Nonlocal Dispersal Equations and Convergence to Random Dispersal Equations by Xiaoxia Xie A dissertation submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 2, 2014 Keywords: nonlocal dispersal, random dispersal, principal spectrum point, principal eigenvalue, KPP type evolution equation, spatial inhomogeneity Copyright 2014 by Xiaoxia Xie Approved by Wenxian Shen, Chair, Professor of Mathematics and Statistics Yanzhao Cao, Professor of Mathematics and Statistics Dmitry Glotov, Associate Professor of Mathematics and Statistics Georg Hetzer, Professor of Mathematics and Statistics Abstract This dissertation is devoted to the study of the dynamics of nonlocal and random disper- sal evolution equations. Dispersal evolution equations are widely used to model the di usions of organisms or individuals in many biological and ecological systems. More precisely, ran- dom and nonlocal dispersal equations arise in modeling the dynamics of di usive systems which exhibit random or local, and nonlocal internal interactions, respectively. In this dis- sertation, we study the dynamics of such equations complemented with Dirichlet, Neumann, and periodic types of boundary condition in a uni ed way. It is mainly concerned with principal spectral theory of nonlocal dispersal operators and the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. Regarding the principal spectral theory of nonlocal dispersal operators, we investigate the dependence of the principal spectrum points of nonlocal dispersal operators on the un- derlying parameters and its applications. In particular, we study the e ects of the spatial inhomogeneity, the dispersal rate, and the dispersal distance on the existence of the principal eigenvalues, the magnitude of the principal spectrum points, and the asymptotic behaviors of the principal spectrum points of time homogeneous nonlocal dispersal operators with Dirich- let type, Neumann type, and periodic boundary conditions. We also discuss the applications of the principal spectral theory of nonlocal dispersal operators to the asymptotic dynamics of two species competition systems. About the approximations of random dispersal operators/equations by nonlocal dis- persal operators/equations, we rst prove that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems converge to the solutions of the corresponding ran- dom dispersal initial-boundary value problems. Next, we prove that the principal spectrum points of time periodic nonlocal dispersal operators with properly rescaled kernels converge ii to the principal eigenvalues of the corresponding random dispersal operators. Thirdly, we prove that the unique positive time periodic solutions of nonlocal dispersal KPP type evolu- tion equations with properly rescaled kernels converge to the unique positive time periodic solutions of the corresponding random dispersal KPP type evolution equations. We also discuss the applications of the approximation results to the e ects of the rearrangements with equimeasurability on principal spectrum point of nonlocal dispersal operators. iii Acknowledgments It would have been next to impossible to nish my dissertation without the guidance of my committee members and support from my family and husband. I would like to gratefully and sincerely thank my adviser, Dr. Wenxian Shen, for her excellent guidance in doing research, patience, and caring. I would also like to thank my committee members, Dr. Yanzhao Cao, Dr. Dmitry Glotov, and Dr. Georg Hetzer for guiding my research for the past several years. And I would like to thank my University Reader, Dr. Rita Graze, for being willing to review and evaluate my dissertation. Finally, I would like to thank my family for their support and encouragement. In particular, my wonderful husband, Hui Yi, always stands by my side, cheering me up and being with me through the good times and the bad. iv Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Notations, Assumptions, De nitions and Main Results . . . . . . . . . . . . . . 15 2.1 Notations, Assumptions and De nitions . . . . . . . . . . . . . . . . . . . . . 15 2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 Solutions of Evolution Equation and Semigroup Theory . . . . . . . . . . . . 28 3.2 Sub- and Super-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Comparison Principle and Monotonicity . . . . . . . . . . . . . . . . . . . . 32 3.4 A Technical Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1 Basic Properties of Principal Eigenvalues/Principal Spectrum Points of Time Homogeneous Dispersal Operators . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 E ects of Spatial Variations and the Proof of Theorem 2.4 . . . . . . . . . . 45 4.3 E ects of Dispersal Rates and the Proof of Theorem 2.6 . . . . . . . . . . . 54 4.4 E ects of Dispersal Distance and the Proof of Theorem 2.8 . . . . . . . . . . 60 4.5 Applications to the Asymptotic Dynamics of Two Species Competition System 65 4.5.1 Asymptotic Dynamics of KPP Type Competition Systems . . . . . . 66 4.5.2 Proof of Theorem 2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 Approximations of Random Dispersal Operators/Equations by Nonlocal Disper- sal Operators/Equations and Applications . . . . . . . . . . . . . . . . . . . . . 72 v 5.1 Approximations of Solutions of Random Dispersal Initial-Boundary Value Problems by Nonlocal Dispersal Initial-Boundary Value Problems . . . . . . 72 5.1.1 Proof of Theorem 2.13 in the Dirichlet Boundary Condition Case . . 72 5.1.2 Proof of Theorem 2.13 in the Neumann Boundary Condition Case . 76 5.1.3 Proof of Theorem 2.13 in the Periodic Boundary Condition Case . . 81 5.2 Approximations of Principal Eigenvalues of Time Periodic Random Dispersal Operators by Time Periodic Nonlocal Dispersal Operators . . . . . . . . . . 82 5.2.1 Basic Properties of Principal Eigenvalues/Principal Spectrum Points of Time Periodic Dispersal Operators . . . . . . . . . . . . . . . . . . 82 5.2.2 Proof of Theorem 2.15 in the Dirichlet Boundary Condition Case . . 85 5.2.3 Proof of Theorem 2.15 in the Neumann Boundary Condition Case . 90 5.2.4 Proof of Theorem 2.15 in the Periodic Boundary Condition Case . . 92 5.3 Approximations of Positive Time Periodic Solutions of Random Dispersal KPP Type Evolution Equations by Nonlocal Dispersal KPP Type Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.1 Asymptotic Behavior of KPP Type Evolution Equations . . . . . . . 93 5.3.2 Proof of Theorem 2.16 in the Dirichlet Boundary Condition Case . . 94 5.3.3 Proof of Theorem 2.16 in the Neumann Boundary Condition Case . 99 5.3.4 Proof of Theorem 2.16 in the Periodic Boundary Condition Case . . . 102 5.4 Applications to the E ect of the Rearrangements with Equimeasurability on Principal Spectrum Point of Nonlocal Dispersal Operators . . . . . . . . . . 102 6 Concluding Remarks, Problems, and Future Plans . . . . . . . . . . . . . . . . . 105 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 vi Chapter 1 Introduction This dissertation is devoted to the study of principal spectral theory of nonlocal disper- sal operators and the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations with di erent boundary conditions in a uni ed way. First, let us introduce the prototype of nonlocal problems that will be considered. Let k : RN !R be a nonnegative, continuous function with unit integral. Nonlocal dispersal evolution equations of the form @tu(t;x) = Z RN k(x y)u(t;y)dy u(t;x) +F(t;x;u); x2 D; (1.1) and variations of it, have been widely used to model di usive processes. More precisely, if u(t;x) is thought of as a density at time t and spatial location x of a species and k(x y) is thought of as the probability distribution of jumping from location y to lo- cation x, then RRN k(x y)u(t;y)dy is the rate at which individuals are arriving at position x from all other places and u(t;x) = RRN k(x y)u(t;x)dy is the rate at which they are leaving location x to travel to all other sites. This consideration leads to the fact that RRN k(x y)u(t;y)dy u(t;x) is a dispersal operator which measures the di usion or redistribution of the species with > 0 being the dispersal rate. In (1.1), F(t;x;u) is the external or internal sources, and D RN is the habitat which is not necessarily bounded. Throughout the dissertation, we have the following assumptions for the kernel function k( ). (H0) k( )2C1c(RN;R+); RRN k(z)dz = 1; and k(0) > 0 1 If there is > 0 such that supp(k( )) B(0; ) := fz 2 RNjkzk < g and for any 0 < ~ < , supp(k( ))\ B(0; )nB(0;~ ) 6= ;, is called the dispersal distance of the nonlocal dispersal operators. The operator u( ) 7! RRN k( y)u(y)dy u( ) (and variations of it), and equation (1.1) (and its variations) are called the nonlocal dispersal operator, and nonlocal dispersal evolution equation, respectively, since the di usion of the density u(t;x) at time t and some location x2 D depends not only on the values of u(t;x) and its derivatives in an immediate neighborhood of x, but also on the values of u(t;y) with y being far away from x through the convolution term RRN k(x y)u(t;y)dy. Thus nonlocal dispersal is widely used to model the population dynamics of a species in which the movements or interactions of the organisms occur between non-adjacent spatial locations. Classically, one assumes that the internal interactions of the organisms or individuals of some species are random and local, which leads to the well-known reaction-di usion equations of the following form, @tu(t;x) = u(t;x) +F(t;x;u); x2D; (1.2) where u 7! u is the so-called Laplacian operator in literature, which characterizes the di usion of organisms moving randomly between adjacent spatial locations. And , D RN, and F(t;x;u) have the same meanings as in (1.1). Thus, (1.2) as well as its variations has been extensively studied in modeling the population dynamics of species. In contrast to the nonlocal counterparts, u7! u (and variations of it) and (1.2) (and its variations) are called random dispersal operator and random dispersal evolution equation, respectively. Both nonlocal and random dispersal evolution equations are then of great interests in their own. And they are related to each other. In order to indicate some relationship between nonlocal and random dispersal operators, we assume that k( ) is of the form, k(z) = k (z) := 1 Nk0 z (1.3) 2 for some k0( ) satisfying that k0( ) is a smooth, nonnegative, and symmetric (in the sense that k0(z) = k0(z0) whenever jzj = jz0j) function supported on the unit ball B(0;1) and R RN k0(z)dz = 1, where (> 0) is the dispersal distance. We also assume that = := C 2; (1.4) where C = 1 2 R RN k0(z)z 2 Ndz 1 . Then for any smooth function u(x), we have Z RN k (x y)[u(y) u(x)]dy = C 2 Z RN 1 Nk0 x y [u(y) u(x)]dy = C 2 Z RN k0(z)[u(x+ z) u(x)]dz = C 2 Z RN k0(z) " (ru(x) z) + 2 2 NX i;j=1 uxixjzizj +O( 3) # dz = u(x) +O( ): Hence, the nonlocal dispersal operator u( )7! RRN k ( y)[u(y) u( )]dy \behaves" the same as the random dispersal operator u7! u for 1. Next, let us consider the general boundary value problems with nonlocal dispersal op- erators in a bounded domain D or unbounded domain RN. For random dispersal evolution equations, the two most common boundary conditions on a bounded domain are Neumann?s and Dirichlet?s. When looking at boundary conditions for nonlocal problems on a bounded domain, one has to modify the usual formulations for random problems. The nonlocal dispersal equation with homogeneous Dirichlet type boundary condition is 8 >>< >>: @tu(t;x) = RRN k(x y)u(t;y)dy u(t;x) +F(t;x;u); x2 D; u(t;x) = 0; x =2D; (1.5) 3 or equivalently @tu(t;x) = Z D k(x y)u(t;y)dy u(t;x) +F(t;x;u); x2 D: (1.6) In the model described by (1.5), di usion takes place in the whole RN, but we assume that u vanishes outside D. The biological interpretation is that we have a hostile environment outside D, and any individual that jumps outside dies instantaneously. This is an analog of what is called homogeneous Dirichlet boundary condition in literature, that is, 8 >>< >>: @tu(t;x) = u(t;x) +F(t;x;u); x2D; t> 0; u(t;x) = 0; x2@D: (1.7) However, the boundary datum is not understood in the classical sense for (1.5), since we are not imposing that uj@D = 0. In the model described by (1.6), the integralRDk(x y)u(t;y)dy takes into account the individuals arriving at position x2 D from other places in D, which indicates that individuals arriving at x2 D are not from the outside of D, because there is nothing living outside of D. However, all individuals can leave D and travel to all other places, which are represented by u(x). That?s why (1.5) and (1.6) are equivalent. The nonlocal dispersal equation with homogeneous Neumann type boundary condition is @tu(t;x) = Z D k(x y)[u(t;y) u(t;x)]dy +F(t;x;u); x2 D: (1.8) In this model, the integral term takes into account the di usion inside D. In fact, as we have explained, the integral RRN k(x y)[u(t;y) u(t;x)]dy takes into account the individuals arriving at or leaving position x from or to other places. Since we are integrating over D, we are assuming that di usion takes place only in D. Biologically, the individuals may not enter or leave the domain D. This is analogous to the so-called homogeneous Neumann boundary 4 condition in the literature, which is 8> >< >>: @tu(t;x) = u(t;x) +F(t;x;u); x2D; @u @n(t;x) = 0; x2@D; (1.9) where n is the exterior unit normal vector of @D. The nonlocal dispersal equation on unbounded domain is prescribed with the periodic boundary condition 8 >>< >>: @tu(t;x) = RRN k(x y)u(t;y)dy u(t;x) +F(t;x;u); x2RN; u(t;x) = u(t;x+pjej); x2RN (1.10) (j = 1;2; ;N), where pj > 0 and ej denotes the vector with a 1 in the jth coordinate and 0?s elsewhere, and F(t;x;u) = F(t;x + pjej;u) for j = 1;2; ;N. We remark that heterogeneities are present in many biological end ecological models. The periodicity of the unbounded domain takes into account the periodic heterogeneities of the media. The random dispersal equation with periodic boundary condition is 8 >>< >>: @tu(t;x) = u(t;x) +F(t;x;u); x2RN; u(t;x) = u(t;x+pjej); x2RN: (1.11) In order to study the three types of boundary condition in a uni ed way, we summarize (1.5) or (1.6), (1.8) and (1.10) as follows: 8> >< >>: @tu(t;x) = RD[Dck(x y)[u(t;y) u(t;x)]dy +F(t;x;u); x2 D; Bn;bu(t;x) = 0; x2Dc (x2RN if D = RN); (1.12) where D is a smooth bounded domain of RN or D = RN; Dc = RNnD or Dc =;. When D is bounded and Dc = RNnD, Bn;bu = Bn;D := u (in such case, Bn;Du = 0 on Dc represents 5 homogeneous Dirichlet type boundary condition); when D is bounded and Dc =;, Bn;bu = 0 on Dc trivially holds (we denote Bn;bu by Bn;Nu for convenience) and indicates that nonlocal di usion takes place only in D (hence Bn;Nu = 0 on Dc represents homogeneous Neumann type boundary condition); when D = RN, it is assumed that F(t;x + pjej;u) = F(t;x;u) and Bn;bu = Bn;Pu := u(t;x+pjej) u(t;x) for j = 1;2; ;N (hence Bn;Pu = 0 represents periodic boundary condition). Analogously, (1.7), (1.9) and (1.11) can be written as 8 >>< >>: @tu(t;x) = u(t;x) +F(t;x;u); x2D; Br;bu(t;x) = 0 x2@D (x2RN if D = RN); (1.13) where D is a smooth bounded domain or D = RN. When D is a bounded domain, Br;bu = Br;Du := u (in such case, Br;Du = 0 on @D represents homogeneous Dirichlet boundary condition) or Br;bu = Br;Nu := @u@n (in such case, Br;Nu = 0 on @D represents homogeneous Neumann boundary condition), and when D = RN, it is assumed that F(t;x;u) is periodic in xj with period pj and Br;bu = Br;Pu := u(t;x+pjej) u(t;x) for j = 1;2; ;N (in such case, Br;Pu = 0 represents periodic boundary condition). Finally, let us recall some existing results, and brie y introduce the main objective of this dissertation. Toward various dynamical aspects of random dispersal evolution equa- tions of the form (1.2), a huge amount of research has been carried out (see [3, 4, 5, 10, 28, 29, 34, 44, 52, 64, 68], etc.). And there are many research works toward var- ious dynamical aspects of nonlocal dispersal evolution equations of the form (1.1) (see [7, 11, 12, 14, 17, 19, 20, 27, 31, 38, 41, 46, 47, 66], etc.). It has been seen that random dispersal evolution equations with Dirichlet, or Neumann, or period boundary condition and nonlocal dispersal evolution equations with the corresponding boundary condition share many similar properties. For example, a comparison principle holds for both equations. There are also many di erences between these two types of dispersal evolution equations. 