---
author:
- |
    Wenxian Shen\
    Department of Mathematics and Statistics\
    Auburn University\
    Auburn, AL 36849, U.S.A.\
    \
    Xiaoxia Xie\
    Department of Applied Mathematics\
    Illinois Institute of Technology\
    Chicago, IL 60616, U.S.A.
title: Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
---

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[**Abstract.**]{} In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.

[**Key words.**]{} Nonlocal dispersal operator, time periodic weight function, principal spectrum point, principal eigenvalue, KPP equations

[**Mathematics subject classification.**]{} 45C05, 45M05, 45M20, 47G10, 92D25.

Introduction
============

Random dispersal operators, such as $u(t, \cdot)\mapsto \Delta u(t, \cdot)$ with proper boundary condition, are often adopted in modeling dissipative systems in applied sciences when the organisms in the systems move randomly between the adjacent spatial locations (see [@BrCoFl; @CaCo; @CaCo1; @CaCo2; @Co; @CoCuPo; @Fisher; @Ha; @He; @He0; @HeKo; @HessWein; @Kolmo; @Zhaox], etc.). Nonlocal dispersal operators, such as $u(t, \cdot)\mapsto  \int_{{\ensuremath{\mathbb{R}}}^N}k(y-\cdot)[u(t, y)-u(t, \cdot)]dy$, are applied in modeling diffusive systems in applied sciences when the systems exhibit long range internal interactions (see [@FeIsPe; @Fif; @GrHiHuMiVi; @HuGr; @HuMaMiVi; @KaLoSh1; @OtDuAl; @ShZh0; @St], etc.). Diffusive evolution equations with both random and nonlocal dispersals have been widely studied on one hand. On the other hand, there are still many important dynamical issues for such systems which are not well understood yet.

The current paper is devoted to the study of principal spectrum points/principal eigenvalues of nonlocal dispersal operators with time periodic indefinite weight functions and Dirichlet type, Neumann type, and periodic boundary conditions. More precisely, the eigenvalue problem subject to Dirichlet type boundary condition considered in this paper reads as $$\label{dirichlet-ori}
\begin{cases}
 -\p_t u(t, x)+\int_D \kappa(y-x)u(t, y)dy-u(t, x)+\lambda_1 m_1(t,x)u(t, x)=0,\quad x\in \bar D,\\
u(t+T, x)=u(t, x),
\end{cases}$$ where $D\subset \R^N$ is a smooth bounded domain and $m_1(t, x)\neq 0$ is a continuous, time periodic weight function; the eigenvalue problem subject to Neumann type boundary condition considered in this paper reads as $$\label{neumann-ori}
\begin{cases}
 -\p_t u(t, x)+\int_D \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_2 m_2(t,x)u(t, x)=0,\quad x\in \bar D,\\
u(t+T, x)=u(t, x),
\end{cases}$$ where $D\in \R^N$ and $m_2(t, x)$ are as ; and the eigenvalue problem subject to periodic boundary conditions considered in this paper reads as $$\label{periodic-ori}
\begin{cases}
 -\p_t u(t, x)+\int_{\R^N} \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_3 m_3(t,x)u(t, x)=0,\quad x\in \R^N,\\
u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x),
\end{cases}$$ where $p_j>0$, ${\bf e_j}=(\delta_{j_1}, \delta_{j_2}, \cdots, \delta_{jN})$ ($\delta_{jk}=1$ if $j=k$ and $\delta_{jk}=0$ if $j\neq k$), and $m_3(t, x)$ is a continuous function with $m_3(t+T, x)=m_3(t, x+p_j{\bf e_j})=m_3(t, x)$ for $j=1, 2, \cdots, N$. Throughout the paper, we assume that the nonlocal kernel function $\kappa(\cdot)$ in - satisfies the following condition \[K\] ()C\_c(\^N),  (0)&gt;0, (-z)=(z), \_[[$\mathbb{R}$]{}\^N]{} (z)dz=1.

The eigenvalue problems , , and can be viewed as the nonlocal dispersal counterparts of the following eigenvalue problems associated to random dispersal operators with time periodic indefinite functions and Dirichlet, Neumann, and periodic boundary conditions, $$\label{random-dirichlet-op}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_1m_1(t,x)u(t, x)=0 \, & x\in  D,\\
 u(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ $$\label{random-neumann-op}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_2 m_2(t,x)u(t, x)=0 \, & x\in  D,\\
 \frac{\p u}{\p {\bf n}}(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ and $$\label{random-periodic-op}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_3 m_3(t,x)u(t, x) =0\, & x\in  \R^N,\\
 u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x),
\end{cases}$$ respectively (see [@ShXi1] and references therein for the relation between nonlocal dispersal operators with Dirichlet type, Neumann type, and periodic boundary conditions, and random dispersal operators with Dirichlet, Neumann, and periodic boundary conditions).

The eigenvalue problems of random dispersal operators with indefinite weight functions have been extensively for more than two decades (see, e.g. [@BeHe; @Boc; @Bo; @BrCoFl; @BrLi; @CaCo1; @Co; @CoCuPo; @He; @He0; @HeKo; @HiKaLa; @KaLoYa; @LoYa; @SeHe] and references therein). For $i=1$ ($i= 2$, or $i= 3$, respectively), recall that a real number $\lambda_1^{r,p}$ ($\lambda_2^{r,p}$ or $\lambda_3^{r,p}$, respectively) is called a [*principal eigenvalue*]{} of (, or , respectively) if (, or , respectively) with $\lambda_1=\lambda_1^{r,p}$ ($\lambda_2=\lambda_2^{r,p}$, or $\lambda_3=\lambda_3^{r,p}$, respectively) has a positive solution (called [*eigenfunction*]{}). The eigenvalue problem , , or are closely related to the following regular eigenvalue problems, $$\label{random-dirichlet-op-eq}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_1m_1(t,x)u(t, x)=\mu_1 u(t, x) \, & x\in  D,\\
 u(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ $$\label{random-neumann-op-eq}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_2 m_2(t,x)u(t, x)=\mu_2 u(t, x) \, & x\in  D,\\
 \frac{\p u}{\p {\bf n}}(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ $$\label{random-periodic-op-eq}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_3 m_3(t,x)u(t, x) =\mu_3u(t, x)\, & x\in  \R^N,\\
 u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x).
\end{cases}$$ For $i=1, 2, 3$, and any given $\lambda_i$, let $\mu^{r, p}_{i}(\lambda_i)$ be the principal eigenvalue of , , and , respectively. Then $\lambda_1^{r,p}$ (resp. $\lambda_2^{r,p}$, $\lambda_3^{r,p}$) is a principal eigenvalues of (resp. , ) if and only if $\mu^r_1(\lambda_1^{r,p})=0$ (resp. $\mu^r_2(\lambda_2^{r,p})=0$, $\mu^r_3(\lambda_3^{r,p})=0$).

