u 0 X p + \ {0}. Hence Φ(t; ξ, µ)u 0 0 for t > 0 and then Φ(t; ξ, µ)u 1 Φ(t; ξ, µ)u 2 for t > 0. (2) Let v(t, x) = u(t, x; u 2 ) u(t, x; u 1 ) for t 0 at which both u(t, x; u 1 ) and u(t, x; u 2 ) exist. Then v(0, ) = u 2 u 1 0 and v(t, x) satisfies v t = k δ (y x)v(t, y)dy v(t, x) + f(x, u(t, x; u 2 ))v(t, x) + [ u(t, x; u 1 ) 1 0 f u (x, su(t, x; u 1 ) + (1 s)u(t, x; u 2 ))ds ] v(t, x), x. (2) then follows from the arguments similar to those in (1). 16

17 2.2 Convergence on compact sets In this subsection, we investigate the convergence of solutions of (1.1) or (2.1) on compact sets. First, we state the following lemma which follows easily. Lemma 2.1. For given ρ 0 0 and {u n } X(ρ 0 ) with u n X(ρ0 ) M for some M > 0 and n = 1, 2,, u n (x) 0 as n uniformly for x in bounded subsets of if and only if u n (x) 0 in X(ρ) as n for every ρ > ρ 0. Proposition 2.3 (Convergence on compact sets). (1) If u n X and u 0 X are such that u n X M for some M > 0 and n = 1, 2,, and u n (x) u 0 (x) as n uniformly for x in bounded subsets of, then ( Φ(t; ξ, µ)un ) (x) ( Φ(t; ξ, µ)u0 ) (x) as n uniformly for (t, x) in bounded subsets of [0, ). (2) If u n X + and u 0 X + are such that u n X M for some M > 0 and n = 1, 2, and u n (x) u 0 (x) as n uniformly for x in bounded subsets of, then u(t, x; u n ) u(t, x; u 0 ) as n uniformly for (t, x) in bounded subsets of [0, ). Proof. (1) First of all, by Lemma 2.1, for every given ρ > 0, u n u 0 X(ρ) 0 as n. By (2.4), Φ(t; ξ, µ)u n Φ(t; ξ, µ)u 0 X(ρ) e ω(µ,ρ)t u n u 0 X(ρ) for t 0 and n = 1, 2,. Then, Φ(t; ξ, µ)u n Φ(t; ξ, µ)u 0 X(ρ) 0 as n uniformly for t in bounded subsets of [0, ). This implies that ( Φ(t; ξ, µ)un ) (x) ( Φ(t; ξ, µ)u0 ) (x) as n uniformly for (t, x) in bounded subsets of [0, ). (2) By Proposition 2.1 (3), u(t, x; u 0 ) and u(t, x; u n ) exist on [0, ) for any n 1 and there is M > 0 such that u(t, ; u 0 ) X M and u(t, ; u n ) X M for t 0 and n = 1, 2,. Let v n (t, x) = u(t, x; u n ) u(t, x; u 0 ) for x, t 0, and n = 1, 2,. Then v n t = k δ (y x)v n (t, y)dy v n (t, x) + a n (t, x)v n (t, x), x, where a n (t, x) = f(x, u(t, x; u n ))+ [ u(t, x; u 0 ) Hence 1 v n (t, ) = e (K δ I)t v n (0, ) + 0 f u (x, su(t, x; u 0 )+(1 s)u(t, x; u n ))ds ] v n (t, x). t 0 e (K δ I)(t s) a n (s, )v n (s, )ds. 17

18 Note that for every ρ > 0 there are ω 0 (ρ) R and L 0 > 0 such that e (K δ I)t v X(ρ) e ω 0(ρ)t v X(ρ) v X(ρ) and a n (t, x) L 0 t 0, x. It then follows from Gronwall s inequality that v n (t, ) X(ρ) e (ω 0(ρ)+L 0 )t v n (0, ) X(ρ). By the arguments in (1), we have v n (t, x) 0 and hence u(t, x; u n ) u(t, x; u 0 ) as n uniformly for (t, x) in bounded subsets of [0, ). 