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
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