Thanks to its applications in the nonlinear mathematical models, in particular, the population dynamics in biology, the existence of positive principal eigenvalues of , , and is of particular interest and has been well studied. The time independent version was first studied by Manes and Micheletti in [@MaMi]. Then, Hess and Kato in [@HeKo] and Brown and Lin in [@BrLi] obtained some results for elliptic operators in Dirichlet boundary case. The proofs of Brown and Lin and Manes and Micheletti are based on the variational characterization of the principal eigenvalue; the proof of Hess and Kato uses Krein-Rutman’s theorem. For more general elliptic operators subject to various boundary conditions, we refer to [@Boc; @Bo; @BrCoFl; @BrLi; @CaCo1; @GoLa; @He; @HeKo; @Lo-Go0; @Lo-Go; @SeHe] and references therein, and for the applications in population dynamics, such as the optimization of spatial arrangement of favorable and unfavorable regions for a species to survive in biological context, we refer to [@CaCo; @CaCo2; @Co; @CoCuPo; @HiKaLa; @KaLoYa; @LoYa], etc.

Among others, in the time independent case, it is proved that with $m_1(t, x)=m_1(x)$ admits a unique positive principal eigenvalue if and only if \[random-auto-diri-condi\] m(x\_0)&gt;0 x\_0D (see [@GoLa; @HeKo; @He]), and with $m_2(t,x)=m_2(x)$ admits a unique positive principal eigenvalue if and only if $$\label{random-auto-neum-condi}
m_2(x_0)>0 \quad\text{for some}\,\, x_0\in D\,\,\, \text{and}\,\,\int_D m_2(x)dx<0$$ (see [@He]). In the time periodic case, it is proved that has a unique positive principal eigenvalue if and only if \[R-D\] \_0\^T\_[x|D]{}m\_1(t, x)dt&gt;0 (see [@He0]), and has a unique positive principal eigenvalue if \[R-N\] \_0\^T\_[x|D]{}m\_2(t, x)dt&gt;0 \_D\_0\^Tm\_2(t, x)dtdx&lt;0 (see [@He0]). There is no result for random dispersal operator subject to the spatial periodic boundary condition so far. But as we can see in the proofs for nonlocal dispersal operator, results in spatial periodic boundary case are very similar to those in Neumann boundary case.

The study of , , and is of great interest in its own and will also have important applications in the study of many nonlinear mathematical models with nonlocal dispersal from applied science, including the following time periodic dispersal evolution equations, $$\label{n-kpp-diri}
 \begin{cases}
 \p_t u=\int_{D} \kappa(y-x)u(t,y)dy-u(t,x)+\lambda_1 uf_1(t,x,u),\quad x\in \bar D,\\
 u(t+T, x)=u(t, x),\\
 u(0, x)\geq 0, u(0, x)\not\equiv 0,
 \end{cases}$$ $$\label{n-kpp-neum}
 \begin{cases}
 \p_t u=\int_{D} k(y-x)[u(t,y)-u(t,x)]dy+\lambda_2 uf_2(t,x,u),\quad x\in \bar D,\\
 u(t+T, x)=u(t, x),\\
 u(0, x)\geq 0, u(0, x)\not\equiv 0,
 \end{cases}$$ and $$\label{n-kpp-peri}
\begin{cases}
 \p_t u=\int_{\R^N} k(y-x)u(t,y)dy-u(t,x)+\lambda_3 uf_3(t,x,u),\quad x\in \R^N,\cr
u(t+T, x+p_i{\bf e_i})=u(t, x+p_i{\bf e_i})=u(t, x),\\
 u(0, x)\geq 0, u(0, x)\not\equiv 0,
\end{cases}$$ where $u(0, x)$ is continuous and bounded, $\lambda_i >0$, and $f_i(t, x, u)$ satisfies the following condition $(i=1,2,3)$,

[**(F)**]{} [*$f_i$ is $C^1$ in $t\in{\ensuremath{\mathbb{R}}}$ and $C^3$ in $(x,u)\in {\ensuremath{\mathbb{R}}}^N\times{\ensuremath{\mathbb{R}}}$; $f_i(t, x, u)<0$ for $u\gg 1$ and $\partial _u f_i(t, x, u)<0$ for $u\geq0$*]{}; $f_i(t+T,x,u)=f_i(t,x,u)$; and when $i=3$, $f_i(t+T, x,u)=f_i(t, x+p_j{\bf e_j}, u)=f(t,x,u)$ for $j=1, 2, \cdots, N$.

In , , and , $u(t, x)$ represents the density of a species at location $x$ and time $t$, and $\lambda_i f(t, x, 0) (i=1, 2, 3)$ represents the intrinsic growth rate of a species. Hence only non-negative solutions of , , and are of interest. Notice that $u\equiv 0$ is a solution to , , and . It is of great interest to know for which $\lambda_1$ (resp. $\lambda_2$, $\lambda_3$), there is a positive solution of (resp. , ). As in the random dispersal case, the spectral theory for the eigenvalue problems , , and will play an important role in the study of positive solutions of , , and ).