2.3 Positive equilibrium In this subsection, we study the existence, uniqueness and stability of positive equilibrium solutions of (1.1). Throughout this subsection, Φ(t; ξ, µ) denotes the solution operator of (2.1) with a(x) = a 0 (x) := f(x, 0) defined in (2.3) and λ(δ, a 0 ) is as in (1.20) with a = a 0. Suppose that u = u is an equilibrium solution of (1.1) in X p + \ {0}. u = u is said to be globally asymptotically stable in X p + \ {0} if for any u 0 X p + \ {0}, u(t, ; u 0 ) u in X p as t. Proposition 2.4. (1) Assume (H1) and (H2). Then u = u + is globally asymptotically stable in X p + \ {0}, u = 0 is linearly unstable in X p (i.e. (H3) holds), and there are no other equilibrium solutions of (1.1) in X p + apart from u = 0 and u = u +. (2) Assume (H1) and (H3). For given δ > 0, assume also that λ(δ, a 0 ) exists. Then (1.1) has an equilibrium solution u = u + (x) in X + p \ {0} (hence (H2) holds). Proof. (1) First, by the arguments in [35, Theorem 3.2], for every u 0 X + p \ {0}, u(t, ; u 0 ) u + ( ) Xp 0 as t. Hence u = u + is globally asymptotically stable in X p + \ {0} and (1.1) has exactly two equilibrium solutions in X p +, that is, u = 0 and u = u +. It remains to prove that u = 0 is linearly unstable (i.e. λ 0 > 0, where λ 0 is as in (H3)). Note that ( (Kδ I)u +) (x) + f(x, u + (x))u + (x) = 0 x. By Proposition 2.2, inf x u + (x) > 0. Then by (H1), there is ɛ 0 > 0 such that f(x, u + (x)) a 0 (x) ɛ 0, x. 18

19 Hence u = u + is a sub-solution of u t = (K δ I)u + (a 0 (x) ɛ 0 )u. This implies that u = e ɛ 0t u + is a sub-solution of the linearization equation (1.17) of (1.1) at u = 0. By Proposition 2.1, Φ(t; ξ, 0)u + e ɛ 0t u + t 0. It then follows from the general spectral theory for bounded linear operators (see [46]) that λ 0 ɛ 0 > 0. (2) First of all, if λ(δ, a 0 ) exists, then by (H3), λ(δ, a 0 ) > 0. Next, by (H1) and Proposition 2.2, there is M + > 0 such that u(t, x; u + 0 ) is strictly decreasing in t (i.e. u(t 1, x; u + 0 ) < u(t 2, x; u + 0 ) for 0 t 1 < t 2, x ), where u + 0 ( ) M + ( X + p ). Now we claim that there exists 0 < u 0 1 such that u(t, x; u 0 ) u 0 for t > 0. In fact, let φ(x) be a positive principal eigenfunction of K δ I + a 0 ( ). Note that for every 0 < ɛ 0 < λ(δ, a 0 ), there is b 0 > 0 such that f(x, u) f(x, 0) ɛ 0 for 0 u b 0, x. (2.7) It is not difficult to see that if u 0 X + is such that 0 u(t, x; u 0 ) b 0 for 0 t t 0 and x, then u(t, x; u 0 ) e ɛ 0t (Φ(t; ξ, 0)u 0 )(x) (2.8) for 0 t t 0 and x. Let u 0 = ηφ with η <. Note that (Φ(t; ξ, max x φ(x) 0)u 0 )(x) = ηe λ(δ,a0)t φ(x). There is t 0 > 0 such that (2.8) holds for u 0 = u 0 and 0 t t 0. Thus, u(t, x; u 0 ) e ɛ0t (Φ(t; ξ, 0)u 0 )(x) = ηe (λ(δ,a 0) ɛ 0 )t φ(x) ηφ(x) = u 0 (x) for 0 t t 0. Then by Proposition 2.1, u(t, x; u 0 ) u 0 for t 0. Finally, it follows from the arguments in [35, Theorem 3.2] that lim t u(t, x; u + 0 ) = lim t u(t, x; u 0 ) and u + (x) := lim t u(t, x; u + 0 ) X p is an equilibrium solution of (1.1) in X p + \ {0}. 2.4 Remarks Due to the spatial inhomogeneity of (1.1), it is sometime important to consider the space shifted equation (1.21) of (1.1) and the following space shifted equation of (2.1), u t = e µ(y x) ξ k δ (y x)u(t, y)dy u(t, x) + a(x + z)u(t, x), (2.9) b 0 19