6 For example, solutions of random dispersal evolution equations have smoothness and cer- tain compactness properties, but solutions of nonlocal dispersal evolution equations do not have such properties. Many fundamental dynamical issues for nonlocal dispersal evolution equations are far away from being well understood. The objective of this dissertation is to investigate two dynamical issues, one is the principal spectral theory of nonlocal dis- persal operators (see Chapter 4), and the other is the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (see Chapter 5). Spectral theory for random and nonlocal dispersal operators is a basic technical tool for the study of nonlinear evolution equations with random and nonlocal dispersals. The following is the eigenvalue problem of time homogeneous nonlocal dispersal operator with Dirichlet, Neumann or periodic types of boundary condition 8 >>< >>: RD[Dck(x y)[u(y) u(x)]dy +a(x)u(x) = u(x); x2 D; Bn;bu(x) = 0; x2@D (x2RN if x = RN); (1.14) where k( ) are as in (H0), and a(x + pjej) = a(x) ( j = 1;2; ;N) in the case of periodic boundary condition. Observe that the eigenvalue problems (1.14) can be viewed as the nonlocal counterpart of the following eigenvalue problems associated with random dispersal operators, 8 >>< >>: u(x) +a(x)u(x) = u(x); x2D; Br;bu(x) = 0; x2@D (x2RN if D = RN); (1.15) where a(x+pjej) = a(x) ( j = 1;2; ;N) in the case of periodic boundary condition. The eigenvalue problem (1.15) and in particular, its associated principal eigenvalue problem, are well understood. For example, it is known that the largest real part, denoted by R( ;a), of the spectrum set of (1.15) is an isolated algebraically simple eigenvalue with 7 a positive eigenfunction, and for any other in the spectrum set, Re < R( ;a) ( R( ;a) is called the principal eigenvalue of the random operator in literature). The principal eigenvalue problem (1.14) has also been studied recently by many people (see [17], [30], [37], [41], [61], [60], and references therein). Let ~ N( ;a) be the largest real part of the spectrum set of (1.14) (in case that the kernel function k( ) depends on , we use ~ N( ;a; )). ~ N( ;a) is called the principal spectrum point of the nonlocal dispersal operator, ~ N( ;a) is also called the principal eigenvalue of (1.14), if it is an isolated algebraically simple eigenvalue with a positive eigenfunction (see De nition 2.1 and Remark 2.2(2) for detail). It is known that a nonlocal dispersal operator may not have a principal eigenvalue (see [17], [61] for examples), which reveals some essential di erence between nonlocal and random dispersal operators. Some su cient conditions are provided in [17], [41], and [61] for the existence of principal eigenvalue of (1.14). Such su cient conditions have been found important in the study of nonlinear evolution equations with nonlocal dispersals (see [17], [35], [37], [41], [42], [45], [61], [62], [63]). However, the understanding is still little to many interesting questions regarding the principal spectrum points/principal eigenvalues of nonlocal dispersal operators, including the dependence of principal spectrum point or principal eigenvalue (if exists) of nonlocal dispersal operators on the underlying parameters. In Chapter 4, we study the e ects of the spatial inhomogeneity, the dispersal rate, and the dispersal distance on the existence of principal eigenvalues, on the magnitude of the principal spectrum points, and on the asymptotic behavior of the principal spectrum points of nonlocal dispersal operators. Among others, we obtain the following: criteria for ~ N( ;a) to be the principal eigenvalue of (1.14) (see Theorem 2.4 (1), (2), Theorem 2.6 (3), and Theorem 2.8 (3) for detail); lower bounds of ~ N( ;a) in terms of ^a (where ^a is the spatial average of a(x)) in the Neumann and periodic boundary cases (see Theorem 2.4 (4) for detail); monotonicity of ~ N( ;a) with respect to a(x) and (see Theorem 2.4 (5) and Theorem 2.6 (1) for detail); 8 limits of ~ N( ;a) as !0 and !1 (see Theorem 2.6 (4), (5) for detail); limits of ~ N( ;a; ) as ! 0 and !1 in the case k( ) = k ( ), where k ( ) is as in (1.3). (see Theorem 2.8 (1), (2) for detail). In Chapter 4, we also discuss the applications of principal spectral theory of nonlocal dispersal operators to the asymptotic dynamics of the following two species competition system, 8 >>< >>: ut = [RDk(x y)u(t;y)dy u(t;x)] +uf(x;u+v); x2 D; vt = RDk(x y)[u(t;y) u(t;x)]dy +vf(x;u+v); x2 D; (1.16) where D and k( ) are as in (1.14) and f( ; ) is a C1 function satisfying that ~ ( ;f( ;0)) > 0, f(x;w) < 0 for w 1, and @2f(x;w) < 0 for w > 0. (1.16) models the population dynamics of two competing species with the same local population dynamics (i.e. the same growth rate function f( ; )), the same dispersal rate (i.e. ), but one species adopts nonlocal dispersal with Dirichlet type boundary condition and the other adopts nonlocal dispersal with Neumann type boundary condition, where u(t;x) and v(t;x) are the population densities of two species at time t and space location x. We show the species di using nonlocally with Neumann type boundary condition drives the species di using nonlocally with Dirichlet type boundary condition extinct (see Theorem 2.12 for detail). As mentioned in the above, nonlocal dispersal operators/equations and random disper- sal operators/equations share many properties and there are also many di erences between them. Thanks to the formal relation between the random operator u7! u and nonlocal dispersal operator u( ) 7! RRN k ( y)[u(y) u( )]dy for su ciently small with k and being as in (1.3) and (1.4), respectively, it is expected that nonlocal dispersal evolution equations with Dirichlet, or Neumann, or periodic boundary condition and small disper- sal distance possess similar dynamical behaviors as those of random dispersal evolution equations with the corresponding boundary condition and that certain dynamics of random dispersal evolution equations with Dirichlet, or Neumann, or periodic boundary condition 9 can be approximated by the dynamics of nonlocal dispersal evolution equations with the corresponding boundary condition and properly rescaled kernels. It is of great theoretical and practical importance to investigate whether such naturally expected properties actually hold or not. Regarding the approximations of dynamics of random dispersal operators or equations by those of nonlocal dispersal operators or equations, we investigate from three di erent points of view, that is, from initial-boundary value problem point of view, from spectral problem point of view, and from asymptotic behavior point of view. To this end, throughout Chapter 5, we assume (H1) D RN is either a bounded C2+ domain for some 0 < < 1 or D = RN; k( ) = k ( ) de ned as in (1.3) and = de ned as in (1.4). We rst explore the approximation in terms of solutions of initial-boundary value prob- lems. Consider (1.13) and (1.12) with the assumption (H1) for random and nonlocal cases, respectively. To be more precise, let F(t;x;u) be C1 in t2R and C3 in (x;u) 2R RN, and F(t;x+pjej;u) = F(t;x;u) (j = 1;2; ;N) in case of D = RN. With an initial value u0(x) at t = s, (1.13) in case of = 1 is 8 >>< >>: @tu(t;x) = u(t;x) +F(t;x;u); x2D; Br;bu(t;x) = 0; x2@D (x2RN if D = RN): (1.17) By general semigroup theory, for any u0 2C( D) with Br;bu0 = 0 on @D, (1.17) has a unique (local) solution, denoted by u(t; ;s;u0), such that u(t; ;s;u0) = u0( ). Similarly, with the same initial value u0(x) at t = s, (1.12) in case of = and k( ) = k ( ) is 8> >< >>: @tu(t;x) = RD[Dck (x y)[u(t;y) u(t;x)]dy +F(t;x;u); x2 D; Bn;bu(t;x) = 0; x2Dc (x2RN if D = RN): (1.18) 10 By general semigroup theory, for any u0 2C( D), (1.18) has a unique (local) solution, denoted by by u (t; ;s;u0), such that u (s; ;s;u0) = u0( ). Among others, we prove that for any u0 2 C3( D) with Br;bu0 = 0, and any T > 0 satisfying that u(t; ;s;u0) and u (t; ;s;u0) exists on [s;s+T], we have lim !0 sup t2[s;s+T] ku (t; ;s;u0) u(t; ;s;u0)kC( D) = 0 (see Theorem 2.13 for details). We remark that Theorem 2.13 is fundamental in the study of approximation results. And in fact, the smoothness of the initial value u0 is not optimal. But as the optimal smoothness is not what we are seeking for, we assume u0 2C3( D) technically. Secondly, we investigate the following eigenvalue problem with time periodic random dispersal 8 >>> >>> < >>> >>> : @tu+ u+a(t;x)u = u; x2D; Br;bu = 0; x2@D (x2RN if D = RN); u(t+T;x) = u(t;x); x2D; (1.19) and the nonlocal counterpart are as follows 8 >>>> >>< >>> >>>: @tu+ RD[Dck (x y) [u(t;y) u(t;x)]dy +a(t;x)u = u; x2 D; Bn;bu = 0; x2Dc (x2RN if D = RN); u(t+T;x) = u(t;x); x2 D; (1.20) where Br;b = Br;D (resp. Bn;b = Bn;D) or Br;b = Br;N (resp. Bn;b = Bn;N) or Br;b = Br;P (resp. Bn;b = Bn;P). We assume that a(t;x) is a C1 function in (t;x)2R RN, a(t+T;x) = a(t;x), and a(t+T;x+pjej) = a(t;x) (j = 1;2; ;N) in case of D = RN. The eigenvalue problem of (1.19) with a(t;x) a(x) reduces to (1.15) with = 1. The principal eigenvalue problem associated to (1.19) has been extensively studied and is quite well understood (see [2, 22, 23, 33, 37, 39, 54, 58], etc.). For example, with any one of the three boundary conditions, it is known that the largest real part, denoted by R(1;a), of the 11 spectrum set of (1.19) is an isolated algebraically simple eigenvalue of (1.19) with a positive eigenfunction, and for any other in the spectrum set of (1.19), Re R(1;a) ( R(1;a) is called the principal eigenvalue in literature). The eigenvalue problem (1.20) with a(t;x) a(x) reduces to (1.14) with = and k( ) = k ( ). The principal spectrum problem associated to (1.20) has also been studied recently by many people (see [8, 17, 39, 56, 58, 61, 62, 59], etc.). The largest real part of the spectrum set of (1.20) with any one of the three boundary conditions, denoted by ~ N( ;a; ) is called the principal spectrum point of (1.20). ~ N( ;a; ) is also called the principal eigenvalue of (1.20), if it is an isolated algebraically simple eigenvalue of (1.20) with a positive eigenfunction (see De nition 2.1 for detail). For simplicity, we put ~ N( ;a; ) = ~ (a)( N( ;a; ) = (a) if N( ;a; ) exists), and R(1;a) = r(a) (see Remark 2.2 and Remark 2.19 for detail) and show that the principal eigenvalue of (1.19) can be approximated by the principal spectrum point of (1.20) in case that goes to zero, that is, lim !0 ~ (a) = r(a) (see Theorem 2.15 for details). We remark that some necessary and su cient conditions are provided in [56] and [57] for the existence of principal eigenvalues of (1.20) (see Remark 2.11 for detail). This together with Theorem 2.15 implies the following remark. Remark 1.1. The principal eigenvalue (a) of (1.20) exists provided 1. We also remark that Theorem 2.15 is another basis for the study of approximations of various dynamics of random dispersal evolution equations by those of nonlocal dispersal evolution equations. Thirdly, we explore the asymptotic dynamics of the following time periodic KPP type evolution equation with random dispersal 8 >>< >>: @tu = u+uf(t;x;u); x2D; Br;bu = 0; x2@D (x2RN if D = RN); (1.21) 12 and the time periodic KPP type evolution equation with nonlocal dispersal 8> >< >>: @tu = RD[Dck (x y)[u(t;y) u(t;x)]dy +uf(t;x;u); x2 D; Bn;bu = 0; x2Dc (x2RN if D = RN): (1.22) We assume the following monostable assumptions on f: (H2)f is C1 in t2R and C3 in (x;u)2R RN; f(t;x;u) < 0 for u 1 and @uf(t;x;u) < 0 for u 0; f(t + T;x;u) = f(t;x;u); and when D = RN, f(t + T;x;u) = f(t;x + pjej;u) = f(t;x;u) for j = 1;2; ;N. (H3) For (1.21), r(f( ; ;0)) > 0, where r(f( ; ;0)) is the principle eigenvalue of (1.19) with a(t;x) = f(t;x;0). (H3) For (1.22), ~ (f( ; ;0)) > 0, where ~ (f( ; ;0)) is the principle spectrum point of (1.20) with a(t;x) = f(t;x;0). Equations (1.21) and (1.22) are widely used to model population dynamics of species exhibiting random interactions and nonlocal interactions, respectively (see [7, 20, 53], etc. for (1.21) and [56] for (1.22)). Thanks to the pioneering works of Fisher [29] and Kolmogorov et al. [44] on the following special case of (1.21), @tu = uxx +u(1 u); x2R; (1.21) and (1.22) are referred to as Fisher type or KPP type evolution equations. The dynamics of (1.21) and (1.22) have been studied in many papers (see [34, 53, 67] and references therein for (1.21), and [56] and references therein for (1.22)). With conditions (H2) and (H3), it is proved that (1.21) has exactly two nonnegative time periodic solutions, one is u 0 which is unstable and the other one, denoted by u (t;x), is asymptotically stable and strictly positive (see [67, Theorem 3.1], see also [53, Theorems 1.1, 1.3]). Similar results for (1.22) under the assumptions (H2) and (H3) are proved in [56, Theorem E]. We 13 denote the strictly positive time periodic solution of (1.22) by u (t;x). In Chapter 5, we show If (H2) and (H3) hold, supt2[0;T]ku (t; ) u (t; )kC( D;R) !0, as !0 (see Theorem 2.16 for detail). Theorems 2.13-2.16 show that many important dynamics of random dispersal equations can be approximated by the corresponding dynamics of nonlocal dispersal equations, which is of both great theoretical and practical importance. At the end of Chapter 5, we apply the approximation theorems to the e ect of rearrangement with equimeasurability on principal spectrum point of nonlocal dispersal operators. The rest of the dissertation is organized as follows. In Chapter 2, we state some standing notations, assumptions, de nitions, and the main results. In Chapter 3, we develop some basic tools for fundamental theory to be used in later Chapters, such as semigroup theory, comparison principle, sub- and super-solutions. We will investigate the spectral theory of time homogeneous nonlocal dispersal operators in Chapter 4. In Chapter 5, we study the approximations of random dispersal evolution operators/equations by the nonlocal dispersal evolution operators/equations. The dissertation will end with concluding remarks, several problems which are not well understood yet, and future plan in Chapter 6. 14 Chapter 2 Notations, Assumptions, De nitions and Main Results In this chapter, we introduce rst the standing notations, assumptions, and the def- initions to be used in the rest of the dissertation. We then state the main results of the dissertation. 