However, little is known about the eigenvalue problems of nonlocal dispersal operators with (time periodic) indefinite weight functions. The objective of this paper is to investigate the principal spectrum points /eigenvalues (if exists) of the nonlocal time-periodic weighted eigenvalue problems with Dirichlet type boundary condition , Neumann type boundary condition and spatial periodic type boundary condition , respectively. Note that, unlike random dispersal operators, a nonlocal dispersal operator may not have a principal eigenvalue (see [@Cov; @ShZh0] for examples). We hence first introduce the notion of the principal spectrum points of , and , which are the natural generalization of notion of the principal eigenvalue of , , and . Moreover, due to the lack of compactness of nonlocal operators, the Krein-Rutman’s theorem is not applicable, and there is in general no variational characterization for time-periodic nonlocal dispersal operators. We overcome the difficulties by developing various new techniques, and prove the necessary and sufficient conditions for the existence of positive principal spectrum points of , , and , respectively. We also investigate the upper bounds of principal spectrum points and the sufficient conditions for the principal spectrum points to be principal eigenvalues. As an application, we study asymptotic dynamics of , , and by the spectral theory of weighted nonlocal dispersal operators.

The rest of this paper is organized as follows. In section 2, we introduce notations, definitions and state the main results of the paper. In section 3, we present some preliminary materials to be used in the proof of the main results. We prove the main results and discuss the application of the main results in section 4.

Notations, Definitions, and Main Results
========================================

In this section, we introduce notations, definitions, and state the main results.

We first introduce some standing notations and the concept of principal spectrum points of nonlocal dispersal operators with time periodic indefinite weight functions.

Let \[space-t12\] X\_1=X\_2={uC(|D, )| u(t+T, x)=u(t, x)} with norm $\|u\|_{\mathcal X_i}=\sup_{t\in \R, x\in \bar D}|u(t, x)| (i=1, 2)$, \[space-t3\] X\_3={uC(\^N, )|u(t+T, x)=u(t, x+p\_j[**e\_j**]{})=u(t, x) j=1,2,,N} with norm $\|u\|_{\mathcal X_3}=\sup_{t\in\R, x\in\R^N}|u(t, x)|$. Set $$\mathcal X_i^+=\{u\in \mathcal X_i|u\geq 0\}$$ and $$\mathcal X_i^{++}=\text{Int}{X_i^+}=\{\phi\in \mathcal X_i, \phi>0\}$$ ($i=1, 2, 3$). We define the integral operator $K_i$, the multiplication operator $B_i$, and the domain $D_i$ $(i=1,2,3)$ as follows, $$\label{Ki}
K_i: \mathcal X_i\to\mathcal X_i,\,\, K_iu(t, x)=
\begin{cases}
\int_D \kappa(y-x)u(t, y)dy\quad &\forall u\in X_i,\quad i=1,2,\\
\int_{{\ensuremath{\mathbb{R}}}^N}\kappa(y-x)u(t, y)dy\quad&\forall u\in X_3,
\end{cases}$$ \[bi\] B\_i: X\_iX\_i, B\_iu=b\_i u   b\_i(x)=

1 & i=1, 3,\
\_D(y-x)dy & i=2,

and $$\label{D-i}
D_i=\begin{cases}  D\quad &{\rm for}\,\, i=1,2,\cr
[0,p_1]\times[0,p_2]\times [0,p_N] &{\rm for}\,\, i=3.
\end{cases}$$ For $1\leq i\leq 3$, set \[Li\] (L\_iu)(t, x)=-\_tu(t, x)+(K\_iu)(t, x)-(B\_iu)(t, x) with domain $$\mathcal D (\mathcal L_i)=\{u\in \mathcal X_i| u \text{ is } C^1 \text{ in } t \text{ and } u_t\in \mathcal X_i\}.$$ Then for $i=1,2,3$, the eigenvalue problems , , and in the space $\mathcal X_i$ can be written uniformly as $$\label{EP-lambdai}
\mathcal L_iu+\lambda_i m_i u=0,$$ Observe that $K_2=K_1$, $B_3=B_1$ and $D_2=D_1$. The introduction of $K_2$, $B_3$, and $D_2$ is for convenience.

For any given real number $\lambda_i$, the weighted eigenvalue problem is closely related to the following regular eigenvalue problem on $\mathcal X_i$, \[EP-mui\] L\_i u+\_i m\_iu=\_i u. Let $\sigma_i(\mathcal{L}_i+\lambda_i m_i)$ be the spectrum set of (i.e. the spectrum set of the operator $\mathcal{L}_i+\lambda_im_i$ in $\mathcal{X}_i$) and $$\ds\mu_i^n(\mathcal{L}_i+\lambda_i m_i)=\sup\{{\rm Re}\mu|\mu\in\sigma_i(\mathcal{L}_i+\lambda_i m_i)\}.$$

\[PSP-def\]

-   $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ is called [*the principal spectrum point*]{} of the regular eigenvalue problem . The principal spectrum point $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ is called [*the principal eigenvalue*]{} of if with $\mu_i=\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ has a positive solution in $\mathcal{X}_i$.

-   A real number $\lambda_i^p(m_i)$ is called a [*principal spectrum point*]{} of the weighted eigenvalue problem if $\mu_i^n(\mathcal{L}_i+\lambda_i^p(m_i) m_i)=0$. When $\lambda_i^p(m_i)$ is a principal spectrum point of and $\mu_i^n(\mathcal{L}_i+\lambda_i^p(m_i) m_i)$ is a principal eigenvalue of , $\lambda_i^p(m_i)$ is also called a [*principal eigenvalue*]{} of .

If not confusion occurs, we may write $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ and $\lambda_i^p(m_i)$ as $\mu_i^n(\lambda_i)$ and $\lambda_i^p$, respectively.