20 where z. Note that if µ = 0 and a(x) = a 0 (x)(:= f(x, 0)), then (2.9) reduces to the space shifted equation of the linearization equation (1.17) of (1.1) at u = 0, u t = k δ (y x)u(t, y)dy u(t, x) + a 0 (x + z)u(t, x), x. (2.10) It is again a consequence of the general semigroup theory that (1.21) has a unique (local) solution u(t, x; u 0, z) with u(0, x; u 0, z) = u 0 (x) (z ) for every u 0 X. Also given u 0 X(ρ) (ρ 0), (2.9) has a unique solution u(t, x; u 0, ξ, µ, z) with u(0, x; u 0, ξ, µ, z) = u 0 (x). We set Φ(t; ξ, µ, z)u 0 = u(t, ; u 0, ξ, µ, z). (2.11) Sub- and super-solutions of (1.21) and (2.9) are defined similarly as those of (2.1) in 2.1. Remark 2.1. Propositions also hold for the space shifted versions (1.21) and (2.9) of equations (1.1) and (2.1), respectively. Remark 2.2. Assume (H1), (H3), and (H4). Then lim (u(t, x; t α+, z) u + (x + z)) = 0 holds uniformly in x and z for every α + > 0. Here α + in u(t, x; α +, z) stands for the constant function with value α +. 3 Principal Eigenvalue Theory In this section, we investigate the eigenvalue problem (1.15) and prove Theorems A and B stated in the introduction and some other related results which are used in the proof of the existence of spreading speeds of (1.1) in later sections. Throughout this section, X p is as in (1.3), a : is a smooth function, a X p, and a max = max x a(x), a min = min x a(x). a( )I : X p X p has the same meaning as in (1.6) with a 0 ( ) being replaced by a( ) and K δ,ξ,µ : X p X p is understood as in (1.16) for δ > 0, ξ S N 1, and µ R. We first introduce in 3.1 some important operators related to K δ,ξ,µ I + a( )I or (1.15) and explore some basic properties of the eigenvalue problems associated with these operators. We then prove Theorems A and B in 3.2 and derive in 3.3 from Theorems A and B some results on the spectral radius of some operator related to K δ,ξ,µ I + a( )I. This section ends with some remarks in

21 3.1 Evolution operators and eigenvalue problems In this subsection, we introduce some evolution operators related to the operator K δ,ξ,µ I + a( )I, explore the basic properties of the eigenvalue problems associated to these operators, and discuss the relations between the eigenvalues of K δ,ξ,µ I + a( )I and its related operators. If no confusion occurs, we may write the principal eigenvalue λ(δ, ξ, µ, a) K δ,ξ,µ I + a( )I (if exists) (see Definition 1.2) as λ(δ, ξ, µ) or λ(ξ, µ). First of all, we introduce a compact operator associated to K δ,ξ,µ based on the perturbation idea in [8]. This operator plays an important role in the proofs of Theorems A and B in the next subsection. For given α > 1 + a max, let U δ,ξ,µ,α : X p X p be defined as follows e µ(y x) ξ k δ (y x)u(y) (U δ,ξ,µ,α u)(x) = dy. (3.1) α + 1 a(y) Observe that U δ,ξ,µ,α is a compact