2.1 Notations, Assumptions and De nitions Throughout this section, we will distinguish the three boundary conditions by i = 1;2;3. We rst introduce the spaces of time independent functions and their norms. Let X1 = X2 = C( D) (2.1) with norm kukXi = maxx2 Dju(x)j for i = 1;2, X3 =fu2C(RN;R)ju(x+pjej) = u(x); x2RN;j = 1;2; ;Ng (2.2) with norm kukX3 = maxx2RNju(x)j. And X+i =fu2Xiju(x) 0; x2 Dg; (2.3) X++i = Int(X+i ) =fu2X+i ju(x) > 0; x2 Dg (2.4) (i = 1;2;3). For u1( );u2( )2Xi, we de ne u1 u2(u1 u2); if u2 u1 2X+i (u1 u2 2X+i ); (2.5) 15 u1 u2(u1 u2); if u2 u1 2X++i (u1 u2 2X++i ) (2.6) (i = 1;2;3). For time periodic functions, we introduce the following spaces, together with their norms. Let X1 =X2 =fu2C(R D;R)ju(t+T;x) = u(t;x)g with norm kukXi = supt2[0;T]ku(t; )kXi(i = 1;2), X3 =fu2C(R RN;R)ju(t+T;x) = u(t;x+pjej) = u(t;x)g with norm kukX3 = supt2[0;T]ku(t; )kX3. And X+i =fu2Xiju(t;x) 0g; (2.7) X++i = Int(X+i ) =fu2X+i ju(t;x) > 0g (2.8) (i = 1;2;3). For u1;u2 2Xi, we de ne u1 u2(u1 u2); if u2 u1 2X+i (u1 u2 2X+i ); (2.9) u1 u2(u1 u2); if u2 u1 2X++i (u1 u2 2X++i ) (2.10) (i = 1;2;3). The introduction of X2 and X2 is for convenience. Next, we introduce the de nitions of principal spectrum point and principal eigenvalues for nonlocal dispersal operators. For i = 1;2;3, let ai( ; ) 2Xi\C1(R RN), i > 0, and Ni( i;ai) : D(Ni( i;ai)) Xi!Xi be de ned as follows, (N1( 1;a1)u)(t;x) = @tu(t;x) + 1 Z D k(x y)u(t;y)dy u(t;x) +a1(t;x)u(t;x) (2.11) 16 for (t;x)2R D, (N2( 2;a2)u)(t;x) = @tu(t;x) + 2 Z D k(x y)[u(t;y) u(t;x)]dy +a2(t;x)u(t;x) (2.12) for (t;x)2R D, and (N3( 3;a3)u)(t;x) = @tu(t;x) + 3 Z RN k(x y)[u(t;y) u(t;x)]dy +a3(t;x)u(t;x) (2.13) for (t;x)2R RN. De nition 2.1 (Principal Eigenvalue). For i = 1;2;3, let (Ni( i;ai)) be the spectrum of Ni( i;ai) on Xi (1) ~ Ni ( i;ai) := supfRe j 2 (Ni( i;ai))g is called the principal spectrum point of Ni( i;ai). (2) A real number Ni ( i;ai) is called the principal eigenvalue of (1.20) orNi( i;ai) if it is an isolated algebraically simple eigenvalue of Ni( i;ai) with an eigenfunction v2X+i , and for every 2 (Ni( i;ai))nf Ni ( i;ai)g, Re Ni ( i;ai). Observe that if the principal eigenvalue Ni ( i;ai) exists, then ~ Ni ( i;ai) = Ni ( i;ai). If k( ) depends on , we put ~ Ni ( i;ai) = ~ Ni ( i;ai; ): (2.14) Remark 2.2. (1) We use the super script N to indicate that both principal eigenvalue and principal spectrum point are for nonlocal operators. If there is no confusion, the notation can be simpli ed. For example, in Chapter 4, we only focus on nonlocal dispersal operators and consider the dependence of their principal spectrum points and principal eigenvalues on underlying parameters i;ai and , so we put ~ Ni ( i;ai; ) = ~ i( i;ai; ) for principal spectrum point and Ni ( i;ai; ) = i( i;ai; ) for principal eigenvalue, respectively. In Chapter 5, we consider the approximation of random dispersal operators by nonlocal dispersal operators as 17 the parameter goes to zero. More precisely, in (1.20), k( ) = k ( ) is de ned as in (1.3) and i = is de ned as (1.4), so we put ~ Ni ( ;a; ) = ~ i(a) for principal spectrum point and Ni ( i;a) = i(a) for principal eigenvalue, respectively. (2) In the case ai(t;x) ai(x) (i = 1;2;3), let Ki : Xi!Xi; (Kiu)(x) = Z D k(x y)u(y)dy 8u2Xi; i = 1;2; (2.15) and K3 : X3 !X3; (K3u)(x) = Z RN k(x y)u(y)dy 8u2X3: (2.16) Let 8 >>>> >>< >>>> >>: h1(x) = 1 +a1(x); h2(x) = 2RDk(x y)dy +a2(x); h3(x) = 3 +a3(x): (2.17) Then we have ~ Ni ( i;ai) = supfRe j 2 ( iKi +hi( )I)g; (2.18) where I is the identity map on Xi. Moreover, a real number 2R is called the principal eigenvalue of iKi +hi( )I if it is an isolated algebraically simple eigenvalue of iKi +hi( )I with a positive eigenfunction and for any 2 ( iKi+hi( )I)nf g, Re < . The principal eigenvalue of Ni( i;ai) exists i the principal eigenvalue of iKi +hi( )I exists. The spectral theory of random dispersal operators is well known. For the time periodic random dispersal operators, leta( ; )2Xi\C1(R RN), andRi(a) :D(Ri( i;ai)) Xi!Xi be de ned as follows, (Ri( i;ai)u)(t;x) = @tu(t;x) + i u(t;x) +ai(t;x)u(t;x) 18 for i = 1;2;3. Note that for u2D(R1( 1;a1)), Br;Du = 0 on @D and for u2D(R2( 2;a2)), Br;Nu = 0 on @D. Let Ri ( i;ai) = supfRe j 2 (Ri( i;ai))g: It is well known that Ri ( i;ai) is an isolated algebraically simple eigenvalue of Ri( i;ai) with a positive eigenfunction (see [33]) and Ri ( i;ai) is called the principal eigenvalue of Ri( i;ai) in literature. Recently, the principal eigenvalue problem for nonlocal dispersal operators has been studied by several authors (see [41] for time homogeneous case; see [39] for time-periodic and almost time-periodic cases; see [58] for general time-periodic cases). Remark 2.3. In Chapter 5, we consider the approximation of principal eigenvalues R(1;a) of random dispersal operators in (1.19) by the principal spectrum point ~ Ni ( ;a; ) (i = 1;2;3) of nonlocal dispersal operators in (1.20). We simpli ed the notation in the nonlocal case in Remark 2.2, so for our convenience, we put Ri (1;a) = ri(a) for i = 1;2;3: (2.19) 2.2 Main Results In this section, we state the main results of this dissertation. We rst state the results of the dependence of principal spectrum points/principal eigen- values on the underlying parameters. In the following, we put D = [0;p1] [0;p2] [0;pN]; (2.20) in periodic boundary condition case. For given ai2Xi, let ^ai = 1jDj Z D ai(x)dx; i = 1;2;3; (2.21) 19 where jDj is the Lebesgue measure of D. Let ai;max = max x2 D ai(x); ai;min = min x2 D ai(x); and hi;max = max x2 D hi(x); hi;min = min x2 D hi(x); where hi( ) is as in (2.17). If no confusion occurs, we put ~ i( i;ai) = ~ Ni ( i;ai) and i( i;ai) = Ni ( i;ai) if Ni ( i;ai) exists. Theorem 2.4 (E ects of spatial variation). Let 1 i 3 and ai( )2Xi be given. (1) (Existence of principal eigenvalues) For given 1 i 2, i( i;ai) exists if ai;max ai;min < i infx2 DRDk(x y)dy. (2) (Existence of principal eigenvalues) For given 1 i 2, i( i;ai) exists if hi( ) is in CN( D), there is some x0 2 Int(D) satisfying that hi(x0) = hi;max, and the partial derivatives of hi(x) up to order N 1 at x0 are zero. (3) (Upper bounds) For given 1 i 3 and ci2R, supf~ i( i;ai)jai2Xi; ^ai = cig=1. (4) (Lower bounds) Assume that k( ) is symmetric with respect to 0 (i.e. k( z) = k(z)) and i = 2. For given ci2R, inff~ i( i;ai)jai2Xi; ^ai = cig= i( i;ci)(= ci) (hence ~ i( i;ai) ~ i( i;^ai)). If the principal eigenvalue of iKi + hi( )I exists, then the in mum is attained by the constant function (i.e. ai( ) ^ai). (5) (Monotonicity) For given a1i;a2i 2 Xi, if a1i(x) a2i(x), then ~ i(a1i; i) ~ i(a2i; i) (i = 1;2;3). 20 Remark 2.5. (1) For the case i = 3, similar result to Theorem 2.4(1) is proved in [61]. To be more precise, it is proved in [61] that if a3;max a3;min < 3, then 3( 3;a3) exists. (2) For the case i = 3, similar result to Theorem 2.4(2) is also proved in [61]. Actually it is proved in [61] that if a3( ) is CN and there is x0 2RN such that a3(x0) = a3;max and the partial derivatives of a3(x) up to order N 1 at x0 are zero, then 3( 3;a3) exists. (3) For one space dimensional random dispersal operators, for given ci2R, supf Ri ( i;ai)jai2X++i ; ^ai = cig<1 (see Remark 4.8 for detail). Theorem 2.4(3) hence re ects some di erence between random dispersal operators and nonlocal dispersal operators. (4) Similar result to Theorem 2.4(4) holds for i = 3. To be more precise, it is proved in [63] that for any given c3 2R, inff~ 3( 3;a3)ja3 2X3; ^a3 = c3g= 3( 3;c3)(= c3): But Theorem 2.4(4) may not hold for the case i = 1 (see Remark 4.8 for detail). Theorem 2.6 (E ects of dispersal rate). Assume that 1 i 3 and k( ) is symmetric with respect to 0. Let ai2Xi be given. (1) (Monotonicity) Assume ai( )6 constant. If 1i < 2i , then ~ i( 1i;ai) > ~ i( 2i;ai). (2) (Existence of principal eigenvalue) If i = 1 or 3 and i( i;ai) exists for some i > 0, then i(~ i;ai) exists for all ~ i > i. (3) (Existence of principal eigenvalue) There is 0i > 0 such that the principal eigenvalue i( i;ai) of iKi +hi( )I exists for i > 0i . (4) (Limits as the dispersal rate goes to 0) lim i!0+ ~ i( i;ai) = ai;max. 21 (5) (Limits as the dispersal rate goes to 1) lim i!1 ~ i( i;ai) = 1 for i = 1 and lim i!1 ~ i( i;ai) = ^ai for i = 2 and 3. Remark 2.7. (1) It is open whether Theorem 2.6 (2) holds for the case i = 2. (2) Theorem 2.6 (3) and (4) still hold if k( ) is not symmetric. In the case that k( ) = k ( ) de ned as in (1.3) for > 0, to indicate the dependence of ~ i( i;ai) on , put ~ i( i;ai; ) = ~ i( i;ai): Theorem 2.8 (E ects of dispersal distance). Suppose that k(z) = k (z), where k (z) is de ned as in (1.3) and k(z) = k( z). Let 1 i 3 and ai2Xi be given. (1) (Limits as dispersal distance goes to 0) lim !0 ~ i( i;ai; ) = ai;max. (2) (Limits as dispersal distance goes to 1) lim !1 ~ 1( 1;a1; ) = 1 +a1;max; lim !1 ~ 2( 2;a2; ) = a2;max; and lim !1 ~ 3( 3;a3; ) = 3( 3;a3); where 3( 3;a3) = maxfRe j 2 ( 3 I+h3( )I)g; and Iu = 1 jDj Z D u(x)dx: (3) (Existence of principal eigenvalue) There is 0 > 0 such that the principal eigenvalue i( i;ai) of iKi +hi( )I exists for 0 < < 0. 22 Remark 2.9. (1) For i = 1 or 3, Theorem 2.8 (1) is proved in [41, Theorem 2.6]. (2) For i = 1 or 3, Theorem 2.8 (3) is proved in [41] (see also [61] for the case i = 3). Corollary 2.10 (Criteria for the existence of principal eigenvalues). Let 1 i 3 and ai2Xi be given. (1) i( i;ai) exists provided that maxx2 Dai(x) minx2 Dai(x) < i infx2 DRDk(x y)dy in the case i = 1;2 and maxx2 Dai(x) minx2 Dai(x) < i in the case i = 3. (2) i( i;ai) exists provided that hi( ) is in CN( D), there is some x0 2 Int(D) satisfying that hi(x0) = hi;max, and the partial derivatives of hi(x) up to order N 1 at x0 are zero. (3) There is 0i > 0 such that the principal eigenvalue i( i;ai) of iKi + hi( )I exists for i > 0i . (4) Suppose that k(z) = k (z), where k (z) is de ned as in (1.3) and ~k( ) is symmetric with respect to 0. Then there is 0 > 0 such that the principal eigenvalue i( i;ai) of iKi +hi( )I exists for 0 < < 0. Proof. (1) and (2) are Theorem 2.4(1) and (2), respectively. (3) is Theorem 2.6(3). (4) is Theorem 2.8(3). Remark 2.11 (Conditions for the existence of principal eigenvalue in time periodic cases). The results of conditions for the existence of principal eigenvalue have been extended to time periodic nonlocal dispersal operators of Ni( i;ai) (i = 1;2;3) in [56]. More precisely, for given 1 i 3, and ai( ; )2Xi, let ai(x) = 1T Z T 0 ai(t;x)dt; b1 = 1; b2 = 2 Z D k(x y)dy; and b3 = 3: 23 The following conditions for the existence of principal eigenvalues of the nonlocal dispersal operators of Ni( i;ai) have already been proved in [56]. (1) (Necessary and su cient condition) ~ Ni ( i;ai) is the principal eigenvalue of Ni( i;ai) if and only if ~ Ni ( i;ai) > max x2 Di (bi(x) + ai(x)); where D1 = D2 = D and D3 = [0;p1] [0;p2] [0;pN] as in (2.20). (2) (Su cient condition) ~ Ni ( i;ai) is the principal eigenvalue of Ni( i;ai), provided that (a) maxx2 D ai(x) minx2 D ai(x) < iInfx2 DRDk(x y)dy in the case of i = 1;2 and maxx2 D ai(x) minx2 D ai(x) minx2 D ai(x) < i in the case i = 3 (which extends the result in Theorem 2.4(1)); or (b) bi(x) + ai(x) is in CN, there is some x0 2 Int(Di) in the case of i = 1;2, and x0 2D3 in the case of i = 3 satisfying that bi(x0) + ai(x0) = maxx2 D(bi(x) + ai(x)), and the partial derivatives of bi(x)+ ai(x) up to order N 1 at x0 is zero(which extends the result in Theorem 2.4(2)); or (c) 0 < 1 for N( i;ai; ), where > 0 is the dispersal distance and k( ) = k ( ) as in (1.3) (which extends the result in Theorem 2.8(3)). The following is an application of the above stated theorems to a two-species competition system. Theorem 2.12. (1) There are u ( ) 2 X++1 and v ( ) 2 X++2 such that (u ( );0) and (0;v ( )) are stationary solutions of (1.16). Moreover, for any (u0;v0)2X+1 X+2 with u0 6= 0 and v0 = 0 (resp. u0 = 0 and v0 6= 0), (u(t; ;u0;v0);v(t; ;u0;v0))!(u ( );0) (resp. (u(t; ;u0;v0);v(t; ;u0;v0))!(0;v ( ))) as t!1. 24 (2) For any (u0;v0) 2 (X+1 nf0g) (X+2 nf0g), limt!1(u(t; ;u0;v0);v(t; ;u0;v0)) = (0;v ( )). Next, we state the main results on the approximations of random dispersal operators or equations by nonlocal dispersal operators or equations. Recall that u (t;x;s;u0) is the solution of (1.18) with u(s;x;s;u0) = u0(x) and u(t;x;s;u0) is the solution of (1.17) with u(s;x;s;u0) = u0(x). Theorem 2.13 (Approximations of initial-boundary value problems). For any given s2R, any u0 2C3( D) with Br;bu0 = 0, and any T > 0 satisfying that u(t;x;s;u0) and u (t;x;s;u0) exist on [s;s+T], lim !0 sup t2[s;s+T] ku (t; ;s;u0) u(t; ;s;u0)kC( D) = 0: Remark 2.14. In the Dirichlet and Neumann boundary condition cases with F(t;x;u) 0 in (1.17) and (1.18), Theorem 2.13 has been proved in [15] and [16], respectively. Theorem 2.15 (Approximation of principal eigenvalues). For given 1 i 3, and a( ; )2 Xi\C1(R RN), lim !0 ~ i(a) = ri(a), where ~ (a) and r(a) are the principal spectrum point of the nonlocal dispersal operator Ni( ;a; )(see Remark 2.2), and the principal eigenvalue of the random dispersal operator Ri(1;a) (see Remark 2.19), respectively. Theorem 2.16. Consider (1.22) and (1.21). If (H2) and (H3) hold, then for any > 0, there exists 0 > 0, such that for all 0 < < 0, we have sup t2[0;T] ku (t; ) u (t; )kC( D;R) ; where u ( ; ) and u ( ; ) are the strictly positive, asymptotically stable, and time periodic solutions of (1.22), and (1.21), respectively. Remark 2.17. 25 (1) The existence, uniqueness, and asymptotic stability of u (t;x) have been proved in [67]. (2) The existence, uniqueness, and asymptotic stability of u (t;x) have been proved in [56]. Finally, we present an application of approximation theorems to the e ect of the re- arrangements with equimeasurability on principal spectrum point of nonlocal dispersal op- erators. Consider the restriction of the eigenvalue problem (1.19) on Xi (i = 1;2;3), that is, 8 >>< >>: u+a(x)u = u; x2D; Br;bu(x) = 0; x2@D(x2RN if D = RN): (2.22) Note that the principal eigenvalues of (1.19) and (2.22) are the same. Consider also the symmetrized problem 8> >< >>: u+a](x)u = u; x2D]; Br;bu(x) = 0; x2@D] x2RN if D = RN ; (2.23) where Br;bu denotes the boundary condition as in (1.19), and D] and a]( ) are the Schwarz symmetrization of D and a( ), respectively (see [1] for details of the Schwarz symmetrization). It is well-known that ri(a]) ri(a); (2.24) which simply follows from the following inequality Z D] a](x)u2](x)dx Z D a(x)u2(x)dx; (2.25) and the variational characterization of ri(a]) and ri(a), where ri(a]) is the principal eigen- value of (2.23), and ri(a) is the principal eigenvalue of (2.22) respectively. What?s more, the \=" in (2.24) holds if and only if both the domain and functions are symmetric, that is D = D], a( ) = a]( ), and u( ) = u]( ) (see [21] for details). 26 By Theorem 2.15, the principal eigenvalues of random dispersal operators can be ap- proximated by the principal spectrum point of nonlocal dispersal operators. So it is natural to expect that the the relation like (2.24) holds for principal spectrum point of nonlocal dis- persal operator. So next, we consider the eigenvalue problems of the nonlocal counterparts of (2.22), 8 >>< >>: hR D[Dck (x y)u(y)dy u(x) i +a(x)u(x) = u(x); x2 D; Bn;bu(x) = 0; x2Dc (x2RN if D = RN); (2.26) and its symmetrized problem 8 >>< >>: hR D][(D])ck (x y)u(y)dy u(x) i +a](x)u(x) = u(x); x2 D]; Bn;bu(x) = 0; x2(D])c (x2RN if D = RN); (2.27) where the kernel function k( ) is symmetric with respect to 0, and Bn;bu denotes the boundary condition as in (1.20), a]( ), k ]( ), and D] are the Schwarz symmetrization of a( ), k ( ) and D, respectively. We denote the principal spectrum point of (2.26), and (2.27) by ~ i(a) and ~ i(a]) for i = 1;2;3, respectively. We have the following comparison relation between ~ i(a) and ~ i(a]). Theorem 2.18. For 1 i 3, assume a( ) 2Xi, k ( ) and are as in (1.3) and (1.4), respectively. Let a]( ), k ] and D] be the Schwarz symmetrization of a( ), k ( ) and D. Then there exists 0 > 0, such that ~ i(a]) ~ i(a) for 0; where ~ i(a) and ~ i(a]) are the principle spectrum points of the eigenvalue problems (2.26) and (2.27), respectively. 27 Chapter 3 Preliminary In this Chapter, we establish some basic properties of solutions of nonlocal evolution equations, including the comparison principle and monotonicity of solutions with respect to initial conditions. 3.1 Solutions of Evolution Equation and Semigroup Theory For given 1 i 3, and ai( ; )2Xi, consider the following evolution equation 8 >>> >>>< >>> >>>: @tu(t;x) = iRD[Dck(x y)[u(t;y) u(t;x)]dy +ai(t;x)u(t;x); x2 D; Bn;bu(t;x) = 0; x2Dc (x2RN if D = RN); u(s;x) = u0(x); (3.1) where D RN, k( ) and Bn;bu(t;x) = 0 are the same as in (1.12). By general linear semigroup theory (see [32] and [55]), for any u0 2Xi with Bn;bu0 = 0 on Dc (Dc = RNn D and b = D when i = 1, Dc =;and b = N when i = 2, and Dc = RN and b = P when i = 3), and s2R, (3.1) has a unique (local) solution, we denote it by uNi (t; ;s;u0; i;ai). We put Ni (t;s; i;ai;u0) = uNi (t; ;s;u0; i;ai); u0 2Xi: Note that if = , ai( ; ) = a( ; ) and k( ) = k ( ), (3.1) is the evolution equation associated to the eigenvalue problem (1.20). For i = 1;2;3, we put ( i(t;s;a)u0)( ) = uNi (t; ;s;u0; ;ai); u0 2Xi: (3.2) 28 For evolution equations with random dispersal operators, let A be with Dirichlet boundary condition acting on X1\C0(D), and put Xr1 =D(A ) (3.3) for some 0 < < 1 such that C1( D) Xr1 with kukXr1 =kA ukX1, and Xri = Xi for i = 2;3 (3.4) with kukXri =kukXi. And Xr;+i =fu2Xriju(x) 0g (i=1, 2, 3). The random counterpart of (3.1) is 8 >>> >>>< >>> >>>: @tu(t;x) = i u(t;x) +ai(t;x)u(t;x); x2 D; Br;bu(t;x) = 0; x2Dc (x2RN if D = RN); u(s;x) = u0(x); (3.5) where D RN and Bn;bu(t;x) are the same as in (1.13). By general linear semigroup theory, for any u0 2Xi, Br;bu0 = 0 on @D (b = D when i = 1, b = N when i = 2, and b = P when i = 3) and s2R, (3.5) has a unique (local) solution, we denote it by uR(t;x;s;u0; i;ai). And we put Ri (t;s; i;ai;u0) = uRi (t; ;s;u0; i;ai); u0 2Xi: Note that if i = 1, and ai( ; ) = a( ; ), (3.5) is the evolution equation associated to the eigenvalue problem (1.19). Similarly, for i = 1;2;3, de ne ri(t;s;a) : Xri !Xri by ( ri(t;s;a)u0)( ) = uRi (t; ;s;u0;1;a); u0 2Xri: 29 By general nonlinear semigroup theory (see [32] and [55]), (1.18) and (1.17) has a unique (local) solution uN(t;x;s;u0) with uN(s;x;s;u0) = u0(x) for every u0 2Xi(i = 1;2;3) and uR(t;x;s;u0) with uR(s;x;s;u0) = u0(x) for every u0 2Xri (i = 1;2;3), respectively. Also by general semigroup theory for equation systems (see [32] and [55]), for any given (u0;v0)2X1 X2, (1.16) also has a unique (local) solution (u(t; ;u0;v0);v(t; ;u0;v0)) with (u(0;x;u0;v0);v(0;x;u0;v0)) = (u0(x);v0(x)). 