Let $$\label{x-d-space}
X_1=X_2=\{u\in C(\bar D, \R)\}$$ with norm $\|u\|_{X_i}=\sup_{x\in\bar D}|u(x)|$ ($i=1,2$), $$\label{x-p-space}
X_3=\{u\in C({\ensuremath{\mathbb{R}}}^N,{\ensuremath{\mathbb{R}}})\,|\, u(x+p_j{\bf e_j})=u(x),\quad x\in\R^N,\, j=1,2,\cdots,N\}$$ with norm $\|u\|_{X_3}=\max_{x\in{\ensuremath{\mathbb{R}}}^N}|u(x)|$. Set $$\label{x-d-positive-cone}
X_i^+=\{u\in X_i\,|\, u\geq 0\},$$ and $$\label{x-d-positive-interior}
X_i^{++}=
\begin{cases}
\{u\in X_i^+\,|\, u(x)>0,\quad x\in \bar D\},\quad & i=1,2,\\
\{u\in X_i^+\,|\, u(x)> 0,\quad x\in{\ensuremath{\mathbb{R}}}^N\},\quad & i=3.
\end{cases}$$ In the case that $m_i(t, x)\equiv m_i(x)$, consider $$\label{autonomous-weight-pev}
K_iu-B_iu+\lambda_i m_iu=0,$$ and $$\label{autonomous-pev}
K_iu-B_iu+\lambda_i m_iu=\mu_i u,$$ in $X_i$ ($i=1,2,3$). Similarly, define [*the principal spectrum point*]{} $\mu_i^n(\lambda_i)$ of the regular eigenvalue problem to be the largest real part of the spectrum set of . We then call a real number $\lambda_i^p$ [*the principal spectrum point*]{} of the weighted eigenvalue problem if $\mu_i^n(\lambda_i^p)=0$.

Note that when $m_i(t,x)\equiv m_i(x)$, the principal spectrum point of and the principal spectrum point of are the same (see Proposition \[spectrum-autonomous\]).

Note also that , as well as its time independent version , is an eigenvalue problem in regular sense for $i=1, 2, 3$, as we can see from Definition \[PSP-def\] (1). Many properties of the principal spectrum point/principal eigenvalue have been studied extensively (see [@BaZh; @Cov; @HuShVi; @RaSh; @ShXi1; @ShZh0] etc.). And we will recall the basic properties and prove some new properties of the principal spectrum point/principal eigenvalue of in subsection 3.3.

However, little is known about the eigenvalue problem . We are the first to study the principal spectrum point/principal eigenvalue of nonlocal dispersal operators with time periodic indefinite weight function. Definition \[PSP-def\] (2) is a natural generalization of the principal eigenvalue of random dispersal operators with time periodic indefinite weight functions. And the necessary and sufficient conditions for the existence of positive principal spectrum point of are in Theorem \[PSP-diri\]-\[PSP-peri\], and the properties of the principal spectrum point of are in Theorem \[upper-bounds\].

For $i=1, 2, 3$, let \[hat-mi\] m\_i(x)=\_0\^T m\_i(t,x)dt, \[mi-max-min\] m\_[i, ]{}=\_[x|D\_i]{}m\_i(x), m\_[i, ]{}=\_[x|D\_i]{}m\_i(x), and \[P\] P(m\_i)=\_0\^T\_[x|D\_i]{}m\_i(t, x)dt.

We now state the main results of the paper. We first state the main results on the necessary and sufficient conditions for the existence and uniqueness of positive principal spectrum points of , , and .

\[PSP-diri\] Suppose $\kappa(\cdot)$ satisfies (K) and $m_1\in\mathcal X_1$. The eigenvalue problem has exactly one positive principal spectrum point, denoted by $\lambda_1^p$, if and only if \[D\] P(m\_1)&gt;0. More precisely, we show the following.

-   There is $\lambda_1^p>0$ such that $\mu_1^n(\lambda_1^p)=0$ if and only if (D) holds.

-   If $\lambda_1^{p, 1},\lambda_1^{p, 2}>0$ are such that $\mu_1^n(\lambda_1^{p, 1})=\mu_1^n(\lambda_1^{p, 2})=0$, then $\lambda_1^{p, 1}=\lambda_1^{p, 2}$.

\[PSP-diri-coro\] $\quad$

-   If $m_1(t, x)\equiv m_1(t)$, then has exactly one positive principal spectrum point $\lambda_1^p$ if and only if $$\int_0^T m_1(t)dt>0.$$ Moreover, $\lambda_1^p$ is a principal eigenvalue of and $\ds \lambda_1^p=\frac{-\bar\lambda_1}{\widehat m_1}$, where $\bar \lambda_1(<0)$ is the principal eigenvalue of $K_1-B_1(=K_1-\mathcal I)$.

-   If $m_1(t, x)\equiv m_1(x)$ , consider $$\label{autonomous-dirichlet-ori}
    \left[\int_D\kappa(y-x)dy-u(x)\right]+\lambda_1m_1(x)u(x)=0,\quad x\in\bar D.$$ There exists a unique positive principal spectrum point $\lambda_1^p$ of if and only if $m_1(x_0)> 0$ for $x_0\in  D_1$.

\[PSP-dirichlet-remark\] Theorem \[PSP-diri\] and Corollary \[PSP-diri-coro\] extend the principal eigenvalue theory for random dispersal operators with time independent or time periodic indefinite weight functions subject to Dirichlet boundary condition to nonlocal dispersal operators with time independent or time periodic indefinite weight functions subject to Dirichlet type boundary condition.

\[PSP-neum\] Suppose that $\kappa(\cdot)$ satisfies (K), $m_2(\cdot, \cdot)\in\mathcal X_2$ and $m_2(t, x)\not \equiv m_2(t)$. The eigenvalue problem has exactly one positive principal spectrum point, denoted by $\lambda_2^p$, if and only if \[N\] P (m\_2)&gt;0 \_[D\_2]{} \_0\^Tm\_2(t, x)dtdx&lt;0. More precisely, we show the following.

-   There is a $\lambda_2^p>0$ such that $\mu_2^n(\lambda_2^p)=0$ if and only if (N) holds.