and positive operator on X p. Let r(u δ,ξ,µ,α ) be the spectral radius of U δ,ξ,µ,α. Proposition 3.1. (1) α > 1 + a max is an eigenvalue of K δ,ξ,µ I + a( )I or (1.15) iff 1 is an eigenvalue of the eigenvalue problem U δ,ξ,µ,α v = λv. (2) For α > 1 + a max, 1 is an eigenvalue of U δ,ξ,µ,α with a positive eigenfunction iff r(u δ,ξ,µ,α ) = 1. (3) If there is α > 1 + a max with r(u δ,ξ,µ,α ) > 1, then there is α 0 > α such that r(u δ,ξ,µ,α0 ) = 1. (4) If α > 1 + a max is an eigenvalue of K δ,ξ,µ I + a( )I or (1.15), then it is the principal eigenvalue of (1.15). Proof. (1) and (2) follow from Proposition 2.1 of [8]. (3) and (4) follow from Theorem 2.2 of [8]. By Proposition 3.1, the spectral radius of U δ,ξ,µ,α provides a useful tool for the investigation of those eigenvalues of K δ,ξ,µ I + a( )I which are greater than 1 + a max. The following proposition shows that if K δ,ξ,µ I + a( )I possesses a principal eigenvalue, then it must be greater than 1 + a max. Proposition 3.2. If λ(ξ, µ) is the principal eigenvalue of K δ,ξ,µ I +a( )I, then λ(ξ, µ) > 1 + a max. 21

22 Proof. Since λ(ξ, µ) is the principal eigenvalue of K δ,ξ,µ I+a( )I, there is an eigenfunction ψ X p + \ {0} such that e µ(y x) ξ k δ (y x)ψ(y)dy ψ(x) + a(x)ψ(x) = λ(ξ, µ)ψ(x), x. (3.2) Note that u(t, x) = e λ(ξ,µ)t ψ(x) is a solution of (2.1). By Proposition 2.2, ψ Int(X + p ). Let x 0 be such that a(x 0 ) = a max. By ψ Int(X + p ), e µ(y x 0) ξ k δ (y x 0 )ψ(y)dy > 0. This together with (3.2) implies that Hence λ(ξ, µ) > 1 + a max. λ(ξ, µ)ψ(x 0 ) > ψ(x 0 ) + a(x 0 )ψ(x 0 ). Next, consider the evolution equation (2.1) associated with the operator K δ,ξ,µ I + a( )I. Let Φ(t; ξ, µ) be the solution operator of (2.1) given in (2.3) and Φ p (t; ξ, µ) : X p X p be defined by Φ p (t; ξ, µ) = Φ(t; ξ, µ) Xp (3.3) for t 0, ξ S N 1 and µ R. Let r(φ p (1; ξ, µ)) and σ(φ p (1; ξ, µ)) be the spectral radius and the spectrum of Φ p (1; ξ, µ), respectively. The following lemma states the relationship between the principal eigenvalue of K δ,ξ,µ I + a( )I and the spectral radius of Φ p (1; ξ, µ) and follows easily (see [26, Theorems and 1.5.3]). Lemma 3.1. The principal eigenvalue λ(ξ, µ) of (1.15) exists if and only if r(φ p (1; ξ, µ)) is an algebraically simple eigenvalue of Φ p (1; ξ, µ) with an eigenfunction in X p + and for every λ σ ( Φ p (1; ξ, µ) ) \{r(φ p (1; ξ, µ))}, λ < r(φ p (1; ξ, µ)). Moreover, if λ(µ, ξ) exists, then λ(ξ, µ) = ln r(φ p (1; ξ, µ)). Therefore, the spectral radius of Φ p (1; ξ, µ) plays an important role in the investigation of the principal eigenvalue of K δ,ξ,µ I + a( )I or (1.15). We next establish some further observations for r(φ p (1; ξ, µ)). Note that Φ(t; ξ, 0) is independent of ξ S N 1. We put Φ(t) = Φ(t; ξ, 0) (3.4) for ξ S N 1. For given u 0 X and µ R, letting u ξ,µ 0 (x) = e µx ξ u 0 (x), then u ξ,µ 0 X( µ ). The following lemma follows directly from the uniqueness of solutions of (2.1). Lemma 3.2. For given u 0 X, ξ S N 1, and µ R, Φ(t; ξ, µ)u 0 = e µx ξ Φ(t)u ξ,µ 0. 22