3.2 Sub- and Super-Solutions De nition 3.1 (Sub- and Super- solutions). A continuous function u(t;x) on [s;s+T) RN is called a sub-solution (super-solution) of (1.12) on (s;s + T) if for any x2 D, u(t;x) is di erentiable on (s;s+T) and satis es that 8 >>>> >>< >>>> >>: @tu(t;x) ( ) RD[Dck(x y)[u(t;y) u(t;x)]dy +F(t;x;u); x2 D;t>s; Bn;bu(t;x) ( )0; x =2D;t>s; u(s;x) ( )u0(x); x2 D; where u0( )2Xi (i = 1;2;3) is the initial value of the solution of (1.12) at t = s. Remark 3.2. The sub- and super-solutions of evolution equation with random operator (1.13) are de ned similarly. Remark 3.3. In the Dirichlet boundary case with nonlocal kernel k( ) being k ( ), we have the following equivalent de nition for a continuous function u(t;x) on [s;s+T) RN to be the super-solution (sub-solution) of (1.5). 30 For any x2 D, u(t;x) is di erentiable on (s;s+T) and satis es that 8 >>>> >>< >>>> >>: @tu(t;x) ( ) RRN k (x y)[u(t;y) u(t;x)]dy +F(t;x;u); x2 D; u(t;x) ( )0; x2Dc; dist(x;@D) ; u(s;x) ( )u0(x); x2 D: (3.6) where is the dispersal distance and Dc = RNnD. In fact, (3.1) and (3.6) are equivalent, since supp(k ( )) B(0; ), and hence k (x y) = 0 for x2Dc\fxjdist(x;@D) g; and y2D: We will use the above de nitions for sub- and super-solutions in the proof of Theorem 2.13. Next, we consider (1.16) and present some basic properties for solutions of the two species competition system. For given (u1;v1);(u2;v2)2X1 X2, we de ne (u1;v1) 1 (u2;v2); if u1(x) u2(x); v1(x) v2(x); and (u1;v1) 2 (u2;v2); if u1(x) u2(x); v1(x) v2(x): De nition 3.4. Let T > 0 and (u(t;x);v(t;x)) 2C([0;T) D;R2) with (u(t; );v(t; )) 2 X+1 X+2 . Then (u(t;x);v(t;x)) is called a super-solution (sub-solution) of (1.16) on [0;T) if 8 >>< >>: @tu(t;x) ( ) [RDk(x y)u(t;y)dy u(t;x)] +u(t;x)f(x;u(t;x) +v(t;x)); x2 D; @tv(t;x) ( ) RDk(x y)[v(t;y) v(t;x)]dy +v(t;x)f(x;u(t;x) +v(t;x)); x2 D; for t2[0;T). 31 3.3 Comparison Principle and Monotonicity We will introduce the comparison principle and strong monotonicity for general linear and nonlinear evolution equations, and systems. Proposition 3.5 (Comparison principle for evolution equations). (1) (Comparison principle for linear evolution equations) If u1(t;x) and u2(t;x) are bounded sub- and super-solution of (3.1) (resp. (3.5)) on (s;s+T), respectively, and u1(s; ) u2(0; ), then u1(t; ) u2(t; ) for t2[s;T). (2) (Comparison principle for nonlinear evolution equations)If u1(t;x) and u2(t;x) are bounded sub- and super-solution of (1.18) (resp. (1.17)), on (s;s + T), respectively, and u1(0; ) u2(0; ), then u1(t; ) u2(t; ) for t2[s;s+T). Proof. It follows from the arguments in [61, Proposition 2.1]. The following remarks follows by the arguments similar to those in Proposition 3.5. Remark 3.6. For given 1 i 3, u0 2X+i , and a1i(t; );a2i(t; )2Xi, if a1i(t; ) a2i(t; ), then uNi (t; ;s;u0; i;a1i) uNi (t; ;s;u0; i;a2i) for t s; where uNi (t; ;s;u0; i;a1i) and uNi (t; ;s;u0; i;a2i) are solutions of (3.1) with uNi (s; ;s;u0; i;a1i) = u0 and uNi (s; ;s;u0; i;a2i) = u0, respectively. And uRi (t; ;s;u0; i;a1i) uRi (t; ;s;u0; i;a2i) for all t>s; where uRi (t; ;s;u0; i;a1i) and uRi (t; ;s;u0; i;a2i) are solutions of (3.5) with uRi (s; ;s;u0; i;a1i) = u0 and uRi (s; ;s;u0; i;a2i) = u0, respectively. Proof. We consider the case i = 1 for (3.1). Other cases can be proved similarly. 32 Note that u1(t;x;s;u0; 1;a21) is a super-solution of (3.1) in the case i = 1 with a1( ; ) being replaced by a11( ; ). Then by Proposition 3.5 (1), uN1 (t; ;s;u0; 1;a11) uN1 (t; ;s;u0; 1;a21) 8t s: Remark 3.7. (1) Suppose that u (t;x) and u+(t;x) are sub-solution and super-solution of (1.17) on (s;s+ T), respectively, then u (t;x) u+(t;x) 8t2[s;s+T); x2 D: (2) Suppose that u (t;x) and u+(t;x) are sub-solution and super-solution of (1.18) on (s;s+ T), respectively, then u (t;x) u+(t;x) 8t2[s;s+T); x2 D: Proof. (1) It follows from comparison principle for parabolic equations. (2) It follows from [56, Proposition 3.1]. Proposition 3.8 (Strong monotonicity). For given 1 i 3, if u1;u2 2Xi, u1 u2 and u1 6 u2, then for all t>s, (1) (Strong monotonicity for linear evolution equations) Ni (t;s; i;ai;u1) Ni (t;s; i;ai;u2), and Ri (t;s; i;ai;u1) Ri (t;s; i;ai;u2). (2) (Strong monotonicity for linear evolution equations) uNi (t; ;s;u1) uNi (t; ;s;u2), and uRi (t; ;s;u1) uRi (t; ;s;u2). 33 Proof. (1) It follows from the arguments in [61, Proposition 2.2]. (2) We show the proof of evolution equations in the Dirichlet boundary condition case with nonlocal dispersal opera- tor. Other cases can be proved similarly . Let v(t;x) = uN1 (t;x;s;u2) uN1 (t;x;s;u1) for t s at which both uN1 (t;x;s;u2)and uN1 (t;x;s;u1) exist. Then v(0; ) = u2 u1 0 and v(t;x) satis es @tv = Z D k(x y)v(t;y)dy v(t;x) +F(t;x;u(t;x;s;u2))v(t;x) + u(t;x;s;u1) Z 1 0 Fu(t;x;su(t;x;s;u1) + (1 s)u(t;x;s;u2))ds v(t;x); x2 D: (2) then follows from the argument similar to those in (1). Proposition 3.9 (Comparison principle for systems). (1) If (0;0) 1 (u0;v0), then (0;0) 1 (u(t; ;u0;v0);v(t; ;u0;v0)) for all t> 0 at which (u(t; ;u0;v0);v(t; ;u0;v0)) exists. (2) If (0;0) 1(ui;vi), for i = 1;2, (u1(0; );v1(0; )) 2 (u2(0; ), v2(0; )), and (u1(t;x);v1(t;x)) and (u2(t;x);v2(t;x)) are a sub-solution and a super-solution of (1.16) on [0;T) respectively, then (u1(t; );v1(t; )) 2 (u2(t; ), v2(t; )) for t2[0;T). (3) If (0;0) 1 (ui;vi), for i = 1;2, and (u1;v1) 2 (u2;v2), then (u(t; ;u1;v1);v(t; ;u1;v1)) 2 (u(t; ;u2;v2);v(t; ;u2;v2)) for all t>0 at which both (u(t; ;u1;v1), v(t; ;u1;v1)) and (u(t; ;u2;v2), v(t; ;u2;v2)) exist. (4) Let (u0;v0)2X+1 X+2 , then (u(t; ;u0;v0);v(t; ;u0;v0)) exists for all t> 0. Proof. It follows from the arguments in Proposition 3.1 in [35]. 3.4 A Technical Lemma The technical lemma is for time homogeneous evolution equations with nonlocal disper- sal operators. However, similar lemma holds in time periodic case (see [56, Lemma 4.2]). 34 Lemma 3.10. Let 1 i 3 and ai2Xi be given. For any > 0, there is a i2Xi such that kai a ik< ; h i(x) = i + a i(x) for i = 1 or 3 and h i(x) = iRDk(x y)dy + a i(x) for i = 2 is in CN, and satis es the following vanishing condition: there is x0 2 Int(D) such that h i(x0) = maxx2 Dh i(x) and the partial derivatives of h i(x) up to order N 1 at x0 are zero. Proof. See Lemma 3.1 in [59]. 35 Chapter 4 Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications In this chapter, we will focus on eigenvalue problems of nonlocal dispersal operators in the time homogeneous case, that is, (1.14) in case of Dirichlet, Neumann, and periodic types of boundary condition. First of all, let us recall some standard notations in Chapter 2, and introduce some basic properties of principal eigenvalues and principal spectrum points of time homogeneous dispersal operators. Next, we will prove Theorem 2.4, Theorem 2.6, and Theorem 2.8 for all the three boundary conditions in a uni ed way. Finally, we apply some results derived from the above theorems and prove Theorem 2.12. Throughout this chapter, we assume ai(t;x) ai(x) 2Xi for i = 1;2;3. Most results in this chapter are included in [59], which has been submitted for publication. 4.1 Basic Properties of Principal Eigenvalues/Principal Spectrum Points of Time Homogeneous Dispersal Operators In the section, we present some basic properties of principal eigenvalue and princi- pal spectrum points of time homogeneous nonlocal dispersal operators. Let us recall that Ni (t;s; i;ai) is the solution operator of (3.1) for i = 1;2;3. Without loss of generality, we set s = 0. Since we only focus on nonlocal dispersal operators in this chapter, we do not need to distinguish between nonlocal operators and random operators. For simplicity, throughout this chapter, we put Ni (t;0; i;ai) = i(t; i;ai) for i = 1;2;3: (4.1) 36 We have the following propositions. Proposition 4.1. Let 1 i 3 be given. (1) For given t> 0, e~ i( i;ai)t = r( i(t; i;ai)): (2) ~ i( i;ai)2 ( iKi +hi( )I). Proof. Observe that iKi +hi( )I : Xi!Xi is a bounded linear operator. Then by spectral mapping theorem, e ( iKi+hi( )I)t = ( i(t; i;ai))nf0g 8t> 0: (4.2) By Proposition 3.7, i(t; i;ai)X+i X+i 8t> 0: (4.3) Hence i(t; i;ai) is a positive operator onXi. Then by [50, Proposition 4.1.1], r( i(t; i;ai))2 ( i(t; i;ai)) for any t> 0. By (4.2), e~ i( i;ai)t = r( i(t; i;ai)) 8t> 0; and hence ~ i( i;ai)2 ( iKi +hi( )I). Proposition 4.2. (1) ~ 1( 1;0) < 0. (2) ~ 2( 2;0) = 0. (3) ~ 3( 3;0) = 0. Proof. (1) Let u0(x) 1. Observe that Z D k(x y)u0(y)dy u0(x) 0; and there is x0 2D such that Z D k(x y0)u0(y)dy u0(x0) < 0: 37 By Proposition 3.7(2), 0 1(t; 1;0)u0 u0 8t> 0; and then k 1(t; 1;0)u0k< 1 8t> 0: Note that for any ~u0 2X1 with k~u0k 1, by Proposition 3.7(2) again, k 1(t; 1;0)~u0k k 1(t; 1;0)u0k< 1 8t> 0: This implies that r( 1(t; 1;0)) < 1 8t> 0; and then ~ 1( 1;0) < 0: (2) Let u0( ) 1. Observe that 2(t; 2;0)u0 = u0 8t 0; and k 2(t; 2;0)~u0k k 2(t; 2;0)u0k= 1 for all t 0 and ~u0 2X2 with k~u0k 1. It then follows that r( 2(t; 2;0)) = 1 8t 0; and then ~ 2( 2;0) = 0: (3) It can be proved by the similar arguments as in (2). Next, we prove some properties of principal spectrum points of nonlocal dispersal op- erators by using the spectral radius of the induced nonlocal operators Uiai; i; i and Viai; i; i 38 (i = 1;2;3), where i > maxx2 Dhi(x) (i = 1;2;3), (Uiai; i; iu)(x) = Z D ik(x y)u(y) i hi(y) dy; i = 1;2; (4.4) (U3a3; 3; 3u)(x) = Z RN 3k(x y)u(y) 3 h3(y) dy; (4.5) and (Viai; i; iu)(x) = i R Dk(x y)u(y)dy i hi(x) = i(Kiu)(x) i hi(x); i = 1;2; (4.6) (V 3a3; 3; 3u)(x) = 3 R RN k(x y)u(y)dy 3 h3(x) = 3(K3u)(x) 3 h3(x): (4.7) Observe that Uiai; i; i and Viai; i; i are positive and compact operators on Xi (i = 1;2;3). Moreover, there is n 1 such that Ui ai; i; i n(X+ i nf0g) X ++ i ; i = 1;2;3; and Vi ai; i; i n(X+ i nf0g) X ++ i ; i = 1;2;3: Then by Krein-Rutman Theorem, r(Uiai; i; i)2 (Uiai; i; i); r(Viai; i; i)2 (Viai; i; i); (4.8) and r(Uiai; i; i) and r(Viai; i; i) are isolated algebraically simple eigenvalues of Uiai; i; i and Viai; i; i with positive eigenfunctions, respectively. Proposition 4.3. (1) i > hi;max is an eigenvalue of iKi + hi( )I with (x) being an eigenfunction i 1 is an eigenvalue of Uiai; i; i with (x) = ( i hi(x)) (x) being an eigenfunction. 39 (2) i > hi;max is an eigenvalue of iKi + hi( )I with (x) being an eigenfunction i 1 is an eigenvalue of Viai; i; i with (x) being an eigenfunction. Proof. It follows directly from the de nitions of Uiai; i; i and Viai; i; i. Proposition 4.4. Let 1 i 3 be given. (a) r(Uiai; i; i) is continuous in i(>hi;max), strictly decreases as i increases, and r(Uiai; i; i)!0 as i!1: (b) r(Viai; i; i) is continuous in i(>hi;max), strictly decreases as i increases, and r(Viai; i; i)!0 as i!1: Proof. We prove (a) in the case i = 1. The other cases can be proved similarly. First, note that r(U1a1; 1; 1) is an isolated algebraically simple eigenvalue of U1a1; 1; 1. It then follows from the perturbation theory of the spectrum of bounded operators that r(U1a1; 1; 1) is continuous in 1(>h1;max). Next, we prove that r(U1a1; 1; 1) is strictly decreasing as 1 increases. To this end, x any 1 >h1;max. Let 1( ) be a positive eigenfunction of U1a1; 1; 1 corresponding to the eigenvalue r(U1a1; 1; 1). Note that for any given ~ 1 > 1, there is 1 > 0 such that ~ 1 1 1 h1(x) > 1 8x2 D: 40 This implies that U1 a1; 1;~ 1 1 (x) = Z D 1k(x y) 1(y) ~ 1 h1(y) dy = Z D 1k(x y) 1(y) 1 h1(y) 1 1 + ~ 1 1 1 h1(y)dy 11 + 1 Z D 1k(x y) 1(y) 1 h1(y) dy = r(U 1 a1; 1; 1) 1 + 1 1(x) 8x2 D: It then follows that r(U1a1; 1;~ 1) r(U 1 a1; 1; 1) 1 + 1 0, there is 1 > 0 such that for 1 > 1, Z D 1k(x y) 1 h1(y)dy< 8x2 D: This implies that kU1a1; 1; 1k< 8 1 > 1: Hence r(U1a1; 1; 1)!0 as 1 !1. Proposition 4.5. Let 1 i 3 be given. (a) If there is i >hi;max such that r(Uiai; i; i) > 1, then ~ i( i;ai) >hi;max. (b) If there is i >hi;max such that r(Viai; i; i) > 1, then ~ i( i;ai) >hi;max. Proof. We prove (b). Part (a) can be proved similarly. 41 Fix 1 i 3. Suppose that there is i > hi;max such that r(Viai; i; i) > 1. Then by Proposition 4.4, there is 0 >hi;max such that r(Viai; i; 0) = 1: (4.9) By Proposition 4.3, 0 2 ( iKi +hi( )I). This implies that ~ i( i;ai) 0 >hi;max: Proposition 4.6 (Necessary and su cient condition). For given 1 i 3, i( i;ai) exists if and only if ~ i( i;ai) >hi;max. Proof. For 1 i 3, iKi is a compact operator. Hence iKi + hi( )I can be viewed as compact perturbation of the operator hi( )I. Clearly, the essential spectrum ess(hiI) of hi( )I is given by ess(hiI) = [hi;min;hi;max]: Since the essential spectrum is invariant under compact perturbations (see [25]), we have ess( iKi +hiI) = [hi;min;hi;max]; where ess( iKi +hiI) is the essential spectrum of iKi +hi( )I. Let disc( iKi +hiI) = ( iKi +hiI)n ess( iKi +hiI): Note that if 2 disc( iKi +hiI), then it is an isolated eigenvalue of nite multiplicity. On the one hand, if ~ i( i;ai) >hi;max(x), then ~ i( i;ai)2 disc( iKi +hiI). By Propo- sition 4.3, 12 Uia i; i;~ i( i;ai) . Hence r Uia i; i;~ i( i;ai) 1: 42 By Proposition 4.4, there is ~~ ~ i( i;ai) such that r Uia i; i;~~ = 1: This together with Proposition 4.3 implies that ~~ is an isolated algebraically simple eigenvalue of iKi +hi( )I with a positive eigenfunction. By De nition 2.1 (2), i( i;ai) exists. On the other hand, if i( i;ai) exists, then ~ i( i;ai) = i( i;ai) 2 disc( iKi + hiI). This implies that ~ i( i;ai) >hi;max(x). Finally, we present some variational characterization of the principal spectrum points of nonlocal dispersal operators when the kernel function is symmetric. In the rest of this subsection, we assume that k( ) is symmetric with respect to 0. Recall K3 : X3 !X3; (K3u)(x) = Z RN k(x y)u(y)dy 8u2X3: For given a2X3, let ^k(z) = X j1;j2; ;jN2Z k(z + (j1p1;j2p2; ;jNpN)); (4.10) where p1;p2; pN are periods of a(x). Then ^k( ) is also symmetric with respect to 0 and (K3u)(x) = Z D ^k(x y)u(y)dy 8u2X3; (4.11) where D = [0;p1] [0;p2] [0;pN] (see (2.20)). Proposition 4.7. Assume that k( ) is symmetric with respect to 0. Then ~ i( i;ai) = sup u2L2(D);kukL2(D)=1 Z D [ i(Kiu)(x)u(x) +hi(x)u2(x)]dx (i = 1;2;3): 43 Proof. First of all, note that iKi +hi( )I is also a bounded operator on L2(D) and iKi is a compact operator on L2(D), where Ki is de ned as in (4.11) when i = 3. Let ( iKi + hiI;L2(D)) be the spectrum of iKi +hi( )I considered on L2(D) and ~ ( i;ai;L2(D)) = supfRe j 2 ( iKi +hiI;L2(D))g: Then we also have ~ ( i;ai;L2(D))2 ( iKi +hiI;L2(D)); [hi;min;hi;max] ( iKi +hiI;L2(D)); and ~ ( i;ai;L2(D)) hi;max: Moreover, if ~ i( i;ai) > hi;max (resp. ~ i( i;ai;L2(D)) > hi;max), then ~ i( i;ai) (resp. ~ i( i;ai;L2(D))) is an eigenvalue of iKi + hiI considered on L2(D) (resp. C( D)) and hence ~ i( i;ai;L2(D)) ~ i( i;ai) (resp. ~ i( i;ai) ~ i( i;ai;L2(D))). We then must have ~ i( i;ai) = ~ i( i;ai;L2(D)): Assume now that k( ) is symmetric with respect to 0, that is, k( z) = k(z) for any z2RN. Then for any u;v2L2(D), in the case i = 1;2, Z D (Kiu)(x)v(x)dx = Z D Z D k(x y)u(y)v(x)dydx = Z D Z D k(x y)u(x)v(y)dxdy = Z D Z D k(x y)v(y)u(x)dydx = Z D (Kiv)(x)u(x)dx; 44 and in the case i = 3, Z D (K3u)(x)v(x)dx = Z D Z D ^k(x y)u(y)v(x)dydx = Z D Z D ^k(x y)u(x)v(y)dxdy = Z D Z D ^k(x y)v(y)u(x)dydx = Z D (K3v)(x)u(x)dx: ThereforeKi : L2(D)!L2(D) is self-adjoint. By classical variational formula (see [24]), we have ~ i( i;ai;L2(D)) = sup u2L2(D);kukL2(D)=1 Z D [ i(Kiu)(x)u(x) +hi(x)u2(x)]dx: The proposition then follows. 