-   If $\lambda_2^{p, 1}, \lambda_2^{p, 2}>0$ are such that $\mu_2^n(\lambda_2^{p, 1})=\mu_2^n(\lambda_2^{p, 2})=0$, then $\lambda_2^{p, 1}=\lambda_2^{p, 2}$.

\[PSP-neum-coro\]

-   If $m_2(t,x)\equiv m_2(t)$, then $$\mu_2^n(\lambda_2)=\lambda_2\widehat m_2.$$ It then follows that, if $\int_0^T m_2(t)dt\not =0$, then there is no positive principal spectrum point of , and if $\int_0^T m_2(t)dt=0$, then any $\lambda_2>0$ is a positive principal spectrum point of .

-   If $m_2(t, x)\equiv m_2(x)$, consider $$\label{autonomous-neumann-ori}
    \int_D\kappa(y-x)[u(y)-u(x)]dy+\lambda_2m_2(x)u(x)=0,\quad x\in\bar D.$$ There is exactly one positive principal spectrum point $\lambda_2^p$ of if and only if $$m_2(x_0)>0\quad \text{for some}\,\, x_0\in D_2\quad {\rm and}\,\,\int_{D_2} m_2(x)dx<0.$$

\[PSP-neum-remark\] Theorem \[PSP-neum\] and Corollary \[PSP-neum-coro\] extend the principal eigenvalue theory for random dispersal operators with time independent or time periodic indefinite weight functions subject to Neumann boundary condition to nonlocal dispersal operators with time independent or time periodic indefinite weight functions subject to Neumann type boundary condition.

\[PSP-peri\] Suppose that $\kappa(\cdot)$ satisfies (K), $m_3(\cdot, \cdot)\in\mathcal X_3$, and $m_3(t, x)\not \equiv m_3(t)$ for $t\in \R$ and $x\in \R^N$. The eigenvalue problem has exactly one positive principal spectrum point, denoted by $\lambda_3^p$, if and only if \[P\] P (m\_3)&gt;0 \_[D\_3]{} \_0\^Tm\_3(t, x)dtdx&lt;0. More precisely, we show the following.

-   There is a $\lambda_3^p>0$ such that $\mu_3^n(\lambda_3^p)=0$ if and only if (P) holds.

-   If $\lambda_3^{p, 1},\lambda_3^{p, 2}>0$ are such that $\mu_3^n(\lambda_3^{p, 1})=\mu_3^n(\lambda_3^{p, 2})=0$, then $\lambda_3^{p, 1}=\lambda_3^{p, 2}$.

\[PSP-peri-coro\]

-   If $m_3(t, x)=m_3(t)$, then $$\mu_3^n(\lambda_3)=\lambda_3 \widehat m_3.$$ It then follows that, if $\int_0^T m_3(t)dt\not =0$, then there is no positive principal spectrum point of , and if $\int_0^T m_3(t)dt=0$, then any $\lambda_3>0$ is a positive principal spectrum point of .

-   If $m_3(t, x)\equiv m_3(x)$, consider $$\label{autonomous-periodic-ori}
    \int_{\R^N}\kappa(y-x)[u(y)-u(x)]dy+\lambda_3m_3(x)u(x)=0.$$ There is exactly one positive principal spectrum point $\lambda_3^p$ of if and only if $$m_3(x_0)>0\quad \text{for some}\,\, x_0\in D_3,\quad {\rm and}\,\,\int_{D_3}  m_3(x)dx<0.$$

Next, we state the main result on the upper bounds of principal spectrum points of , , and , and sufficient conditions for the principal spectrum points of , , and to be principal eigenvalues.

Consider the eigenvalue problem with indefinite weight function $\hat m_i$, $$\label{average-weight-pev}
K_iu-B_iu+\lambda_i \widehat m_i u=0,$$ and the regular eigenvalue problem $$\label{average-pev}
K_iu-B_iu+\lambda_i\widehat m_i u=\mu_i(\lambda_i) u$$ in $X_i$.

\[upper-bounds\] $\quad$

-   (Upper bounds) If has a unique positive principal spectrum point $\lambda_i^p(\widehat m_i)$, then has also a unique positive principal spectrum point $\lambda_i^p(m_i)$ and $$\lambda_i^p(m_i)\le \lambda_i^p(\widehat m_i).$$

-   (Sufficient conditions of the existence of positive principal eigenvalues)

-   Assume that $N=1$ or $2$, $b_i+\widehat m_i$ is $C^N$, and has a unique positive principal spectrum point $\lambda_i^p$, then $\lambda_i^p$ is a positive principal eigenvalue of .

-   For $N\geq 1$, and $i=1$ or $3$, assume that $\ds\int_{D_i}\frac{1}{\widehat m_{i,\max} -\widehat m_i(x)}dx=\infty$ and has a unique positive principal spectrum point $\lambda_i^p$, then $\lambda_i^p$ is a positive principal eigenvalue of .

\[average-pev-remark\] Assume that $\widehat m_i(x_0)>0$ for some $x_0\in D_i$, and in addition, $\int_{D_i}\widehat m_i(x)dx<0$ in the case $i=2,3$. Then has a unique positive principal spectrum point $\lambda_i^p(\widehat m_i)$.

Finally, we consider the applications of the principal spectrum point theory of nonlocal dispersal operators with time periodic indefinite weight function to the KPP type equations , , and . And we prove the following result.

\[KPP\] Assume (K) and (F) hold. Denote the principal spectrum point of the eigenvalue problem with $m_i(t, x)= f_i(t, x, 0)$ by $\mu_i^n(\lambda_i)$. Assume in addition that $f_1(t, x, 0)$, $ f_2(t, x, 0)$, and $ f_3(t, x, 0)$ satisfies (D), (N), (P), respectively. Then ( or ) admits a unique positive time-periodic solution $u^*(t, x)$ if and only if $$\lambda_i>\lambda_i^p,$$ where $\lambda_i^p$ is the positive number such that $\mu_i^n(\lambda_i^p)=0$.