23 Observe that for each x, there is a measure m(x; y, dy) such that ( Φ(1)u 0 )(x) = u 0 (y)m(x; y, dy). (3.5) Moreover, by ( Φ(1)u 0 ( p i e i ))(x) = ( Φ(1)u 0 ( ))(x p i e i ) for x and i = 1, 2,, N, u 0 (y)m(x p i e i ; y, dy) = u 0 (y p i e i )m(x; y, dy) = u 0 (y)m(x; y + p i e i, dy) and hence m(x p i e i ; y, dy) = m(x; y + p i e i, dy) (3.6) for i = 1, 2,, N. By Lemma 3.2, we have (Φ(1; ξ, µ)u 0 )(x) = e µ(x y) ξ u 0 (y)m(x; y, dy), u 0 X. Proposition 3.3. For every u Int(X p + ), e µ(x y) ξ u(y)m(x; y, dy) R inf N r(φ p (1; ξ, µ)) x u(x) sup x e µ(x y) ξ u(y)m(x; y, dy). u(x) Proof. By [21, Theorems 3.6 and 4.3], the spectral radius of the nonnegative operator Φ p (1; ξ, µ) is bounded by the lower and upper Collatz-Wielandt numbers of u for every u Int(X + p ), which are defined by sup{λ 0 : λu Φ p (1; ξ, µ)u} and inf{λ 0 : λu Φ p (1; ξ, µ)u}, respectively. (2) then follows. In proving the existence of spreading speeds of (1.1) in section 5, properly truncated operators of Φ(1; ξ, µ) are used. We therefore introduce them next. Let ζ : R [0, 1] be a smooth function satisfying that { 1 for s 1 ζ(s) = (3.7) 0 for s 2. For a given B > 0, define Φ B (1; ξ, µ) : X X by (Φ B (1; ξ, µ)u 0 )(x) = e µ(x y) ξ u 0 (y)ζ( y x /B)m(x; y, dy). (3.8) Define Φ p B (1; ξ, µ) : X p X p by Φ p B (1; ξ, µ) = Φ B(1; ξ, µ) Xp. (3.9) Similarly, let r(φ p B (1; ξ, µ)) and σ(φp B (1; ξ, µ)) be the spectral radius and the spectrum of (1; ξ, µ), respectively. Φ p B 23

24 Lemma 3.3. Φ p B (1; ξ, µ) Φp (1; ξ, µ) Xp 0 as B uniformly for µ in bounded sets and ξ S N 1. Proof. It suffices to prove that e µ y x m(x; y, dy) 0 as B y x B uniformly for µ in bounded sets and for x. For given µ 0 > 0 and n N, let u n X(µ 0 + 1) be such that { e µ 0 x for x n u n (x) = 0 for x n 1 and 0 u n (x) e µ 0n for x n. Then u n X(µ0 +1) 0 as n. Therefore, Φ(1)u n X(µ0 +1) 0 as n. This together with Lemma 2.1 implies that u n (y)m(x; y, dy) 0 as n uniformly for x in bounded subsets of and then e µ 0 y m(x; y, dy) 0 as n y n uniformly for x in bounded subsets of. The later implies that e µ y x m(x; y, dy) 0 as n y x n uniformly for µ µ 0 and x in bounded subset of. By (3.6), for every 1 i N, e µ y (x+p ie i ) m(x + p i e i ; y, dy) = e µ y x m(x + p i e i ; y + p i e i, dy) y (x+p i e i ) n y x n = e µ y x m(x; y, dy). We then have y x n y x n e µ y x m(x; y, dy) 0 as n uniformly for µ µ 0 and x. The lemma now follows. 24