4.2 E ects of Spatial Variations and the Proof of Theorem 2.4 In this section, we investigate the e ects of spatial variations on the principal spectrum points/principal eigenvalues of nonlocal dispersal operators and prove Theorem 2.4. First of all, for given 1 i 3 and ci2R, let Xi(ci) =fai2Xij^ai = cig (see (2.21) for the de nition of ^ai). For given x0 2RN and > 0, let B(x0; ) =fy2RNjkx y0k< g: Proof of Theorem 2.4. (1) We rst prove the case i = 1. Let x0 2 D be such that h1(x0) = h1;max: 45 Note that there is 0 > 0 such that 0 a1(x0) a1(x) < 1 inf x2 D Z D k(x y)dy 0 1 Z D k(x y)dy 0 8x2 D: For any 0 < < 0, put = h1(x0) + (= 1 +a1(x0) + ): Then 1RDk(x y)dy h1(x) = 1RDk(x y)dy a1(x0) a1(x) + 1 R Dk(x y)dy 1RDk(x y)dy + 0 > 1 8x2 D: This implies r(V 1a1; 1; ) > 1 8 0 < 1: Then by Proposition 4.5 (b), ~ 1( 1;a1) >h1;max. By Proposition 4.6, 1( 1;a1) exists. We now prove the case i = 2. Similarly, let x0 2 D be such that h2(x0) = h2;max: Note that there is 0 > 0 such that 0 a2(x0) a2(x) < 2 inf x2 D Z D k(x y)dy 0 2 Z D k(x y0)dy 0: For any 0 < < 0, put = h2(x0) + (= 2 Z D k(x y0)dy +a2(x0) + ): 46 Then 2RDk(x y)dy h2(x) = 2RDk(x y)dy a2(x0) 2RDk(x y0)dy + 2RDk(x y)dy a2(x) + 2 R Dk(x y)dy 2RDk(x y)dy + 0 > 1 8x2 D: This again implies that r(V 2a2; 2; ) > 1 8 0 < 1: Then by Proposition 4.5 (b), ~ 2( 2;a2) >h2;max. By Proposition 4.6, 2( 2;a2) exists. (2) It can be proved by the similar arguments as in [61, Theorem B(2)]. For the com- pleteness, we provide a proof below. Let x0 2Int(D) be such that hi(x0) = hi;max and the partial derivatives of hi(x) up to order N 1 at x0 are zero. Then there is M > 0 such that hi(x0) hi(y) Mjjx0 yjjN 8 y2D: Fix > 0 such that B(x0;2 ) D and B(0;2 ) b supp(k( )). Let v 2X+i be such that v (x) = 8> >< >>: 1; x2B(x0; ); 0; x2DnB(x0;2 ): Clearly, for every x2DnB(x0;2 ) and > 1, we have (Uiai; i;hi(x0)+ v )(x) v (x) = 0 8 > 0: (4.12) 47 Note that there is ~M > 0 such that for any x2B(x0;2 ), k(x y) ~M 8 y2B(x0; ): It then follows that for x2B(x0;2 ) (Uiai; i;hi(x0)+ v )(x) = Z D ik(x y)v (y) hi(x0) + hi(y)dy Z B(x0; ) ik(x y) Mjjx0 yjjN + dy Z B(x0; ) i ~M Mjjx0 yjjN + dy: Notice that RB(x0; ) ~MMjjx0 yjjNdy =1. This implies that for 0 < 1, there is > 1 such that (Uiai; i;hi(x0)+ v )(x) > v (x) 8 x2B(x0;2 ): (4.13) By (4.12) and (4.13), Uiai; i;hi(x0)+ v (x) v (x) 8 x2D: Hence, r(Uiai; i;hi(x0)+ ) > 1. By Proposition 4.5(a), ~ i( i;ai) >hi(x0) = hi;max. By Proposi- tion 4.6, the principle eigenvalue i( i;ai) exists. (3) Recall that ~ i( i;~a) = supfRe j 2 ( iKi + ~hi( )I)g with ~hi(x) = i + ~a(x) for i = 1;3 and ~hi(x) = 2RDk(x y)dy + ~a(x) for i = 2. By the arguments of Proposition 4.6, ess( iKi + ~hiI) = [min x2 D ~hi(x);max x2 D ~hi(x)]: Note that sup ~a2Xi(ci) (max x2 D ~a(x)) =1: Then sup ~a2Xi(ci) ~ i( i;~a) sup ~a2Xi(ci) (max x2D ~hi(x)) i + sup ~a2Xi(ci) (max x2D ~a(x)) =1: 48 (4) We rst assume that the principal eigenvalue 2( 2;a2) exists. Suppose that u2(x) is a strictly positive principal eigenfunction with respect to the eigenvalue 2( 2;a2). We divide both sides of (1.2) by u2(x) and integrate with respect to x over D to obtain Z D 2[ R Dk(x y)(u2(y) u2(x))dy] +a2(x)u2(x) u2(x) dx = Z D 2( 2;a2)dx; or 2( 2;a2) = 2jDj Z D Z D k(x y)u2(y) u2(x)u 2(x) dydx+ 1jDj Z D a2(x)dx = 2jDj Z D Z D k(x y)u2(y) u2(x)u 2(x) dydx+ ^a2: By the symmetry of k( ), Z D Z D k(x y)u2(y) u2(x)u 2(x) dydx = 12 Z Z D D k(x y)u2(y) u2(x)u 2(x) dydx+ 12 Z Z D D k(x y)u2(y) u2(x)u 2(x) dydx = 12 Z Z D D k(x y)u2(y) u2(x)u 2(x) dydx+ 12 Z Z D D k(x y)u2(x) u2(y)u 2(y) dydx = 12 Z Z D D k(x y)(u2(y) u2(x)) 2 u2(x)u2(y) dydx 0: (4.14) So, inff 2( 2;a2)ja2 2X2;^a2 = c2g ^a2 = c2: And clearly, 2( 2;^a2) = ^a2. Hence, inff 2( 2;a2)ja2 2X2;^a2 = c2g= 2( 2;^a2) = c2: 49 Second, by Lemma 3.1, for any > 0, there is a 2 2X2\CN, such that ka2 a 2k< ; and h 2( )2CN(= 2RDk(x y)dy + a 2) satis es the vanishing condition in Theorem 2.1 (2). So, the principal eigenvalue 2( 2;a 2) exists and ~ 2( 2;a 2) = 2( 2;a 2). By the above arguments, ~ 2( 2;a 2) = 2( 2;a 2) 2( 2;^a 2) = ^a 2: (4.15) We claim that lim !0 ~ 2( 2;a 2) = ~ 2( 2;a2): In fact, ka 2 a2k , that is a2(x) a 2(x) a2(x) + 8 x2 D: Note that 2(t; 2;a2 + )u0 = e t 2(t; 2;a2)u0, where 2(t; 2;a2)u0 is the solution of (3.2) with the initial value u0( ). Similarly, we have 2(t; 2;a2 )u0 = e t 2(t; 2;a2)u0. So r( 2(t; 2;a2 )) = e tr( 2(t; 2;a2)): Hence ~ 2( 2;a2 ) = ~ 2( 2;a2) : (4.16) By Remark 3.6, we have 2(t; 2;a2 )u0 2(t; 2;a 2)u0 2(t; 2;a2 + )u0: Hence r( 2(t; 2;a2 )) r( 2(t; 2;a 2)) r( 2(t; 2;a2 + )): 50 By(4.16), ~ 2( 2;a2 ) ~ 2( 2;a 2) ~ 2( 2;a2 + ): Taking the limit of (4.15) as !0, we have ~ 2( 2;a2) ^a2 So, inff~ 2( 2;a2)ja2 2X2;^a2 = c2g= 2( 2;c2)(= c2). When the principal eigenvalue exists, it is not di cult to prove that the in mum is attained by the constant function a2( ) c2. In fact, suppose that 2( 2;a2) exists and u2( ) is a corresponding positive eigenfunction. By (4.14), 2( 2;a2) = ^a2(= c2) i u2(x) = u2(y) for all x;y2 D. Hence 2( 2;a2) = ^a2(= c2) i u2( ) constant, which implies that a2(x) = 2( 2;a2) = ^a2. (5) Suppose that a1i;a2i 2Xi and a1i a2i. By Remark 3.7, for any u0 2X+i and t 0, i(t; i;a1i)u0 i(t; i;a2i)u0: This implies that r( i(t; i;a1i)) r( i(t; i;a2i)): By Proposition 4.1, we have ~ i( i;a1i) ~ i( i;a2i): Remark 4.8. (1) Theorem 2.1 (3) is not true in the random dispersal case when the space dimension is one. In fact, for 1 i 3, we have R;i ci + ci2L2 for any ai( ) 2X++i , ^ai = ci and D = (0;L). For the periodic boundary case, see Lemma 4.1 in [48]. The proof of Neumann or Dirichlet boundary case is similar to that of the periodic boundary case. 51 We give a proof for the Neumann boundary case. Let (x) be the eigenvalue function of the operator + a2( )I de ned on C2([0;L]) with Neumann boundary condition. So (x) > 0 and we have 8 >>< >>: 00(x) +a2(x) (x) = R;2 (x); x2(0;L); @ @n(x) = 0; x = 0 or L: Multiplying this by (x) and integrating it from 0 to L, we have Z L 0 02(x)dx+ Z L 0 a2(x) 2(x)dx = R;2 Z L 0 2(x)dx: Hence R;2 = RL 0 02(x)dx+RL 0 a2(x) 2(x)dx RL 0 2(x)dx : Take x1;x2 2[0;L), we have 2(x2) 2(x1) = Z x2 x1 2 (x) 0(x)dx: Hence, for any positive number k> 0, 2(x2) 2(x1) 1k Z L 0 02(x)dx+k Z L 0 2(x)dx: Multiplying the above inequality by a2(x2) and integrating it with respect to x1 2[0;L) and x2 2[0;L), we get L Z L 0 a2(x2) 2(x2)dx2 c2L Z L 0 2(x1)dx1 c2L2 1 k Z L 0 02(x)dx+k Z L 0 2(x)dx ; 52 where c2 = RL0 a2(x)dx. This is equivalent to L Z L 0 a2(x) 2(x)dx c2L Z L 0 2(x)dx c2L2 1 k Z L 0 02(x)dx+k Z L 0 2(x)dx : Letting k = c2L, we obtain Z L 0 02(x)dx+ Z L 0 a2(x) 2(x)dx (c2 +c22L2) Z L 0 2(x)dx: So, we have R;2 c2 +c22L2: (2) Theorem 2.1 (4) may not be true for the Dirichlet type boundary condition. That is, ~ 1( 1;a1) 1( 1;^a1) may not be true, where a1 2X1. In the random dispersal case, there is an example in [60] which shows that the principal eigenvalue R;1( 1;a1) of (1.4) is smaller than the principal eigenvalue R;1( 1;c1) of (1.4) with a1(x) being replaced by c1(= ^a1). It is proved in Theorem 2.15 that ~ 1( 1;a1; )! R;1( 1;a1) as ! 0. So, for any 0 < 1, ~ 1( 1;a1; ) is close to R;1( 1;a1), and ~ 1( 1;c1; ) is close to R;1( 1;c1). Hence ~ 1( 1;a1; ) can be smaller than ~ 1( 1;c1; ) = 1( 1;c1; ) for 1. (3) Theorem 2.1 (4) holds for periodic case (see [63]). When i( i;ai) does not exist (i = 2;3), we may have ~ i( i;ai) = ^ai, but ai( ) is not a constant function. For example, let X3 =fu(x)2C(RN;R)ju(x+ ej) = u(x));x2RN;j = 1;2; ;Ng, and q2X3 with q(x) = 8 >>< >>: e kxk2 kxk2 2 if kxk< ; 0 if kxk 12: 53 ThenK3+h3( )I with k(z) = k (z) has no principal eigenvalue for M > 1, 0 < 1, 1 and h3(x) = 1 +Mq(x) where x2RN and N 3 (see [61]). Hence ~ 3 = maxx2 Dh3(x) = 1+M maxx2 Dq(x) = 1+M. Choosing M = 11 ^q, we have M^q = 1+M, that is ^a3 = ~ 3, but a3(x) = Mq(x) is not a constant function. 4.3 E ects of Dispersal Rates and the Proof of Theorem 2.6 In this section, we investigate the e ects of the dispersal rates on the principal spectrum points and the existence of principal eigenvalues of nonlocal dispersal operators and prove Theorem 2.6. Proof of Theorem 2.6. (1) Assume that k( ) is symmetric. Observe that for any u( ) 2 L2(D), Z Z D D k(x y)u(x)u(y)dydx Z D u2(x)dx Z D Z D k(x y)u(y)u(x)dydx Z D Z D k(x y)dyu2(x)dx = Z D Z D k(x y)(u(y) u(x))u(x)dydx = 12 Z Z D D k(x y)(u(y) u(x))u(x)dydx+ 12 Z Z D D k(x y)(u(y) u(x))u(x)dydx = 12 Z Z D D k(x y)(u(y) u(x))u(x)dydx+ 12 Z Z D D k(x y)(u(x) u(y))u(y)dydx = 12 Z Z D D k(x y)(u(y) u(x))2dydx 0: Then (1) follows from the following facts: 8 i > 0, ~ i( i;ai) = sup u2L2(D);jjujjL2(D)=1 i Z D Z D k(x y)u(y)u(x)dydx Z D u2(x)dx + Z D ai(x)u2(x)dx 54 in the case i = 1, ~ i( i;ai) = sup u2L2(D);jjujjL2(D)=1 i2 Z Z D D k(x y)(u(y) u(x))2dydx+ Z D ai(x)u2(x)dx in the case i = 2, and ~ i( i;ai) = sup u2L2(D);jjujjL2(D)=1 i Z D Z D ^k(x y)u(y)u(x)dydx Z D u2(x)dx + Z D ai(x)u2(x)dx in the case i = 3 (see (4.11)). (2) We prove the case i = 1. The case i = 3 can be proved similarly. Without loss of generality, assume a1(x) > 0 for x2 D. Assume that 1 > 0 is such that 1( 1;a1) exists and ~ 1 > 1. By proposition 4.6, 1( 1;a1) > maxx2 Dh1(x), that is, 1( 1;a1) > max x2 D ( 1 +a1(x)): Let 1( ) be a positive principal eigenfunction with jj 1jjL2(D) = 1. Then 1( 1;a1) = 1 Z Z D D k(x y) 1(y) 1(x)dydx 1 + Z D a1(x) 21(x)dx> max x2 D ( 1 +a1(x)): By Proposition 4.7, ~ 1(~ 1;a1) ~ 1 Z Z D D k(x y) 1(y) 1(x)dydx ~ 1 + Z D a1(x) 21(x)dx = 1( 1;a1) + (~ 1 1) Z Z D D k(x y) 1(y) 1(x)dydx+ 1 ~ 1 > max x2 D ( 1 +a1(x)) + 1 ~ 1 + (~ 1 1) Z Z D D k(x y) 1(y) 1(x)dydx > max x2 D ( ~ 1 +a1(x)): By proposition 4.6 again, 1(~ 1;a1) exists. (3) It follows from Theorem 2.1(1) and can also be proved as follows. 55 To show i( i;ai) exists, we only need to show ~ i( i;ai) > maxx2 Dhi(x), where hi(x) = i +ai(x) for i = 1 and 3 and hi(x) = iRDk(x y)dy+ai(x) for i = 2. In the case i = 2 or 3, ~ i( i;ai) ^ai by theorem 2.4(4). This implies that ~ i( i;ai) >hi;max 8 i 1: In the case i = 1, note that 1(1;0) exists and 1 < 1(1;0) < 0: This implies that 1(1;a1 1 ) exists for 1 1 and then 1( 1;a1) exists for 1 1. (4) On the one hand, we have ~ i( i;ai) hi;max i +ai;max: On the other hand, for any > ai;max, I ai( )I has bounded inverse. This implies that ai;max + > ~ i( i;ai) 8 0 < i 1: Therefore, lim i!0 ~ i( i;ai) = ai;max: (5) We prove the cases i = 1 and i = 2. The case i = 3 can be proved by the similar arguments as in the case i = 2. First, we prove the case i = 1. By Proposition 4.2, ~ 1(1;0) < 0: 56 Observe that ~ 1( 1;a1) = 1~ 1 1;a1 1 and ~ 1 1;a1 1 ! ~ 1(1;0) as 1 !1. It then follows that ~ 1( 1;a1) 1 2 ~ 1(1;0) 8 1 1: This implies that lim 1!1 ~ 1( 1;a1) = 1: Second, we prove the case i = 2. By (3), 2( 2;a2) exists for 2 1. In the following, we assume 2 1 such that 2( 2;a2) exists. Let 2; 2(x) be a positive principal eigenfunction with RD 22; 2(x)dx = 1. Note that ^a2 2( 2;a2) a2;max; and 2 Z D Z D k(x y)( 2; 2(y) 2; 2(x)) 2; 2(x)dydx+ Z D a2(x) 22; 2(x)dx = 2( 2;a2): This implies that 2 2 Z D Z D k(x y)( 2; 2(y) 2; 2(x))2dydx = Z D a2(x) 22; 2(x)dx 2( 2;a2) a2;max ^a2; and then Z D Z D k(x y)( 2; 2(y) 2; 2(x))2dydx 2(a2;max ^a2) 2 : (4.17) Let 2; 2(x) = 2; 2(x) ^ 2; 2. Then 2 Z D Z D k(x y)( 2; 2(y) 2; 2(x))dydx+ Z D a2(x) 2; 2(x)dx = Z D a2(x)( 2; 2(x)+ ^ 2; 2)dx; 57 and hence 2( 2;a2) Z D 2; 2(x)dx = ^ 2; 2 Z D a2(x)dx+ Z D a2(x) 2; 2(x)dx: This implies that 2( 2;a2)^ 2; 2 = ^a2 ^ 2; 2 + 1jDj Z D a2(x) 2; 2(x)dx: (4.18) To show 2( 2;a2) ! ^a2 as 2 !1, we rst show that RDa2(x) 2; 2(x)dx ! 0 as 2 !1. Note that ~ 2(1;0) = 0 and ~ 2(1;0) is the principal eigenvalue ofK2+b0( )Iwith ( ) 1 being a principal eigenfunction, where b0(x) = Z D k(x y)dy: Moreover, ~ 2(1;0) is also an isolated algebraically simple eigenvalue ofK2 +b0( )I on L2(D). Note also that Z D ( K2 b0I)u (x)u(x)dx = 12 Z D Z D k(x y)(u(y) u(x))2dydx 0 (4.19) for any u( )2L2(D) and K2 b0( )I is a self-adjoint operator on L2(D). Then there is a bounded linear operator A : L2(D)!L2(D) such that Z D ( K2 b0I)u (x)u(x)dx = Z D (Au)(x)(Au)(x)dx 8u2L2(D): (4.20) Let E1 = spanf ( )g; 58 and E2 =fu( )2L2(D)j Z D u2(x)dx = 0g: Then L2(D) = E1 E2 and K2 +b0( )I (E2) E2: Moreover, (K2 +b0( )I)jE2 is invertible. We claim that there is C > 0 such that Z D (Au)(x)(Au)(x)dx C Z D u2(x)dx 8u2E2: (4.21) For otherwise, there is un2E2 with RDu2n(x)dx = 1 such that Z D (Aun)(x)(Aun)(x)dx!0 as n!1. It then follows that 02 ((K2 +b0( )I)jE2), a contradiction. Hence (4.21) holds. By (4.19), (4.20) and (4.21), for any 2 1, Z D 22; 2(x)dx 12C Z D Z D k(x y)( 2; 2(y) 2; 2(x))2dydx: (4.22) Observe that Z D Z D k(x y)( 2; 2(y) 2; 2(x))2dydx = Z D Z D k(x y)( 2; 2(y) 2; 2(x))2dydx: This together with (4.17) and (4.22) implies that Z D 22; 2(x)dx!0 as 2 !1; 59 and then Z D a2(x) 2; 2(x)dx!0 as 2 !1: Second, assume 2( 2;a2) 6! ^a2 as 2 !1. By (4.18), we must have ^ 2; 2;n ! 0 for some sequence 2;n!1. This and (4.17) implies that Z D 22; 2;n(x)dx C0 Z D Z D k(x y) 22; 2;n(x)dydx = C0 Z D Z D k(x y)( 22; 2;n(x) 2; 2;n(x) 2; 2;n(y))dydx +C0 Z D Z D k(x y) 2; 2;n(y) 2; 2;n(x)dydx C02 Z D Z D k(x y)( 2; 2;n(y) 2; 2;n(x))2dydx+jDj2C0M ^ 2; 2;n ^ 2; 2;n C0(a2;max ^a2) 2 +jDj2C0M ^ 2; 2;n ^ 2; 2;n where C0 = (minx2 DRDk(x y)dy) 1 and M = supx;y2 Dk(x y). That is Z D 22; 2;n(x)dx!0 as 2;n!1: This is a contradiction. Therefore 2( 2;a2)!^a2 as 2 !1. 4.4 E ects of Dispersal Distance and the Proof of Theorem 2.8 In this section, we investigate the e ects of the dispersal distance on the principal spectrum points and the existence of principal eigenvalues and prove Theorem 2.8. 60 Proof of Theorem 2.8. (1) As mentioned in Remark 2.9, the cases i = 1 and 3 are proved in [41, Theorem 2.6]. The case i = 2 can be proved by the similar arguments as in [41, Theorem 2.6]. For completeness, we provide a proof for the case i = 2 in the following. By Proposition 4.7, ~ i( i;ai; ) = sup u2L2(D);kukL2(D)=1 Z D i Z D k (x y)(u(y) u(x))dy +ai(x)u(x) u(x)dx: On the one hand, ~ i( i;ai; ) = sup u2L2(D);kukL2(D)=1 Z D i Z D k (x y)(u(y) u(x))dy +ai(x)u(x) u(x)dx = sup u2L2(D);kukL2(D)=1 i2 Z D Z D k (x y)(u(y) u(x))2dydx+ Z D Z D ai(x)u2(x)dx ai;max: On the other hand, assume that x0 2 D is such that ai(x0) = ai;max. Then for any 0 < < 1, there are 0 > 0 and x 0 2IntD such that B(x 0; 0) D and ai(x0) ai(x) < =2 for x2B(x 0; 0): Let u0( ) be a smooth function with supp(u0( ))\D B(x 0; 0) and ku0kL2(D) = 1. Then ~ i( i;ai; ) Z D i Z D k (x y)(u0(y) u0(x))dy +ai(x)u0(x) u0(x)dx i Z D Z D k (x y)(u0(y) u0(x))dy u0(x)dx+ ai;max 2 : Note that Z D k (x y)(u0(y) u0(x))dy!0 8x2Int(D) 61 as !0. And Z D k (x y)(u0(y) u0(x))dy 2 max y2 D ju0(y)j 8x2D: Hence, there exists 0 > 0, such that for any < 0, we have i Z D Z D k (x y)(u0(y) u0(x))dy u0(x)dx 2: It then follows that ai;max ~ i( i;ai; ) ai;max : This implies that ~ i( i;ai; )!ai;max as !0. (2) First, for i = 1, Z D k (x y)u(y)dy kuk Z D k (x y)dy!0; as !1 uniformly in u2X1 with kuk 1. Therefore, ~ 1( 1;a1; )!supfRe j 2 (( 1 +a1( ))I)g= 1 +a1;max; as !1. For i = 2, Z D k (x y)(u(y) u(x))dy 2kuk Z D k (x y)dy!0; as !1 uniformly in u2X2 with kuk 1. Hence ~ 2( 2;a2; )!supfRe j 2 (a2( )I)g= a2;max; as !1. 62 For i = 3, recall that 3( 3;a3) = supfRe j 2 ( 2 I+h3( )I)g; where Iu = 1 p1p2 pN Z p1 0 Z p2 0 Z pN 0 u(x)dx: We rst assume that a3( ) satis es the conditions in Remark 2.5 (2). Then by similar arguments as in Theorem 2.4 (2), 3( 3;a3) is the principal eigenvalue of 3 I+ h3( )I. Let 3( ) be the positive principal eigenfunction of 3 I + h3( )I with ^ 3 = 1jDjRD 3(x)dx = 1. We then have 3( 3;a3) >h3;max and 1 jDj Z D 3 3(x) 3( 3;a3) + 3 a3(x)dx = 1; (4.23) where 3(x) = ( 3( 3;a3) + 3 a3(x)) 3(x): Fix 0 < < 3( 3;a3) hi;max. Then 1 jDj Z D 3 3(x) 3( 3;a3) + 3 a3(x)dx> 1: (4.24) Observe that for any k = (k1;k2; ;kN)2ZNnf0g, Z RN ~k(z) cos NX i=1 kipixi + NX i=1 kipizi dz!0; and Z RN ~k(z) sin NX i=1 kipixi + NX i=1 kipizi dz!0 63 as !1. This implies that for any a2X3, Z RN ~k(z)a(x+ z)dz!^a as !1 and then Z RN 3k (x y) 3(y) 3( 3;a3) + 3 a3(y)dy = Z RN 3~k(z) 3(x+ z) 3( 3;a3) + 3 a3(x+ z)dz ! 