Preliminary
===========

In this section, we present some notations and preliminary materials to be used in the proofs of the main results in Section 4. We first present a technical lemma in subsection 3.1. Then we present a comparison principle for solutions of some linear nonlocal evolution equations in subsection 3.2. Finally we present some basic properties of principal spectrum points of the regular eigenvalue problem in subsection 3.3. Throughout this section, for $1\leq i\leq 3$, let \[hi\] h\_i(t, x; \_i)=-b\_i(x)+\_i m\_i(t, x), and \[hathi\] h\_i(x;\_i)=-b\_i(x)+\_im\_i(x), where $\widehat m_i$ is as in . For $D_i$ defined in , set \[hathi-min-max\] h\_[i, ]{}(\_i)=\_[x|D\_i]{} h\_i(x;\_i), h\_[i, ]{}(\_i)=\_[x|D\_i]{}h\_i(x;\_i). If there is no confusion, we may omit $\lambda_i$, and abbreviate the notations in , , and as $h_i(t, x)$, $\widehat h_i(x)$ and $\widehat h_{i, \max}$, respectively.

A technical lemma
-----------------

In this subsection, we provide a useful technical lemma.

\[technical-lm\] Let $1\le i\le 3$. For any fixed $0<\lambda_i\in \R$, any $m_i\in \mathcal X_i$ and any $\epsilon>0$, there is $m_{i,\epsilon}\in\mathcal X_i$ satisfying that $$\|m_i-m_{i,\epsilon}\|_{\mathcal X_i}<\epsilon,$$ $b_i(x)+\lambda_i\widehat m_{i, \epsilon}(x)$ is $C^N$, $b_i(x)+\lambda_i\widehat m_{i, \epsilon}(x)$ attains its maximum at some $x_0\in {\rm Int}(D_i)$, and the derivatives of $b_i+\lambda_i\widehat m_{i, \epsilon}(x)$ up to order $N-1$ at $x_0$ are zero.

It follows from [@RaSh Lemma 4.1]. For the self-completeness, we provide a proof in the following.

We prove the case $i=1$ or $2$. The case $i=3$ can be proved similarly. Without loss of generality, we assume $\lambda_i=1$, and recall that $\widehat h_i(x)=-b_i(x)+\widehat m_i(x)$ (see ).

First, let $\tilde x_0\in \bar D_i$ be such that $$\widehat h_i(\tilde x_0)=\max_{x\in\bar D_i}  \widehat h_i(x).$$ For any $\epsilon>0$, there is $\tilde x_\epsilon \in {\rm Int}(D_i)$ such that $$\label{eq1}
\widehat h_i(\tilde x_0)-\widehat h_i(\tilde x_\epsilon)<\frac{\epsilon}{3}.$$ Let $\tilde\sigma>0$ be such that $$B(\tilde x_\epsilon,\tilde\sigma)\Subset D_i,$$ where $B(\tilde x_\epsilon,\tilde\sigma)$ denotes the open ball with center $\tilde x_\epsilon$ and radius $\tilde\sigma$.

Note that there is $\xi_i(\cdot)\in C(\bar D_i)$ such that $0\leq \xi_i(x)\leq 1$, $ \xi_i(\tilde x_\epsilon)=1$, and ${\rm supp}( \xi_i)\subset B(\tilde x_\epsilon,\tilde\sigma)$. Let $$\overline m_{i, \epsilon}(t, x)=m_i(t, x)+\frac{\epsilon}{3} \xi_i(x),$$ and $$\label{eq0}
 \overline h_{i,\epsilon}(x)=-b_i(x)+\widehat m_{i}(x)+\frac{\epsilon}{3} \xi_i(x).$$ Then $\overline m_{i, \epsilon}(t, \cdot)$ and $\overline h_{i,\epsilon}(\cdot)$ is continuous on $\bar D_i$, \[overline-ai\] m\_[i, ]{}(t, )-m\_i(t, ) and $\overline h_{i,\epsilon}(\cdot)$ attains its maximum in ${\rm Int}(D_i)$.

Let $\widetilde D_i\subset {\ensuremath{\mathbb{R}}}^N$ be such that $D_i\Subset \widetilde D_i$. Note that $\overline h_{i,\epsilon}(\cdot)$ can be continuously extended to $\widetilde D_i$. Without loss of generality, we may then assume that $\overline h_{i,\epsilon}(\cdot)$ is a continuous function on $\tilde D_i$ and there is $x_0\in {\rm Int}(D_i)$ such that $\overline h_{i,\epsilon}(x_0)=\sup_{x\in\widetilde D_i}\overline h_{i,\epsilon}(x)$. Observe that there is $\sigma>0$ and $\widetilde h_{i,\epsilon}(\cdot)\in C(\widetilde D_i)$ such that $B(x_0,\sigma)\Subset D_i$, $$\label{eq3}
0\leq\widetilde h_{i,\epsilon}(x)-\overline h_{i,\epsilon}(x)\le\frac{ \epsilon}{3}\quad \forall \,  x\in\widetilde D_i,$$ and $$\widetilde h_{i,\epsilon}(x)=\widehat h_{i,\epsilon}(x_0)\quad \forall \, x\in B(x_0,\sigma).$$