25 3.2 Proofs of Theorems A and B In this subsection, we prove Theorems A and B. Throughout this subsection, U δ,ξ,µ,a is understood as in (3.1), and r(u δ,ξ,µ,a ) denotes the spectral radius of U δ,ξ,µ,a. We may simply write U α for U δ,ξ,µ,a if no confusion can occur. Proof of Theorem A. (1) We prove the existence of a δ 0 > 0 and the existence of a principal eigenvalue λ(δ, ξ, µ, a) for all 0 < δ < δ 0, ξ S N 1 and µ R. By Proposition 3.1, it suffices to prove the existence of δ 0 > 0 such that for each 0 < δ < δ 0, ξ S N 1, and µ R, there exists an α > 1 + a max such that r(u α ) > 1. Let ( 2(amax a min + 1) ) 1/2(> M 0 = inf 0). ξ S N 1 (z ξ) R 2 k(z)dz N We first prove the existence of an α > 1 + a max such that r(u α ) > 1 for every ξ S N 1, δ > 0, and µ R with µ > M 0. δ In fact, for v(x) 1 and every 0 < ɛ < 1, we have e µ(y x) ξ k δ (y x) (U 1+amax+ɛv)(x) = dy R a N max + ɛ a(y) e µ(y x) ξ k δ (y x) dy R a N max a min + ɛ 1 ( µ 2 δ 2 = 1 + (z ξ) 2 k(z)dz a max a min + ɛ 2! + µ4 δ 4 (z ξ) 4 k(z)dz + ) 4! µ2 δ 2. (3.10) M0 2 Hence if µ δ > M 0, then for 0 < ɛ 1, there is γ > 1 such that (U 1+amax+ɛv)(x) > γv(x) x. This implies that r(u 1+amax+ɛ) > 1. We then only need to prove that there is δ 0 > 0 and an α > 1 + a max with r(u α ) > 1 for all 0 < δ < δ 0, ξ S N 1, and µ R with µδ M 0. We prove this by applying similar arguments as in [35, Theorem 2.6]. Let D = [0, p 1 ] [0, p 2 ] [0, p N ]. Assume that x 0 D is such that a(x 0 ) = a max. Without loss of generality, we may assume that x 0 Int(D). Then for every 0 < ɛ < 1, there is some η > 0 such that a(x 0 ) a(x) < ɛ for x B(η, x 0 ) D, where B(η, x 0 ) = {x x x 0 < η}. Let v( ) X p be such that 25

26 v(x) = ψ( x x 0 ) for x D, where cos( πr) if 0 r η 2η ψ(r) = 0 if r > η Let 0 < δ < η and 0 < ɛ 2 1 < 1. Let D 1 = B( η, x 2 0), D 2 = B(η, x 0 ) \ D 1. For x D 2, let D(δ, x) = B(δ, x) B( x x 0, x 0 ). Observe that for x B( η, x 2 0), v(x) 2. For x D 2 2 and y D(δ, x), v(y) v(x). For x D \ B(η, x 0 ), v(x) = 0. Observe also that there are C > 0 (independent of ɛ) and δ 1 > 0 such that inf e x D 1 B(η/2,x µ(y x) ξ k δ (x y)dy C, 0 ) inf e µ(y x) ξ k δ (x y)dy C x D 2 D(δ,x) for 0 < δ < δ 1, ξ S N 1, and 0 µ δ M 0. Clearly, for each γ > 1, (U amax ɛ 1 v)(x) γv(x) for x D \ B(η, x 0 ). (3.11) If x D 1, we have e µ(y x) ξ k δ (x y)v(y) (U amax ɛ1 v)(x) dy D 1 a(y) + a max ɛ 1 1 e µ(y x) ξ k δ (x y)v(y)dy 1 ɛ 1 + ɛ B(η,x 0 ) 2 e µ(y x) ξ k δ (x y)dy 2(1 ɛ 1 + ɛ) B(η/2,x 0 ) 2C 2(1 ɛ 1 + ɛ) 2C v(x). (3.12) 2(1 ɛ 1 + ɛ) If x D 2, we have e µ(y x) ξ k δ (x y)v(y) (U amax ɛ1 v)(x) dy D 1 a(y) + a max ɛ 1 1 e µ(y x) ξ k δ (x y)v(y)dy 1 ɛ 1 + ɛ D v(x) e µ(y x) ξ k δ (x y)dy 1 ɛ 1 + ɛ D(δ,x) Cv(x) 1 ɛ 1 + ɛ. (3.13) 26

27 Let M = 2C 2(1 ɛ 1. By (3.11)-(3.13) and the periodicity of v, we obtain +ɛ) (U amax ɛ 1 v)(x) Mv(x) for all x. Choose ɛ and ɛ 1 such that 0 < ɛ < 2 2 C and 1 > ɛ 1 > 1 + ɛ 2C 2. Let δ 0 = min{δ 1, η 2 }, then M > 1 and r(u amax ɛ 1 ) M > 1, thus (1) is proved. (2) Fix ξ S N 1. By Lemma 3.1, ˆλ(µ i ) := r(φ p (1; ξ, µ i )) is an eigenvalue of Φ p (1; ξ, µ i ) with a positive eigenfunction ψ i (i = 1, 2). Hence ˆλ(µ i ) = (Φp (1; ξ, µ i )ψ i )(x) e µ i(x y) ξ ψ R = N i (y)m(x; y, dy) x ψ i (x) ψ i (x) for i = 1, 2. For given 0 t 1, let ψ 3 = ψ1ψ t 2 1 t. By