1jDj Z D 3 3(x) 3( 3;a3) + 3 a3(x)dx as !1 uniformly in x2RN. This together with (4.24) implies that Z RN 3k (x y) 3(y) 3( 3;a3) + 3 a3(y)dy> 1 8x2RN; 1: It then follows that ~ 3( 3;a3; ) > 3( 3;a3) >hi;max 8 1 (4.25) and 3( 3;a3; ) exists for 1. Now for any > 0, by (4.23), 1 jDj Z D 3 3(x) 3( 3;a3) + + 3 a3(x)dx< 1: (4.26) Then by the similar arguments in the above, ~ 3( 3;a3; ) < 3( 3;a3) + 8 1: (4.27) By (4.25) and (4.27), ~ 3( 3;a3; )! 3( 3;a3) as !1: 64 Now for general a3 2X3, and for any > 0, there is a3; 2X3 such that ka3 a3; k< 8x2RN; and a3; ( ) satis es the conditions in Remark 2.5 (2). By Theorem 2.4 (5), ~ 3( 3;a3; ; ) ~ 3( 3;a3; ) ~ 3( 3;a3; ; ) + : By the above arguments, 3( 3;a3) 3 3( 3;a3; ) 2 ~ 3( 3;a3; ) 3( 3;a3; )+2 3( 3;a3)+3 8 1: We hence also have ~ 3( 3;a3; )! 3( 3;a3) as !1: (3) By (1), for any > 0, ~ i( i;ai; ) >ai;max 8 0 < 1: This implies that there is 0 > 0 such that ~ i( i;ai; ) >hi;max 8 0 < < 0: Then by Proposition 4.7, i( i;ai) exists for 0 < < 0. 4.5 Applications to the Asymptotic Dynamics of Two Species Competition Sys- tem In this section, we consider the asymptotic dynamics of the two species competition system (1.16) and prove Theorem 2.12 by applying some of the principal spectrum properties 65 developed in previous sections. Throughout this section, we assume that k( z) = k(z), ~ 1( ;f( ;0)) > 0, f(x;w) < 0 for w 1, and @2f(x;w) < 0 for w 0. 4.5.1 Asymptotic Dynamics of KPP Type Competition Systems In this subsection, we present some basic properties about the asymptotic dynamics of the time homogeneous two species competition system (1.16). Throughout this subsection, we assume that k( z) = k(z), ~ 1( ;f( ;0)) > 0, f(x;w) < 0 for w 1 and @2f(x;w) < 0 for w> 0. Proposition 4.9. For any given > 0 and a2X1(= X2), ~ 1( ;a) ~ 2( ;a) and if 1( ;a) exists, then ~ 1( ;a)(= 1( ;a)) < ~ 2( ;a) Proof. First, assume that 1( ;a) exists. Let ( ) be the positive principal eigenfunction of K1 I+a( )I with k k= 1. Then 1(t; ;a) = e 1( ;a)t ; and 2(t; ;a) = e~ 2( ;a)t 8t> 0: By Remark 3.7, 2(t; ;a) 1(t; ;a) 8t> 0: This implies that ~ 2( ;a) > 1( ;a): In general, by Lemma 3.10 and Theorem 2.4 (2), for any > 0, there is a 2X1 such that 1( ;a ) exists and a (x) a(x) a (x) + : 66 By the above arguments, ~ 2( ;a ) > 1( ;a ): Observe that ~ 2( ;a) ~ 2( ;a ) and 1( ;a ) ~ 1( ;a) : Hence ~ 2( ;a) ~ 1( ;a) 2 : Letting !0, we have ~ 2( ;a) ~ 1( ;a): Consider ut = Z D k(x y)u(t;y)dy u(t;x) +u(t;x)g(x;u(t;x)); x2 D (4.28) and vt = Z D k(x y)[v(t;y) v(t;x)]dy +v(t;x)g(x;v(t;x)); x2 D; (4.29) where g is a C1 function, g(x;w) < 0 for w 1, and @wg(x;w) < 0 for w 0. Proposition 4.10. (1) If 1( ;g( ;0)) > 0, then there is u 2X++1 such that u = u is a stationary solution of (4.28) and for any solution u(t;x) of (4.28) with u(0; )2X+1 nf0g, u(t; )!u ( ) in X1. (2) If 2( ;g( ;0)) > 0, then there is v 2X++2 such that v = v is a stationary solution of (4.29) and for any solution v(t;x) of (4.29) with v(0; )2X+2 nf0g, v(t; )!v ( ) in X2. Proof. It follows from [56, Theorem E]. 67 4.5.2 Proof of Theorem 2.12 In this subsection, we prove Theorem 2.12. Proof of Theorem 2.12. (1) By ~ 1( ;f( ;0)) > 0 and Proposition 4.9, we have ~ 2( ;f( ;0)) > 0. Then by Lemma 4.10, there are u 2X++1 and v 2X++2 such that (u ;0) and (0;v ) are stationary solutions of (1.16). Moreover, for any (u0;v0) 2 X+1 X+2 with u0 6= 0 and v0 = 0 (resp. u0 = 0 and v0 6= 0), (u(t; ;u0;v0);v(t; ;u0;v0)) ! (u ( );0) (resp. (u(t; ;u0;v0);v(t; ;u0;v0))!(0;v ( ))) as t!1. (2) Observe that Z D k(x y)u (y)dy u (x) +f(x;u (x))u (x) = 0; x2 D: (4.30) This implies that 1( ;f( ;u ( ))) exists and 1( ;f( ;u ( ))) = 0. By Proposition 4.9, we have ~ 2( ;f( ;u ( ))) > 0: By Lemma 3.10, there are > 0 and a2X1 such that 2( ;a) exists, a(x) f(x;u (x)) ; 2( ;a) > 0; and ~ 2( ;f( ;u ( ) + )) > 0: Let ( ) be the positive eigenfunction of K2 b( )I + a( )I with k k = 1, where b(x) = RDk(x y)dy. Let u (x) = u (x) + 2 and v (x) = (x): 68 Then 0 = Z D k(x y)u (y)dy u (x) +u (x)f(x;u (x)) = Z D k(x y)u (y)dy u (x) +u (x)f(x;u (x) +v (x)) + 2 1 Z D k(x y)dy 2f(x;u (x)) +u [f(x;u (x)) f(x;u (x) +v (x))] Z D k(x y)u (y)dy u (x) +u (x)f(x;u (x) +v (x)) for 0 < 1, and 0 2( ;a)v (x) = Z D k(x y)[v (y) v (x)]dy +a(x)v (x) Z D k(x y)[v (y) v (x)]dy + [f(x;u (x)) ]v (x) = Z D k(x y)[v (y) v (x)]dy +v (x)f(x;u (x) +v (x)) +v (x) [f(x;u (x)) f(x;u (x) +v (x)) ] Z D k(x y)[v (y) v (x)]dy +v (x)f(x;u (x) +v (x)) for 0 < 1. It then follows that for 0 < 1, (u (x);v (x)) is a super-solution of (1.16). By Proposition 3.9, (u(t2; ;u ;v );v(t2; ;u ;v )) 2 (u(t1; ;u ;v );v(t1; ;u ;v )) 8 0 >< >>: [RDk(x y)u (y)dy u (x)] +u (x)f(x;u (x) +v (x)) = 0; x2 D; RDk(x y)[v (y) v (x)]dy +v (x)f(x;u (x) +v (x)) = 0; x2 D (4.32) (see the arguments in [35, Theorem A]). Multiplying the rst equation in (4.32) by v (x), second equation by u (x), and integrating over D, we have Z D u (x)v (x)dx = Z D Z D k(x y)dy u (x)v (x)dx: This together with v (x) (x) > 0 implies that 1 Z D k(x y)dy u (x) = 0 8x2 D: Note that RDk(x y)dy < 1 for x near @D. This together with the rst equation in (4.32) implies that u (x) = 0 for all x2 D. We then must have v (x) = v (x) for all x2 D. Moreover, by (4.31) and Dini?s theorem, limt!1(u(t; ;u ;v );v(t; ;u ;v )) = (0;v ( )) in X1 X2: (4.33) Now, for any (u0;v0)2(X+1 nf0g) (X+2 nf0g), there is M0 > 0 such that (u0;v0) 2 (M;0): Then by Proposition 3.9, (u(t; ;u0;v0);v(t; ;u0;v0)) 2 (u(t; ;M;0);v(t; ;M;0)) 8t> 0: 70 Since (u(t; ;M;0);v(t; ;M;0)) ! (u ( );0) in X1 X2 for 0 < 1, there is T > 0 such that (u(t; ;u0;v0);v(t; ;u0;v0)) 2 (u ( );0) 8t T: Then v(t; ;u0;v0) satis es vt(t;x) Z D k(x y)[v(t;y) v(t;x)]dy +v(t;x)f(x;u (x) + +v(t;x)) for t T. Note that ~ 2( ;f( ;u ( ) + )) > 0. By Lemma 4.10, for 0 < 1, there is ~T T such that v(t; ;u0;v0) v ( ) 8t 0: We then have (u(t+ ~T; ;u0;v0);v(t+ ~T; ;u0;v0)) 2 (u(t; ;u ;v );v(t; ;u ;v )) 8t 0: By (4.33), limt!1(u(t; ;u0;v0);v(t; ;u0;v0)) = (0;v ( )): The theorem is thus proved. 71 Chapter 5 Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal Operators/Equations and Applications In this chapter, we prove Theorem 2.13, Theorem 2.15, and Theorem 2.16 with Dirich- let, Neumann, and periodic types of boundary condition by making use of the comparison principle and other results in the Preliminary. In particular, Theorem 2.13 is fundamental to Theorem 2.15 and Theorem 2.16. Finally, we apply the above approximation results to prove Theorem 2.18. Most results in this chapter are included in [60], which has been submitted for publication. 5.1 Approximations of Solutions of Random Dispersal Initial-Boundary Value Problems by Nonlocal Dispersal Initial-Boundary Value Problems In this section, we explore the approximation of solutions to (1.17) by the solutions to (1.18). We rst present some basic properties of solutions to (1.17) and (1.18). Then we prove Theorem 2.13. Though the ideas of the proofs of Theorem 2.13 for di erent types of boundary conditions are the same, di erent techniques are needed for di erent boundary conditions. We hence give proofs of Theorem 2.13 for di erent boundary conditions in di erent subsections. 5.1.1 Proof of Theorem 2.13 in the Dirichlet Boundary Condition Case In this subsection, we prove Theorem 2.13 in the Dirichlet boundary case. Throughout this subsection, we assume (H0), and Br;bu = Br;Du in (1.17), and Dc = RN n D and Bn;bu = Bn;Du in (1.18). Without loss of generality, we assume s = 0. 72 Proof of Theorem 2.13 in the Dirichlet boundary condition case. Letu0 2C3( D) withu0(x) = 0 for x2@D. Let u 1(t;x) be the solution of (1.18) with s = 0 and u1(t;x) be the solution of (1.17) with s = 0. Suppose that u1(t;x) and u 1(t;x) exist on [0;T]. By regularity of solutions for parabolic equations, u1 2C2+ ;1+ 2 ( D (0;T])\C2+ ;0( D [0;T]). Let ~u1 be an extension of u1 to RN [0;T] satisfying that ~u1 2C2+ ;0(RN [0;T]). De ne L (z)(t;x) = Z D[Dc k (x y)[z(t;y) z(t;x)]dy: Let G(t;x) = ~u1(t;x). Then ~u1 veri es 8 >>> >>> < >>> >>> : @t~u1(t;x) = L (~u1)(t;x) +F (t;x) +F(t;x; ~u1(t;x)); x2 D; t2(0;T]; ~u1(t;x) = G(t;x); x2Dc;t2[0;T]; ~u1(0;x) = u0(x); x2 D; where F (t;x) = ~u1(t;x) L (~u1)(t;x) = ~u1(t;x) Z D[Dc k (x y)(~u1(t;y) ~u1(t;x))dy: Let w 1 = ~u1 u 1. We then have 8 >>>> >>< >>>> >>: @tw 1(t;x) = L (w 1)(t;x) +F (t;x) +a1(t;x)w 1(t;x); x2 D; t2(0;T]; w 1(t;x) = G(t;x); x2Dc; t2[0;T]; w 1(0;x) = 0; x2 D; (5.1) where a1(t;x) = R10 Fu[t;x;u 1(t;x) + (~u1(t;x) u 1(t;x))]d . 73 We claim that 8 >>< >>: supt2[0;T]kF (t; )kX1 = O( ); supt2[0;T];x2RNn D;dist(x;@D) jG(t;x)j= O( ): (5.2) In fact, ~u1(t;x) Z D[Dc k (x y)(~u1(t;y) ~u1(t;x))dy = ~u1(t;x) Z RN 1 Nk0 x y (~u1(t;y) ~u1(t;x))dy = ~u1(t;x) Z RN k0(z)(~u1(t;x+ z) ~u1(t;x))dz = ~u1(t;x) Z RN k0(z) 2z2 N 2! ~u1(t;x) +O( 2+ ) dz = ~u1(t;x) 2 Z RN k0(z)z 2 N 2 dz ~u1(t;x) +O( ) = ~u1(t;x) ~u1(t;x) +O( ) = O( ) 8x2 D; and jG(t;x)j=j~u1(t;x)j sup t2[0;T];x2RNnD;z2@D;dist(x;z) j~u1(t;x) u1(t;z)j = O( ) 8x2Dc; dist(x;@D) : Therefore, (5.2) holds. Next, let w be given by w(t;x) = eAt(K1 t) +K2 ; 74 where A = max x2 D;t2[0;T] a1(t;x). By direct calculation, we have 8 >>> >>>< >>> >>>: @t w(t;x) = L ( w) +a1(t;x) w + F (t;x) x2 D; t2(0;T]; w(t;x) = eAt(K1 t) +K2 ; x2Dc; t2[0;T]; w(0;x) = K2 ; x2 D; (5.3) where F (t;x) = eAtK1 + [A a1(t;x)]eAtK1 t a1(t;x)K2 : By (5.2), there are 0 > 0 and K1;K2 > 0 such that 8 >>< >>: F (t;x) F (t;x); x2 D; t2[0;T]; G(t;x) eAt(K1 t) +K2 ; x2Dc; dist(x;@D) ; t2[0;T]; (5.4) when 0 < < 0. By (5.1), (5.3), (5.4), and Remark 3.7, we obtain w (t;x) w(t;x) = eAt(K1 t) +K2 8x2 D; t2[0;T] (5.5) for 0 < < 0. Similarly, let w(t;x) = eAt( K1 t) K2 . We can prove that for 0 < < 0, w (t;x) w(t;x) = eAt(K1 t) K2 8x2 D; t2[0;T]: (5.6) By (5.5) and (5.6) we have jw (t;x)j eAtK1 t+K2 8x2 D; t2[0;T]; 75 which implies that there is C(T) > 0 such that for any 0 < < 0, sup t2(0;T] ku1( ;t) u 1( ;t)kX1 C(T) : Theorem 2.13 in the Dirichlet boundary condition case then follows. Remark 5.1. If the homogeneous Dirichlet boundary conditions Br;Du = u = 0 on @D and Bn;Du = u = 0 on Dc = RN n D are changed to nonhomogeneous Dirichlet boundary conditions Br;Du = u = g(t;x) on @D and Bn;Du = u = g(t;x) on Dc = RNn D, Theorem 2.13 also holds, which can be proved by the similar arguments as above. 5.1.2 Proof of Theorem 2.13 in the Neumann Boundary Condition Case In this subsection, we prove Theorem 2.13 in the Neumann boundary condition case. Throughout this subsection, we assume (H1), and Br;bu = Br;Nu in (1.17), and Dc =; and Bn;bu = Bn;Nu in (1.18). Without loss of generality, we assume s = 0. We rst introduce two lemmas. To this end, for given > 0 and d0 > 0, let D =fz2 Djdist(z;@D) 0 and > 0 such that for < , Z RNnD k (x y)n( x)x y dy K Z RNnD k (x y)dy: Proof. See [15, Lemma 4]. Proof of Theorem 2.13 in the Neumann boundary condition case. Suppose thatu0 2C3( D). Let u 2(t;x) be the solution to (1.18) with s = 0 and u2(t;x) be the solution to (1.17) with s = 0. Assume that u2(t;x) and u 2(t;x) exist on [0;T]. Then u2 2C2+ ;1+ 2 ( D (0;T]). Let ~u2 be an extension of u2 to RN [0;T] satisfying that ~u2 2C2+ ;1+ 2 (RN (0;T])\ C2+ ;0(RN [0;T]). De ne L (z)(t;x) = Z D k (x y)(z(t;y) z(t;x))dy; and ~L (z)(t;x) = Z RN k (x y)(z(t;y) z(t;x))dy: Set w 2 = u 2 ~u2. Then @tw 2(t;x) = @tu 2(t;x) @t~u2(t;x) = [L (u 2)(t;x) +F(t;x;u 2)] [ ~u2(t;x) +F(t;x; ~u2)] = L (w 2)(t;x) +a2(t;x)w 2(t;x) +F (t;x); where a2(t;x) = R10 Fu(t;x; ~u2(t;x) + (u 2(t;x) ~u2(t;x)))d and F (t;x) = ~L (~u2)(t;x) ~u2(t;x) Z RNnD k (x y)(~u2(t;y) ~u2(t;x))dy: 77 Hence w 2 veri es 8> >< >>: @tw 2(t;x) = L (w 2)(t;x) +a2(t;x)w 2(t;x) +F (t;x); x2 D; w 2(0;x) = 0; x2 D: (5.7) To prove the theorem, let us pick an auxiliary function v as a solution to 8 >>>> >>< >>>> >>: @tv(t;x) = v(t;x) +a2(t;x)v +h(t;x); x2D; t2(0;T]; @v @n(t;x) = g(t;x); x2@D; t2[0;T]; v(0;x) = v0(x); x2D for some smooth functions h(t;x) 1, g(t;x) 1 and v0(x) 0 such that v(t;x) has an extension ~v(t;x)2C2+ ;1+ 2 (RN (0;T])\C2+ ;0(RN [0;T]). Then v is a solution to 8 >>< >>: @tv(t;x) = L (v)(t;x) +a2(t;x)v(t;x) +H(t;x; ); x2 D;t2(0;T]; v(0;x) = v0(x); x2 D;t2[0;T]; (5.8) where H(t;x; ) = ~v(t;x) ~L (v)(t;x) + Z RNnD k (x y)(~v(t;y) ~v(t;x))dy +h(t;x): 78 By Lemma 5.2 and the rst estimate in (5.2), we have the following estimate for H(x;t; ): H(t;x; ) = ~v(t;x) ~L (v)(t;x) + C 2 Z RNnD k (x y)(~v(t;y) ~v(t;x))dy +h(t;x) C Z RNnD k (x y)n( x)x y g( x;t)dy +C Z RNnD k (x y) X j j=2 D ~v 2 ( x;t) " y x x x # dy + 1 C1 C g( x;t) Z RNnD k (x y)n( x)x y dy D1C Z RNnD k (x y)dy + 12 (5.9) for some constants D1 and C1 and su ciently small such that C1 12. Then Lemma 5.3 implies that there exist C0> 0 and 0 such that 1 Z RNnD k (x y)n( x)x y dy C 0 Z RNnD k (x y)dy; if < 0. This implies that H(x;t; ) CC0g( x;t) D1 Z RNnD k (x y)dy + 12; (5.10) if < 0. 79 We estimate now F (t;x). By Lemmas 5.2, 5.3, the rst estimate in (5.2), and the fact that @~u2@n = 0, we have F (t;x) = O( ) + Z RNnD k (x y)(~u2(t;y) ~u2(t;x))dy = O( ) +C Z RNnD k (x y) X j j=2 D 2 ( x;t) " y x x x # dy C2 +D1C Z RNnD k (x y)dy = C2 +D2 Z RNnD k (x y)dy (5.11) for some C2 > 0 and D2 > 0. Given > 0, let v = v. By (5.8), v satis es 8 >>< >>: @tv (t;x) L (v )(t;x) a(t;x)v (t;x) = H(t;x; ); x2 D; v (0;x) = v0(x); x2 D: (5.12) By (5.10) and (5.11), there exist C3 > 0 and 0 0 such that for 0 < 0, F (t;x) C +D2 Z RNnD k (x y)dy 2 + C3 Z RNnD k (x y)dy = H(x;t; ) 8x2 D; t2[0;T]: (5.13) Then by (5.7), (5.12), (5.13), and Remark 3.7, we have M v w 2 v M 8 0; 80 where M = max t2[0;T];x2 D v(t;x). This implies sup t2[0;T] ku 2(t; ) u2(t; )kX2 !0; as !0: Theorem 2.13 in the Neumann boundary condition is thus proved. 5.1.3 Proof of Theorem 2.13 in the Periodic Boundary Condition Case In this subsection, we prove Theorem 2.13 in the periodic boundary condition case. Throughout this subsection, we assume (H1), and Br;bu = Br;Pu in (1.17), and Bn;bu = Bn;Pu in (1.18). Without loss of generality again, we assume s = 0. Proof of Theorem 2.13 in the periodic boundary case. Suppose that u0 2X3\C3(RN). Let u 3(t;x) be the solution to (1.18) with s = 0 and u3(t;x) be the solution to (1.17) with s = 0. Suppose that u3(t;x) and u 3(t;x) exist on [0;T]. Set w 3 = u 3 u3. Then w 3 satis es 8 >>> >>> >>>> < >>> >>> >>> >: @tw 3(t;x) = RRN k (x y)(w 3(t;y) w 3(t;x))dy +a3(t;x)w 3(t;x) +F (t;x); x2RN; t2(0;T]; w 3(t;x) = w 3(t;x+pjej); x2RN; t2[0;T]; w 3(0;x) = 0; x2RN; (5.14) where a3(t;x) = R10 Fu(t;x;u3(t;x) + (u 3(t;x) u3(t;x)))d and F (t;x) = RRN k (x y)[u3(t;y) u3(t;x)]dy u3. Let w(t;x) = eAt(K1 t) +K2 ; where A = max x2RN;t2[0;T] a3(t;x). Applying the similar approach as in the Dirichlet boundary condition case, we can show that there are K1 > 0, K2 > 0, and 0 > 0 such that for 81 0 < < 0, w(t;x) w 3(t;x) w(t;x) 8x2RN; t2[0;T]: Theorem 2.13 in the periodic boundary condition case then follows. 5.2 Approximations of Principal Eigenvalues of Time Periodic Random Disper- sal Operators by Time Periodic Nonlocal Dispersal Operators In this section, we investigate the approximation of principal eigenvalues of time peri- odic random dispersal operators by the principal spectrum points of time periodic nonlocal dispersal operators. We rst recall some basic properties of principal eigenvalues of time periodic random dispersal operators, and basic properties of principal spectrum points of time periodic nonlocal dispersal operators to be used in the proof of Theorem 2.15. 5.2.1 Basic Properties of Principal Eigenvalues/Principal Spectrum Points of Time Periodic Dispersal Operators In this subsection, for i = 1;2;3, we focus on the time-periodic evolution equations (3.1) with i = and k( ) = k ( ), and (3.5) with i = 1. First of all, let us recall that i(t;s;a) is the solution operator of (3.1) with i = , k( ) = k ( ) and ai( ; ) = a( ; ) for i = 1;2;3. And let r( i(T;0;a)) be the spectral radius of i(T;0;a), and ~ i(a) be the principal spectrum point of Ni( ;a; ), respectively. We have the following propositions. Proposition 5.4. Let 1 i 3 be given. Then r( i(T;0;a)) = e~ i(a)T: Proof. See [59, Proposition 3.3]. We remark that Proposition 4.1 (1) is a special case of Proposition 5.4. 