Let $$\eta(x)=\begin{cases} C\exp(\frac{1}{\|x\|^2-1})\quad &{\rm if}\,\ \|x\|<1,\cr\cr
0\quad &{\rm if}\,\ \|x\|\geq 1,
\end{cases}$$ where $C>0$ is such that $\int_{{\ensuremath{\mathbb{R}}}^N}\eta(x)dx=1$. For given $\delta>0$, set $$\eta_\delta(x)=\frac{1}{\delta^N}\eta\left(\frac{x}{\delta}\right).$$ Let $$h_{i,\epsilon,\delta}(x)=\int_{\widetilde D_i}\eta_\delta(y-x)\widetilde h_{i,\epsilon}(y)dy.$$ By [@Ev Theorem 6, Appendix C], $ h_{i,\epsilon,\delta}(\cdot)$ is in $C^\infty(\widetilde D_i)$ and when $0<\delta\ll 1$, $$\label{eq4}
|h_{i,\epsilon,\delta}(x)-\widetilde h_{i,\epsilon}(x)|<\frac{\epsilon}{3}\quad \forall \, x\in\bar D_i.$$ It is not difficulty to see that for $0<\delta\ll 1$, $$h_{i,\epsilon,\delta}(x)\leq
\widetilde h_{i,\epsilon}(x_0)\quad \forall x\in B(x_0,\sigma),$$ and $$h_{i,\epsilon,\delta}(x)=\widetilde h_{i,\epsilon}(x_0)\quad
\forall x\in B(x_0,\sigma/2).$$ Fix $0<\delta\ll 1$. Let $$\widehat h_{i, \epsilon}(x)=h_{i,\epsilon,\delta}(x).$$ Then $\widehat h_{i, \epsilon}(\cdot)$ attains its maximum at some $x_0\in{\rm Int}(D_i)$, and the partial derivatives of $\widehat h_{i,\epsilon}(\cdot)$ up to order $N-1$ at $x_0$ are zero. Let $$m_{i, \epsilon}(t, x)= \overline m_{i, \epsilon}(t, x)   +\widehat h_{i, \epsilon}(x)-\overline h_{i,\epsilon}(x).$$ Then $m_{i, \epsilon}\in{\mathcal X}_2$, $$\|m_i-m_{2,\epsilon}\|\leq \|m_i-\overline m_{i, \epsilon}\|+\|\widehat h_{i, \epsilon}-\overline h_{i, \epsilon}\|\leq\|m_i-\overline m_{i, \epsilon}\|+\|\widehat h_{i, \epsilon}-\widetilde h_{i, \epsilon}\|+\|\widetilde h_{i, \epsilon}-\overline h_{i, \epsilon}\|\leq \epsilon,$$ and $$-b_i(x)+\widehat m_{i,\epsilon}(x)=\widehat h_{i, \epsilon}(x).$$ Therefore, $-b_i+\widehat m_{i, \epsilon}$ is $C^N$, attains its maximum at some point $x_0\in {\rm Int}D_i$, and the partial derivatives of $-b_i+\widehat m_{i, \epsilon}$ up to order $N-1$ at $x_0$ are zero. The lemma is thus proved.

Comparison Principle
--------------------

In this subsection, we present the comparison principle for the solutions to the following evolution equations associated to the eigenvalue problem with $\lambda_i\ge 0$, $$\label{dirichlet-ori-evol}
 \p_t u(t, x)=\int_D \kappa(y-x)u(t, y)dy-u(t, x)+ \lambda_1 m_1(t,x)u(t, x) \quad \text{ in } \bar D,$$ $$\label{neumann-ori-evol}
 \p_t u(t, x)=\int_D \kappa(y-x)[u(t, y)-u(t, x)]dy+ \lambda_2 m_2(t,x)u(t, x) \quad \text{ in } \bar D,$$ and $$\label{periodic-ori-evol}
 \p_t u(t, x)=\int_{\R^N} \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_3 m_3(t, x)u(t, x) \quad \text{ in } \R^N,$$ where $m_i(t, x)\in \mathcal X_i(i=1, 2, 3)$.

By general semigroup theory, (resp. , ) generates evolution families $\{\Phi_1(t, s; m_1)\}$ (resp. $\{\Phi_2(t, s; m_2), \Phi_3(t, s; m_3)\}$) on $X_1$ (resp. $X_2$, $X_3$), that is, for any $u_0\in X_1$ (resp. $u_0\in X_2, u_0\in X_3$), $u(t, x; s, u_0, m_1):=(\Phi_1(t, s; m_1)u_0)(x)$ (resp. $u(t, x; s, u_0, m_2):=(\Phi_2(t, s; m_2)u_0)(x)$, $u(t, x; s, u_0, m_3):=(\Phi_3(t, s; m_3) u_0)(x)$) is the unique solution of (resp. , ) with $u(s, x; s, u_0, m_1)=u_0(x)$ (resp. $u(s, x; s, u_0, m_2)=u_0(x)$, $u(s, x; s, u_0, m_3)=u_0(x)$).

\[super-sub-sol\] A bounded measurable function $u(t, x)$ on $[0, T)\times \bar D$ is called a [*super-solution*]{} (or [*sub-solution*]{}) of if for any $x\in \bar D$, $u(t, x)$ is differentiable for all but finite many $t$’s in $[0, T)$ and satisfies that for each $x\in \bar D$, $$\p_t u\geq (\text{ or } \leq) \int_D \kappa(y-x)u(t, y)dy-u(t, x)+ m_1(t,x)u(t, x)$$ for all but finite many $t$’s in $[0, T)$.

[*Super-solutions and sub-solutions*]{} of and $\eqref{periodic-ori-evol}$ are defined in an analogous way.

\[comparison\] $\quad$

-   If $u^1(t, x)$ and $u^2(t, x)$ are bounded sub- and super- solution of (resp. , ) on $[s, T)$, respectively, and $u^1(s, \cdot)\leq u^2(s, \cdot)$, then $u^1(t, \cdot)\leq u^2(t, \cdot)$ for $t\in [s, T)$.

-   If $u^1, u^2\in X_i$, $u^1\leq u^2$ and $u^1\neq u^2$, then $$\Phi_i(t, s; m_i) u^1\ll \Phi_i(t, s; m_i) u^2 \text{ for all } t>s.$$

-   If $u_0\in X_i^+$, and $m_i^1, m_i^2\in \mathcal X_i$, if $m_i^1\leq m_i^2$, then $$\Phi_i(t, s; m_i^1)u_0\leq \Phi_i(t, s; m_i^2)u_0 \text{ for all } t>s.$$

\(1) follows from the arguments in [@ShZh0 Proposition 3.1 (1)].

\(2) follows from the arguments in [@ShZh0 Proposition 3.1 (2)].

\(3) We consider the case $i=1$. Other cases ca be proved similarly.