Hölder s inequality, [ˆλ(µ 1 )] t [ˆλ(µ 2 )] 1 t e µ 1(x y) ξ ψ R = [ N 1 (y)m(x; y, dy) ] t e µ 2(x y) ξ ψ R [ N 2 (y)m(x; y, dy) ] 1 t ψ 1 (x) ψ 2 (x) Applying Proposition 3.3, we get [ˆλ(µ 1 )] t [ˆλ(µ 2 )] 1 t sup x Thus, By Lemma 3.1 again, eµ1(x y) ξ ψ 1 (y) [ R ψ N 1 (x) e (tµ 1+(1 t)µ 2 )(x y) ξ ψ R = N 3 (y)m(x; y, dy) ψ 3 (x) ] t [ eµ 2(x y) ξ ψ 2 (y) ] 1 t m(x; y, dy) ψ 2 (x) e (tµ 1+(1 t)µ 2 )(x y) ξ ψ 3 (y)m(x; y, dy) ψ 3 (x) x. ln[ˆλ(µ 1 )] t [ˆλ(µ 2 )] 1 t ln(r(φ p (1; ξ, tµ 1 + (1 t)µ 2 )). tλ(δ, ξ, µ 1, a) + (1 t)λ(δ, ξ, µ 2, a) λ(δ, ξ, tµ 1 + (1 t)µ 2, a), that is, λ(δ, ξ, µ, a) is convex in µ. r(φ p (1; ξ, tµ 1 +(1 t)µ 2 ). (3) Note that λ(δ, ξ, µ, a) λ(δ, ξ, µ, a min ) (it follows from Proposition 3.1) and λ(δ, ξ, µ, a min ) = e µy ξ k δ (y)dy 1 + a min with 1 as an eigenfunction. Let m n (ξ) = ( y ξ) n k n! δ (y)dy. Note that m n (ξ) > 0 if n is even and 0 if n is odd. Then RN e µy ξ ( µy ξ) n k δ (y)dy 1 + a min = k δ (y)dy 1 + a min R n! N n=0 = m 2 (ξ)µ 2 + m 2n (ξ)µ 2n + a min n=2 27

28 Let m δ := inf ξ S N 1 m 2 (ξ)(> 0) and µ 0 > 0 be such that n=2 m 2n(ξ)µ 2n > a min for µ µ 0. Then m δ and µ 0 have the required property. (4) By (3), λ(δ,ξ,µ,a) µ as µ. By λ(δ, ξ, 0, a) > 0, λ(δ,ξ,µ,a) µ as µ 0+. This implies that there is µ (ξ) > 0 such that (1.18) and (1.19) hold. Proof of Theorem B. (1) We only prove that if a max a min < 1, then the principal eigenvalue λ(δ, ξ, µ, a) exists for every δ > 0, ξ S N 1, and µ R. The other statements can be proved by exactly the same arguments as in Theorem A. By the arguments in (3.10), we have for v(x) 1 and every 0 < ɛ < 1 that e µ(y x) ξ k δ (y x) (U 1+amax+ɛv)(x) = dy R a N max + ɛ a(y) 1 a max a min + ɛ for all δ > 0, ξ S N 1, and µ R. Hence if a max a min < 1, then for 0 < ɛ 1, there is γ > 1 such that (U 1+amax+ɛv)(x) > γv(x). This implies that r(u 1+amax+ɛ) > 1. It then follows from Proposition 3.1 that the principal eigenvalue λ(δ, ξ, µ, a) of (1.15) exists every δ > 0, ξ S N 1 and µ R. (2) As in (1), we only prove that the principal eigenvalue λ(δ, ξ, µ, a) exists for all δ > 0, ξ S N 1, and µ R. The other statements can be proved by exactly the same arguments as in Theorem A. Let x 0 D be such that a(x 0 ) = a max. Also, without loss of generality, we may assume that x 0 Int(D). Since the partial derivatives of a(x) up to order N 1 at x 0 are zero, there is M > 0 such that a(x 0 ) a(y) M x 0 y N for y. (3.14) Fix δ > 0, ξ S N 1, and µ R. Let σ > 0 be such that σ < δ 2 and B(2σ, x 0) D. Let v X + p be such that v (x) = 1 if x B(σ, x 0 ) and v (x) = 0 if x D \ B(2σ, x 0 ). Clearly, for every x D \ B(2σ, x 0 ) and γ > 1, For x B(2σ, x 0 ), there is M > 0 such that (U 1+amax+ɛv )(x) > γv (x) = 0. (3.15) e µ(y x) ξ k δ (y x) M for y B(σ, x 0 ). It then follows that for x B(2σ, x 0 ) (U 1+amax+ɛv e µ(y x) ξ k δ (y x) )(x) M x 0 y N + ɛ dy B(σ,x 0 ) B(σ,x 0 ) M dy. (3.16) M x 0 y N + ɛ 28