82 Next, for 1 i 3, recall that ri(t;s;a) is the solution operator of (3.5) with i = 1 and ai( ; ) = a( ; ). Similarly, let r( ri(T;0;a)) be the spectral radius of ri(T;0;a) and ri(a) be the principal eigenvalue of Ri(1;a). Note that Xri is a strongly ordered Banach space with the positive cone C = fu2Xri ju(x) 0g and by the regularity, a priori estimate, and comparison principle for parabolic equations, ri(T;0;a) : Xri !Xri is strongly positive and compact. Then by the Kre n-Rutman Theorem (see [65]), r( ri(T;0;a)) is an isolated algebraically simple eigenvalue of ri(T;0;a) with a positive eigenfunction uri( ) and for any 2 ( ri(T;0;a))nfr( ri(T;0;a))g, Re 0 and > 0 such that for any wi 2Zi, there holds k i(nT;0;a)wikXri k i(nT;0;a)urikXri Me nT: Proposition 5.6. For given 1 i 3 and a1;a2 2Xi\C1(R RN), j~ i(a1) ~ i(a2)j max x2 D;t2[0;T] ja1(t;x) a2(t;x)j; (5.15) 83 and j ri(a1) ri(a2)j max x2 D;t2[0;T] ja1(t;x) a2(t;x)j: (5.16) Proof. Let a0 = maxx2 D;t2[0;T]ja1(t;x) a2(t;x)j and a 1 (t;x) = a1(t;x) a0: It is not di cult to see that i(t;s;a 1 ) = e a0(t s) i(t;s;a1): It then follows that r( i(T;0;a 1 )) = e(~ i(a1) a0)T: (5.17) Observe that by Remark 3.7, for any u0 2Xr;+i , i(T;0;a 1 )u0 i(T;0;a2)u0 i(T;0;a+1 )u0: This implies that r( i(T;0;a 1 )) r( i(T;0;a2)) r( i(T;0;a+1 )): This together with (5.17) implies that ~ i(a1) a0 ~ i(a2) ~ i(a1) +a0; (5.18) that is, (5.15) holds. Similarly, we can prove that (5.16) holds. 84 5.2.2 Proof of Theorem 2.15 in the Dirichlet Boundary Condition Case In this subsection, we prove Theorem 2.15 in the Dirichlet boundary condition case. Throughout this subsection, we assume Br;bu = Br;Du in (1.19), and Dc = RN n D and Bn;bu = Bn;Du in (1.20). Note that for any a2X1\C1(R RN), there are an2X1\C3(R RN) such that supt2[0;T]kan(t; ) a(t; )kX1 ! 0 as n!1. By Proposition 5.6, without loss of generality, we may assume that a2X1\C3(R RN). Proof of Theorem 2.15 in the Dirichlet boundary condition case. First of all, for the simplic- ity in notation, we put r(T;0) = r1(T;0;a); r = r1(a); and (T;0) = 1(T;0;a); ~ = ~ 1(a): Let ur( ) be a positive eigenfunction of r(T;0) corresponding to r( r(T;0)). Without loss of generality, we assume that kurkXr1 = 1. We rst show that for any > 0, there is 1 > 0 such that for 0 < < 1, ~ r : (5.19) In order to do so, choose D0 D and u0 2Xr1\C3( D) such that u0(x) = 0 for x2DnD0, and u0(x) > 0 for x2 IntD0. By Proposition 5.5, there exist > 0, M > 0, and u02Z1, such that u0(x) = ur(x) +u0(x); (5.20) and k r(nT;0)u0kXr1 Me nTe rnT: (5.21) 85 By Theorem 2.13, there is 0 > 0 such that for 0 < < 0, there hold (nT;0)ur (x) r(nT;0)ur (x) C1(nT; ); (5.22) and (nT;0)u0 (x) r(nT;0)u0 (x) +C2(nT; ); (5.23) where Ci(nT; )!0 as !0 (i = 1;2). Hence for 0 < < 0, (nT;0)u 0 (x) = (nT;0)ur (x) + (nT;0)u0 (x) r(nT;0)ur (x) C1(nT; ) C2(nT; ) k r(nT;0)u0kXr1 e rnTur(x) C1(nT; ) C2(nT; ) Me nTe rnT =e( r )nTe nT( ur(x) Me nT) C1(nT; ) C2(nT; ): (5.24) Note that there exists m> 0 such that ur(x) m> 0 for x2 D0: Hence for any 0 < < , there is n1 > 0 such that for n n1, e nT( ur(x) Me nT) u0(x) + 1 for x2 D0; (5.25) and there is 1 0 such that for 0 < < 1, C1(n1T; ) +C2(n1T; ) e( r )n1T: (5.26) 86 Note that u0(x) = 0 for x2DnD0 and (n1T;0)u0 (x) 0 for all x2 D. This together with (5.24)-(5.26) implies that for < 1, (n 1T;0)u0 (x) e( r )n 1Tu0(x); x2 D: (5.27) By (5.27) and Remark 3.7, for any 0 < < 1 and n 1, ( (nn1T;0)u0)( ) e( r )nn1Tu0( ): This together with Proposition 5.4 implies that for 0 < < 1, e~ T = r( (T;0)) e( r )T: Hence (5.19) holds. Next, we prove that for any > 0, there is 2 > 0 such that for 0 < < 2, ~ r + : (5.28) To this end, rst, choose a sequence of smooth domains fDmg with D1 D2 D3 Dm D, and \1m=1Dm = D. Consider the following evolution equation 8 >>< >>: @tu(t;x) = u(t;x) +a(t;x)u(t;x); x2Dm; u(t;x) = 0; x2@Dm: (5.29) Let X1;m =fu2C( Dm;R)g; and Xr1;m =D(A m); 87 where Am is with Dirichlet boundary condition acting on X1;m\C0(Dm) and 0 < < 1. We denote the solution of (5.29) by um(t; ;s;u0) = ( rm(t;s)u0)( ) with u(s; ;s;u0) = u0( )2 Xr1;m. By Proposition 5.5, we have r( rm(T;0)) = e rmT; where rm is the principal eigenvalue of the following eigenvalue problem, 8 >>> >>> < >>> >>> : @tu+ u+a(t;x)u = u; x2Dm; u(t+T;x) = u(t;x); x2Dm; u(t;x) = 0; x2@Dm: By the dependence of the principle eigenvalue on the domain perturbation (see [22]), for any > 0, there exists m1 such that rm1 r + 2: (5.30) Second, leturm1( ) be a positive eigenfunction of rm1(T;0) corresponding tor( rm1(T;0)). By regularity for parabolic equations, urm1 2C3( Dm1). Let ( m1(t;0)urm1)(x) be the solution to 8 >>< >>: ut = hR Dm1 k (x y)u(t;y)dy u(t;x) i +a(t;x)u(t;x); x2 Dm1; u(0;x) = urm1(x): (5.31) Then by Theorem 2.13, m1(nT;0)u r m1 (x) m1(nT;0)urm1 (x) +C(nT; ) 8x2 D m1; 88 where C(nT; )!0 as !0. By Remark 3.7, (nT;0)ur m1j D (x) m1(nT;0)u r m1 (x) 8x2 D: It then follows that for x2 D, (nT;0)ur m1j D (x) r m1(nT;0)u r m1 (x) +C(nT; ) = e rm1nTurm1(x) +C(nT; ) e( r+ 2)nTurm1(x) +C(nT; ) = e( r+ )nTe 2nTurm1(x) +C(nT; ): (5.32) Note that min x2 D urm1(x) > 0: Hence for any > 0, there is n2 1 such that e 2n2T 12; (5.33) and there is 2 > 0 such that for 0 < < 2, C(n2T; ) 12e( r+ )n2Turm1(x) 8x2 D: (5.34) By (5.32)-(5.34), (n 2T;0)urm1j D (x) e( r+ )n 2Turm 1(x) 8x2 D: This together with Remark 3.7 implies that for 0 < < 2, (nn 2T;0)urm1j D (x) e( r+ )nn 2Turm 1(x) 8x2 D: (5.35) 89 This together with Proposition 5.4 implies that ~ r + for 0 < < 2, that is, (5.28) holds. Theorem 2.15 in the Dirichlet boundary condition case then follows from (5.19) and (5.28). 5.2.3 Proof of Theorem 2.15 in the Neumann Boundary Condition Case Proof of Theorem 2.15 in the Neumann boundary condition case. We assume Br;bu = Br;Nu in (1.19), and Dc = ; and Bn;bu = Bn;Nu in (1.20). The proof in the Neumann boundary condition case is similar to the arguments in the Dirichlet boundary condition case (it is simpler). For the completeness, we give a proof in the following. Without loss of generality, we may also assume that a2X2\C3(R RN). For the simplicity in notation, put r(nT;0) = r2(nT;0;a); r = r(a); and (nT;0) = 2(nT;0;a); ~ = ~ (a): By Propositions 5.4 and 5.5, r( r(T;0)) = e rT; (5.36) and r( (T;0)) = e~ T: (5.37) 90 Let ur( ) be a positive eigenfunction of r(T;0) corresponding to r( r(T;0)). By regu- larity for parabolic equations, ur2C3( D). By Theorem 2.13, we have k (nT;0)ur r(nT;0)urkX2 C(nT; ); where C(nT; )!0 as !0. This implies that for all x2 D, (nT;0)ur (x) r(nT;0)ur (x) C(nT; ) = e rnTur(x) C(nT; ) = e( r )nTe nTur(x) C(nT; ); (5.38) and (nT;0)ur (x) r(nT;0)ur (x) +C(nT; ) = e rnTur(x) +C(nT; ) = e( r+ )nTe nTur(x) +C(nT; ): (5.39) Note that min x2 D ur(x) > 0: (5.40) Hence for any > 0, there is n1 > 1 such that 8> >>> >>< >>> >>> : e n1Tur(x) 32ur(x) 8x2 D; e n1Tur(x) 12ur(x) 8x2 D; (5.41) 91 and there is 0 > 0 such that for any 0 < < 0, C(n1T) < 12e( r )n1Tur(x) 8x2 D: (5.42) By (5.38)-(5.42), we have that for any 0 < < 0, e( r )n1Tur(x) (n1T;0)ur (x) e( r+ )n1Tur(x) 8x2 D: This together with Remark 3.7 implies that for all n 1, e( r )n1nTur(x) (n1nT;0)ur (x) e( r+ )n1nTur(x) 8x2 D: It then follows that for any 0 < < 0, e( r )T r( (T;0)) e( r+ )T: By Proposition 5.4, we have j~ rj< 80 < < 0: Theorem 2.15 in the Neumann boundary condition case is thus proved. 5.2.4 Proof of Theorem 2.15 in the Periodic Boundary Condition Case Proof of Theorem 2.15 in the periodic boundary condition case. We assume D = RN, and Br;bu = Br;Pu in (1.19), and Bn;bu = Bn;Pu in (1.20). It can be proved by the same arguments as in the Neumann boundary condition case. 92 5.3 Approximations of Positive Time Periodic Solutions of Random Dispersal KPP Type Evolution Equations by Nonlocal Dispersal KPP Type Evolution Equations In this section, we study the approximation of the asymptotic dynamics of time periodic KPP type evolution equations with random dispersal by those of time periodic KPP type evolution equations with nonlocal dispersal. We rst recall the existing results about time periodic positive solutions of KPP type evolution equations with random as well as nonlocal dispersal. Then we prove Theorem 2.16. Throughout this section, we assume that D RN is a bounded C2+ domain or D = RN, and (H2), (H3) and (H3) hold. 5.3.1 Asymptotic Behavior of KPP Type Evolution Equations In this subsection, we present some basic known results for (1.21) and (1.22). Let Xr1 and Xri (i = 2;3) be de ned as in (3.3) and (3.4), respectively. For u0 2 Xri , let (U(t;0)u0)( ) = u(t; ;u0), where u(t; ;u0) is the solution to (1.21) with u(0; ;u0) = u0( ) and Br;bu = Br;Du when i = 1, Br;bu = Br;Nu when i = 2, and Br;bu = Br;Pu when i = 3. Similarly, for u0 2 Xi, let (U (t;0)u0)( ) = u (t; ;u0), where u (t; ;u0) is the solution to (1.22) with u (0; ;u0) = u0( ) and Dc = RN n D, Bn;bu = Bn;Du when i = 1, Dc = ;, Bn;bu = Bn;Nu when i = 2, and Bn;bu = Bn;Pu when i = 3. Proposition 5.7. (1) If u0 0, solution u(t; ;u0) to (1.21) with u(0; ;u0) = u0( ) exists for all t 0 and u(t; ;u0) 0 for all t 0. (2) If u0 0, solution u(t; ;u0) to (1.22) with u(0; ;u0) = u0( ) exists for all t 0 and u(t; ;u0) 0 for all t 0. Proof. (1) Note that u( ) 0 is a solution of (1.21) and u( ) M is a super-solution of (1.21) for M 1. Then by Remark 3.7, there is M 1 such that 0 u(t;x;u0) M 8x2 D; t2(0;tmax); 93 where (0;tmax) is the existence interval of u(t; ;u0). This implies that we must have tmax =1 and hence (1) holds. (2) It can be proved by similar arguments as in (1). Proposition 5.8. (1) (1.21) has a unique globally stable positive time periodic solution u (t;x). (2) (1.22) has a unique globally stable time periodic positive solution u (t;x). Proof. (1) See [67, Theorem 3.1] (see also [53, Theorems 1.1, 1.3]). (2) See [56, Theorem E]. Remark 5.9. By Proposition 5.8(2), if there is u0 2X+i nf0g such that (U (nT;0)u0)( ) u0( ) for some n 1, then we must have limn!1(U (nT;0)u0)( ) = u (0; ) and hence (U (nT;0)u0)( ) u (0; ): Similarly, if there is u0 2X+i nf0g such that (U (nT;0)u0)( ) u0( ) for some n 1, then (U (nT;0)u0)( ) u (0; ): 5.3.2 Proof of Theorem 2.16 in the Dirichlet Boundary Condition Case In this subsection, we prove Theorem 2.16 in the Dirichlet boundary condition case. Throughout this subsection, we assume that Br;bu = Br;Du in (1.21), and Dc = RNn D and Bn;bu = Bn;Du in (1.22). Proof of Theorem 2.16 in the Dirichlet boundary condition case. First of all, note that it suf- ces to prove that for any > 0, there is 0 > 0 such that for 0 < < 0, u (t;x) u (t;x) u (t;x) + 8t2[0;T]; x2 D: 94 We rst show that for any > 0, there is 1 > 0 such that for 0 < < 1, u (t;x) u (t;x) + 8t2[0;T]; x2 D: (5.43) To this end, choose a smooth function 0 2C1c (D) satisfying that 0(x) 0 for x2D and 0( )6 0. Let 0 < 1 be such that u (x) := 0(x) 0 such that u (0;x) u (x) + 0 for x2supp( 0): (5.44) By Proposition 5.8, there is N 1 such that U(NT;0)u (x) u (NT;x) 0=2 = u (0;x) 0=2 8x2 D: By Theorem 2.13, there is 1 > 0 such that for 0 < < 1, we have U (NT;0)u (x) U(NT;0)u (x) 0=2 8x2 D: Hence for 0 < < 1, U (NT;0)u (x) u (0;x) 0 8x2 D: (5.45) Note that U (NT;0)u (x) 0 8x2 D: 95 It then follows from (5.44) and (5.45) that for 0 < < 1, U (NT;0)u (x) u (x) 8x2 D: This together with Proposition 5.8 (2) implies that U (NT;0)u (x) u (0;x) 8x2 D (5.46) (see Remark 5.9). By Proposition 5.8 (1) again, for n 1, u (t;x) (U(nNT +t;0)u )(x) + =2 8t2[0;T]; x2 D: (5.47) Fix an n 1 such that (5.47) holds. By Theorem 2.13, there is 0 < ~ 1 1 such that for 0 < < ~ 1, (U(nNT +t;0)u )(x) (U (nNT +t;0)u )(x) +C1( ); (5.48) where C1( )!0 as !0. By (5.46), Remark 3.7, and Proposition 5.8 (2), U (nNT +t;0)u (x) U (t;0)u (0; ) (x) = u (t;x) (5.49) for t2[0;T] and x2 D. Let 0 < 1 ~ 1 be such that C1( ) < =2 8 0 < < 1: (5.50) (5.43) then follows from (5.47)-(5.50). 96 Next, we need to show for any > 0, there is 2 > 0 such that for 0 < < 2, u (t;x) u (t;x) 8t2[0;T]; x2 D: (5.51) To this end, choose a sequence of open sets fDmg with smooth boundaries such that D1 D2 D3 Dm D, and D = \m2NDm. According to Corollary 5.11 in [2], Dm!D regularly and all assertions of Theorem 5.5 in [2] hold. Consider 8 >>< >>: @tu = u+uf(t;x;u); x2Dm; u(t;x) = 0; x2@Dm: (5.52) Let Um(t;0)u0 = u(t; ;u0), where u(t; ;u0) is the solution to (5.52) with u(0; ;u0) = u0( ). By Proposition 5.8, (5.52) has a unique time periodic positive solution u m(t;x). We rst claim that limm!1u m(t;x)!u (t;x) uniformly in t2[0;T] and x2 D: (5.53) In fact, it is clear that u 2C(R D;R) and u m 2C(R Dm;R). By [22, Theorem 7.1], sup t2R ku m(t; ) u (t; )kLq(D) !0 as m!1 for 1 q <1. Let a(t;x) = f(t;x;u (t;x)) and am(t;x) = f(t;x;u m(t;x)). Then u (t;x) and u m(t;x) are time periodic solutions to the following linear parabolic equations, 8 >>< >>: ut = u+a(t;x)u; x2D; u(t;x) = 0; x2@D; (5.54) and 8 >>< >>: ut = u+am(t;x)u; x2Dm; u(t;x) = 0; x2@Dm; (5.55) 97 respectively. Observe that there isM > 0 such thatkakL1(D) 0, x m 1 such that u (t;x) u m(t;x) =3 8t2[0;T]; x2 D: (5.56) Choose M 1 such that for 0 < 1, Mu m(t;x) u (t;x) 8t2[0;T]; x2 D: (5.57) Let u+m(x) = Mu m(0;x); u+(x) = u+m(x)j D: By Proposition 5.8, for xed m and , there exists N 1, such that u m(t;x) Um(NT +t;0)u+m (x) =3 8t2[0;T]; x2 D: (5.58) By Theorem 2.13, there is 0 < ~ 2 < 1 such that for 0 < < ~ 2, (Um(NT +t;0)u+m)(x) (U m(NT +t;0)u+m)(x) C2( ) 8t2[0;T]; x2Dm; (5.59) 98 where C2( )!0 as !0 and (U m(t;0)u0)( ) = u(t; ;u0) is the solution to 8 >>< >>: ut(t;x) = hR Dmk (x y)u(t;y)dy u(t;x) i +u(t;x)f(t;x;u(t;x)); x2 Dm u(0;x) = u0(x); x2 Dm: Let 0 < 2 < ~ 2 be such that for 0 < < 2, C2( ) < =3: (5.60) By Remark 3.7, for x2 D we have (U m(NT +t;0)u+m)(x) (U (NT +t;0)u+)(x); and (U (NT +t;0)u+)(x) = (U (t;0)U (NT;0)u+)(x) (U (t;0)u (0; ))(x) = u (t;x): This together with (5.56), (5.58), (5.59), and (5.60) implies (5.51). So, for any > 0, there exists 0 = minf 1; 2g, such that for any < 0, we have ju (t;x) u (t;x)j uniformly in t> 0 and x2 D: 5.3.3 Proof of Theorem 2.16 in the Neumann Boundary Condition Case We assume Br;bu = Br;Nu in (1.19), and Dc = ; and Bn;bu = Bn;Nu in (1.20). The proof in the Neumann boundary condition case is similar to the arguments in the Dirichlet boundary condition case (it is indeed simpler). For completeness, we provide a proof. 99 Proof of Theorem 2.16 in the Neumann boundary condition case. For completeness, we pro- vide a proof. First, we show that for any > 0, there is 1 > 0 such that u (t;x) u (t;x) + 8t2[0;T]; x2 D: (5.61) Choose a smooth function u 2C1( D) with u ( ) 0 and u ( )6 0 such that u (x) 0 such that u (0;x) u (x) + 0 8x2 D: (5.62) By Proposition 5.8 (1), there is N 1 such that U(NT;0)u (x) u (0;x) 0=2 8x2 D: (5.63) By Theorem 2.13, there is 1 > 0 such that for 0 < < 1, (U (NT;0)u )(x) (U(NT;0)u )(x) 0=2 8x2 D: (5.64) By (5.62), (5.63) and (5.64), U (NT;0)u (x) u (x) 8x2 D; and then by Proposition 5.8 (2), U (NT;0)u (x) u (0;x) 8x2 D: (5.65) 100 By Proposition 5.8 (1) again, for any given > 0, n 1, and 0 < < 1, u (t;x) (U(nNT +t;0)u )(x) + =2 8t2[0;T]; x2 D: (5.66) By Theorem 2.13, there is 0 < 1 1 such that for < 1, (U(nNT +t;0)u )(x) (U (nNT +t;0)u )(x) + 2 8t2[0;T]; x2 D: (5.67) By Remark 3.7 and (5.65), we have (U (nNT +t;0)u )(x) = (U (t;0)U (nNT;0)u )(x) (U (t;0)u (t; ))(x) = u (t;x) (5.68) for t2[0;T] and x2 D. (5.61) then follows from (5.66)-(5.68). Next, we show that for any > 0, there is 2 > 0 such that for 0 < < 2, u (t;x) u (t;x) 8t2[0;T]; x2 D: (5.69) Choose M 1 such that f(t;x;M) < 0 for t2R and x2 D. Put u+(x) = M 8x2 D: Then for all > 0, u (0;x) u+(x) 8x2 D: (5.70) By Proposition 5.8, there is N 1 such that u (t;x) (U(NT +t;0)u+)(x) =2 8t2[0;T]; x2 D: (5.71) 101 By Theorem 2.13, there are 2 > 0 such that for 0 < < 2, (U(NT +t;0)u+)(x) (U (NT +t;0)u+)(x) 2 8t2[0;T]; x2 D: (5.72) By (5.70), (U (NT +t;0)u+)(x) = (U (t;0)U (NT;0)u+)(x) (U (t;0)u (t; ))(x) = u (t;x) (5.73) for t2[0;T] and x2 D. (5.69) then follows from (5.71)-(5.73). So, for any > 0, there exists 0 = minf 1; 2g, such that for any < 0, we have ju (t;x) u (t;x)j uniform in t> 0 and x2 D: 5.3.4 Proof of Theorem 2.16 in the Periodic Boundary Condition Case Proof of Theorem 2.16 in the periodic boundary condition case. We assume D = RN, and Br;bu = Br;Pu in (1.19), and Bn;bu = Bn;Pu in (1.20). It can be proved by the similar arguments as in the Neumann boundary condition case. 5.4 Applications to the E ect of the Rearrangements with Equimeasurability on Principal Spectrum Point of Nonlocal Dispersal Operators In this section, we will apply the approximation results established in this Chapter to the e ect of the rearrangements with equimeasurability on principal spectrum point of nonlocal dispersal operators. First, we show the proof of Theorem 2.18. 102 Proof of Theorem 2.18. In the case of D = D], a( ) = a]( ), and u( ) = u]( ), Theorem 2.18 holds trivially. Otherwise, by (2.24), we have ~ ri(a]) > ~ ri(a) for 0: And by Theorem 2.15, we have lim !0 ~ i(a) = ri(a) and lim !0 ~ i(a]) = ri(a]): Hence, we have ~ i(a]) ~ i(a) for 0: Remark 5.10 (E ect of the rearrangements with equimeasurability on principal spectrum point of general nonlocal dispersal operators). (1) Consider (2.26) and (2.27) for general kernel k( ) and dispersal rate in the Dirichlet boundary condition case. We denote the principal spectrum point of (2.26) (independent of ) and (2.27) (independent of ) by ~ 1(a) and ~ 1(a]). Assume that k( ) is symmetric with respect to 0. Let a]( ), k]( ) and D] be the Schwarz symmetrization of a( ), k( ) and D, respectively. Then we have ~ 1(a) ~ 1(a]): (5.74) 103 In fact, by Proposition 4.7 and rearrangement inequalities (see [1] for detail), we have ~ 1(a) = sup fujkukX1=1g Z Z D D k(x y)u(y)u(x)dydx + Z D a(x)u2(x)dx sup fu]jku]kX1=1g Z Z D] D] k(x y)u](y)u](x)dydx + Z D] a](x)u2](x)dx = ~ 1(a]): (2) For (2.26) and (2.27) with general kernels k( ) and dispersal rate in the Neumann boundary condition case, we have similar result as in the Dirichlet boundary condition case. (3) For (2.26) and (2.27) with general kernels k( ) and dispersal rate in the periodic bound- ary condition case, it is open to get similar result as in (5.74). 104 Chapter 6 Concluding Remarks, Problems, and Future Plans In this dissertation, I studied two dynamical issues. One is about the principal spectrum of nonlocal dispersal operators and its applications in nonlocal dispersal evolution equations, and the other is about the approximations of random dispersal operators and equations by nonlocal dispersal operators and equations from three points of view. Both are theoretically and practically important. The results of eigenvalue problems of nonlocal dispersal operators are applied to a two species competition system, the approximation results are applied to the e ects of rearrangement with nonlocal dispersals. The two applications cast a new light on di usive systems arising in ecology or biology. More precisely, regarding to the rst dynamical issue, we prove Theorem 2.4, Theorem 2.6, Theorem 2.8 and Theorem 2.12 as an application. Although the semigroups generated by nonlocal operators are not compact, we are able to convert the time homogeneous non- local operator into a compact operator and study the existence of its principal eigenvalue. There are examples showing that there is no principal eigenvalue to some nonlocal operator. However, in some circumstances, the principal spectrum plays the same role as the principal eigenvalue. So we focus on the dependence of the principal spectrum points of nonlocal dis- persal operators on the underlying parameter with Dirichlet, Neumann, and periodic types of boundary condition in a uni ed way. Finally, in the model of population dynamics of two species competing system, we show that the species di using nonlocally with Neumann type boundary condition drives the species adopting Dirichlet type boundary condition ex- tinct. Biologically, individuals di using inside D (Neumann type boundary condition) are more likely to survive than those living in a habitat surrounded by a hostile environment (Dirichlet type boundary condition). 105 On the second dynamical issue, we prove Theorem 2.13, Theorem 2.15, Theorem 2.16 and Theorem 2.18 as an application. From the formal relation between nonlocal operators and Laplacian operators, we are inspired to study the approximation of random dispersal equations by its nonlocal counterparts from other perspectives. Theorem 2.13 is fundamental to the investigation of other approximations, since theorem 2.13 build the connection of solution operators with random dispersal and nonlocal dispersal. By the spectral mapping theorem, the principal spectrum points and principal eigenvalues are related to the solution operators. Hence, we have the approximations of principal eigenvalues of random dispersal operators by principal spectrum points of nonlocal dispersal operators. Next, based on the previous two theorems, we show the approximation of asymptotic dynamics of KPP type evolution equations with random dispersal by that with nonlocal dispersal. Finally, to see the advantage of approximation results, we apply them to the e ect of the rearrangements on principal spectrum point of nonlocal dispersal operators, and prove Theorem 2.18. Hence, as long as we know some results in the random models, we should have the similar results in the nonlocal models, when the dispersal distance of the nonlocal kernel is small. Along the line of my dissertation, there are several important problems which are not well understood yet. We discuss the following three problems. Problem 1 In [57], the authors proved the spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, so it is natural to ask whether the results hold in a cylindrical domain, such that in one direction, it is periodic and in the other direction, either Dirichlet or Neumann type boundary condition is prescribed. It seems like there should be no di culty in extending the results to the cylinder domain. But it will be interesting to prove the existence of traveling waves with speed c = c ( ) and uniqueness and stability of traveling waves in the case that f is both space and time periodic. Problem 2 In [43], the authors studied the principal eigenvalue of a general random operator with inde nite weight on cylindrical domains. Biologically, this problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions 106 for a species to survive. So it will be worthwhile to study the principal spectrum point of a nonlocal operator and nd the optimal spatial arrangement for a species to survive. The principal spectrum point plays the same role as the principle eigenvalue in some situations. The survival of a species is determined by the magnitude of the principal spectrum point of nonlocal dispersal operators. Problem 3 In [40], authors study an evolution equation with nonlinear nonlocal operators as follows @tu = Z RN k(x y)ju(t;y) u(t;x)jp 2(u(t;y) u(t;x))dy; x2RN; (6.1) and they study the decay estimates for (6.1) in the whole space. We can consider the random counterparts @tu =r jru(t;x)jp 2; x2RN; (6.2) and investigate the approximations of nonlinear random dispersal operators/equations by nonlinear nonlocal dispersal operators/equations from many other points of view. 107 Bibliography [1] A. Alvino, G. Trombetti, P.-L. Lions, and S. Matarasso, Comparison results for solutions of elliptic problems via symmetrization, Ann. Inst. H. Poincar e Anal. Non Lin eaire 16 (1999), no. 2, 167-188. [2] W. Arendt and D. Daners Uniform convergence for elliptic problems on varying domains, Math. Nachr. 280 (2007), no. 1-2, 28-49. [3] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694. [4] D. G. Aronson and H. F. Weinberger, Nonlinear di usion in population genetics, com- bustion, and nerve pulse propagation, Partial Di erential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), pp. 5-49. Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975. [5] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear di usion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33-76. [6] P. Bates and F. Chen, Spectral analysis of traveling waves for nonlocal evolution equa- tions, SIAM J. Math. Anal. 38 (2006), pp. 116-126. [7] P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. 332 (2007), no. 1, 428-440. [8] P. Bates and G. Zhao, Spectral convergence and Turing patterns for nonlocal di usion systems, preprint. [9] R. B urger, Perturbations of positive semigroups and applications to population genetics, Math. Z. 197 (1988), pp. 259-272. [10] R. S. Cantrell and C. Cosner, Spatial Ecology via reaction-di usion Equations. Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chich- ester, 2003. [11] R. S. Cantrell, C. Cosner, Y. Lou, and D. Ryan, Evolutionary stability of ideal free dispersal strategies: a nonlocal dispersal model, Can. Appl. Math. Q. 20 (2012), no. 1, 15-38. [12] E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal di usion equations, J. Math. Pures Appl. (9), 86 (2006), no. 3, 271-291. 108 [13] F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity, Discrete Contin. Dyn. Syst. 24 (2009), pp. 659-673. [14] C. Cosner, J. D avila, and S. Mart nez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn. 6 (2012), no. 2, 395-405. [15] C. Cortazar, M. Elgueta, and J. D. Rossi, Nonlocal di usion problems that approximate the heat equation with Dirichlet boundary conditions, Israel J. of Math., 170 (2009), 53-60. [16] C. Cortazar, M. Elgueta, J. D. Rossi, and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal di usion problems, Arch. Ration. Mech. Anal. 187 (2008), no. 1, 137-156. [17] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Di erential Equations 249 (2010), no. 11, 2921-2953. [18] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction di usion equation, Annali di Matematica 185(3) (2006), pp. 461-485 [19] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non-local reaction- di usion equations, Nonlinear Anal. 60 (2005), no.5, 797 - 819. [20] J. Coville, J. D avila, and S. Mart nez, Existence and uniqueness of solutions to a non- local equation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008), no. 5, 1693-1709. [21] D. Daners, An isoperimetric inequality related to a Bernoulli problem, Calc. Var. Partial Di erential Equations 39 (2010), no. 3-4, 547-555. [22] D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Dif- ferential Equations 129 (1996), no. 2, 358-402. [23] D. Daners, Existence and perturbation of principal eigenvalues for a periodic-parabolic problem, Proceedings of the Conference on Nonlinear Di erential Equations (Coral Gables, FL, 1999), 51-67, Electron. J. Di er. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000. [24] M. D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigen- value for operators with maximum principle, Proc. Nat. Acad. Sci. USA 72 (1975) pp. 780-783. [25] D. E. Edmunds and W. D. Evans, Spectral theory and di erential operators, The Claren- don Press Oxford University Press, New York, 1987. [26] L. C. Evans, Partial Di erential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998. [27] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003. 109 [28] P. C. Fife, Mathematical aspects of reacting and di using systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979. [29] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 335- 369. [30] J. Garc a-Mel an and J. D. Rossi, On the principal eigenvalue of some nonlocal di usion problems, J. Di erential Equations, 246 (2009), pp. 21-38. [31] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers, Non-local disper- sal, Di erential Integral Equations 18 (2005), no. 11, 1299-1320. [32] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math- ematics, 840. Springer-Verlag, Berlin-New York, 1981. [33] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, 247, Longman Scienti c & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [34] P. Hess and H. Weinberger, Convergence to spatial-temporal clines in the Fisher equa- tion with time-periodic tnesses, J. Math. Biol. 28 (1990), no. 1, 83-98. [35] G. Hetzer, T. Nguyen, and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Communications on Pure and Applied Anal- ysis, 11 (2012), pp. 1699-1722. [36] G. Hetzer, T. Nguyen, and W. Shen, E ects of small variation of the reproduction rate in a two species competition model, Electron. J. Di erential Equations , (2010), No. 160, 17 pp. [37] G. Hetzer, W. Shen, and A. Zhang, E ects of spatial variations and dispersal strate- gies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, Rocky Mountain J. Math. 43 (2013), no. 2, 489-513. [38] V. Hutson, S. Martinez, K. Mischaikow, and G.T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), no. 6, 483-517. [39] V. Hutson, W. Shen and G.T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math. 38 (2008), no. 4, 1147-1175. [40] L. I. Ignat, D. Pinasco, J.D. Rossi, and A. San Antolin, Decay estimates for nonlinear nonlocal di usion problems in the whole space, J. Anal. Math. 122 (2014), 375-401. [41] C.-Y. Kao, Y. Lou, and W. Shen, Random dispersal vs non-local dispersal, Discrete and Contin. Dyn. Syst. 26 (2010), no. 2, 551-596. [42] C.-Y. Kao, Y. Lou, and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), pp. 2047-2072. 110 [43] C.Y. Kao, Y. Lou, and E. Yanagida, Principal eigenvalue for an elliptic problem with inde nite weight on cylindrical domains, Math. Biosci. Eng. 5 (2008), no. 2, 315-335. [44] A. Kolmogorov, I. Petrowsky, and N. Piscunov, A study of the equation of di usion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos, Univ., 1(6): 1-25 (1937). [45] L. Kong and W. Shen, Positive stationary solutions and spreading speeds of KPP equa- tions in locally spatially inhomogeneous media, Methods and Applications of Analysis, 18 (2011), pp. 427-456. [46] F. Li, Y. Lou, and Y. Wang, Global dynamics of a competition model with non-local dispersal I: The shadow system, J. Math. Anal. Appl. 412 (2014), no. 1, 485-497. [47] W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, 11 (2010), pp. 2302-2313. [48] X. Liang, X. Lin, and H. Matano, A variational problem associated with the mini- mal speed of travelling waves for spatially periodic reaction-di usion equations, Trans. Amer. Math. Soc., 362 (2010), no. 11, pp. 5605-5633. [49] G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction di usion systems, J. Math. Anal. Appl. 385 (2012), pp. 1094-1106. [50] P. Meyre-Nieberg, Banach Lattices, Springer-Verlag, 1991. [51] S. Pan, W.-T. Li, and G. Lin, Existence and stability of traveling wavefronts in a nonlo- cal di usion equation with delay, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), pp. 3150-3158. [52] J. D. Murray, Mathematical Biology, Biomathematics 19, Springer-Verlag, Berlin, 1989. [53] G. Nadin, Existence and uniqueness of the solutions of a space-time periodic reaction- di usion equations, J. Di erential Equation 249 (2010), no. 6, 1288-1304. [54] G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl. (4) 188 (2009),no. 2, 269-295. [55] A. Pazy, Semigroups of linear operators and applications to partial di erential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [56] N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigen- values of time periodic nonlocal dispersal operators and applications, J. Dynam. Dif- ferential Equations 24 (2012), no. 4, 927-954. [57] N. Rawal, W. Shen, and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, submitted. [58] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dis- persal evolution operators, J. Di erential Equations, 235 (2007), no. 1, 262-297. 111 [59] W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, submitted. [60] W. Shen and X. Xie, Approximations of random dispersal operators/equations by non- local dispersal operators/equations, submitted. [61] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dis- persal in space periodic habitats, J. Di erential Equations 249 (2010), no. 4, 747-795. [62] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monos- table equations, Comm. Appl. Nonlinear Anal. 19 (2012), no. 3, 73-101. [63] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monos- table equations in space periodic habitats, Proc. AMS, 140 (2012), pp. 1681-1696. [64] J. G. Skellam, Random dispersal in theoretical populations, Biometrika 38, (1951) 196- 218. [65] P. Tak a c, A short elementary proof of the Kre n-Rutman theorem, Houston J. Math., 20 (1994), no. 1, 93-98. [66] G.-B. Zhang, W.-T. Li, and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Di erential Equations 252 (2012), no. 9, 5096-5124. [67] X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems, Bull. Austral. Math. Soc. 53 (1996), no. 2, 305-324. [68] X.-Q. Zhao, Dynamical Systems in population biology, CMS Books in Mathematics, 16. Springer-Verlag, New York, 2003. 112