Note that $u_1(t, x; s, u_0, m_1^2)$ is a supersolution of with $m_1$ being replaced by $m_1^1$. Then by (1), $$u_1(t, \cdot; s, u_0, m_1^1)\leq  u_1(t, \cdot; s, u_0, m_1^2)  \text{ for all } t>s.$$

For simplicity in notation, put $\Phi_i(T; m_i)=\Phi_i(T, 0; m_i) (i=1, 2, 3)$, and let $r(\Phi_i(T;m_i))$ be the spectral radius of $\Phi_i(T, 0; m_i)$.

Basic properties of principal spectrum points/principal eigenvalues
-------------------------------------------------------------------

Our objective in this subsection is to study some basic properties of the principal spectrum point of the regular eigenvalue problem .

Recall that in , $\mathcal (\mathcal L_i+\lambda_i m_i)u=-\p_t u-B_i u+K_iu+\lambda_i m_i u$, and we may use $\mu_i^n(\mathcal L_i+\lambda_i m_i)$ or $\mu_i^n(\lambda_i)$ to denote the principal spectrum point of , if no confusion occurs. And when $\lambda_i=0$, we use $\mu_i^n(0)$ to denote the principal $\mathcal L_i$.

\[spectrum-sol-operator\] For $1\leq i\leq 3$, $\mu_i^n (\mathcal L_i+\lambda_i m_i)=\frac{\ln r(\Phi_i(T; m_i))}{T}$.

It follows from [@RaSh Proposition 3.10].

\[spectrum-limit\] Let $1\le i\le 3$. Given any $u_0\in X_i^+\setminus\{0\}$, $$\mu_i^n(\mathcal{L}_i+\lambda_i m_i)=\lim_{t\to\infty}\frac{\ln\|\Phi_i(t,0;m_i)u_0\|}{t}.$$

It follows from the arguments in [@HuShVi Proposition 2.5 and Theorem 3.2].

\[spectrum-autonomous\] Let $1\le i\le 3$.

-   If $m_i(t,x)\equiv m_i(x)$, then the principal spectrum point of equals to the principal spectrum point of .

-   If $m_i(t,x)\equiv m_i(t)$, then $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)=\mu_i^n(0)+\lambda_i  \widehat m_i$.

\(1) It follows from [@ShXi1 Proposition 3.3] and Proposition \[spectrum-limit\].

\(2) It follows from Proposition \[spectrum-limit\] and the fact that $$\Phi_i(t,0;m_i)=e^{\lambda_i \int_0^t m_i(s)ds}\Phi_i(t,0;0).$$

Recall that $h_i(t, x)$ is defined in . Set \[hi(x)\] H\_i: (\_i)X\_i, (H\_i u)(t, x)=-\_t u(t, x)+h\_i(t, x)u(t, x) with $\mathcal D(\mathcal H_i)=\mathcal D(\mathcal L_i)$. Note that $\mathcal H_i+K_i=\mathcal L_i+\lambda_i m_i$. Hence, we may use $\mu_i^n(\mathcal H_i+K_i)$ or equivalently $\mu_i^n(\mathcal L_i+\lambda_i m_i)$ in different situations to denote the principal spectrum point of the eigenvalue problem . We use $\mathcal Re\{\sigma(\mathcal H_i)\}$ to denote the real part of the spectrum set of $\mathcal{H}_i$.

\[spectrum-Hi\] Let $1\le i\le 3$.

-   $\mathcal Re\{\sigma(\mathcal H_i)\}=[\widehat h_{i,\min},\widehat h_{i,\max}]$.

-   $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)\ge \mu_i^n(\mathcal{L}_i+\lambda_i \widehat m_i)\ge \widehat h_{i,\max}$.

-   $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ is the principal eigenvalue if and only if $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)>
    \widehat h_{i,\max}$.

\(1) It follows from Lemma 3.7 in [@HuShVi].

\(2) $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)\ge \mu_i^n(\mathcal{L}_i+\lambda_i \widehat m_i)$ follows from [@RaSh Theorem C], and $\mu_i^n(\mathcal{L}_i+\lambda_i \widehat m_i)\ge \widehat h_{i,\max}$ follows from Proposition \[spectrum-autonomous\] (1) and [@ShXi0 Proposition 3.9].

\(3) It follows from [@RaSh Theorem A].

For fixed $\lambda_i$, let (S1), (S2), and (S3) be the following standing assumptions.

[**(S1)**]{} [*For $1\leq i\leq 3$, $\widehat h_i(\cdot;\lambda_i)$ is $C^N$, there is some $x_0\in {\rm{Int}} D_i$ in the case $i=1, 2$ and $x_0\in D_3$ in the case of $i=3$ satisfying that $\widehat h_i(x_0;\lambda_i)=\widehat h_{i, \max}$, and the partial derivatives of $\widehat h_i(x;\lambda_i)$ up to order $N-1$ at $x_0$ are zero.* ]{}

[**(S2)**]{} [ *$| \lambda_i|( \widehat m_{i,\max}-\widehat m_{i, \min})<\inf_{x\in \bar D_i}\int_{D_i}\kappa(y-x)dy$ in the case of $i= 2$, and $|\lambda_i|(\widehat m_{i,\max}-\widehat m_{i, \min})<1$ in the case of $i=1, 3$.* ]{}

[**(S3)**]{} [*$\ds\int_{D_i}\frac{1}{\widehat h_{i,\max}-\widehat h_i(x;\lambda_i)}dx=\infty$ for $i=1, 2, 3$.*]{}

Note that, if $|\lambda_i|\ll 1$, then the condition (S2) is automatically satisfied for $i=1, 2, 3$. Note also that (S1) implies (S3), and when $i=1$ or $3$, (S3) holds if and only if $\ds\int_{D_i}\frac{1}{\widehat m_{i,\max}-\widehat m_i(x)}dx=\infty$.

\[pv-existence\]

-   For given $\lambda_i\in{\ensuremath{\mathbb{R}}}$, if (S1) or (S2) or (S3) is satisfied, then $\mu_i^n(\lambda_i)$ is the principal eigenvalue of $\mathcal{L}_i+\lambda_i m_i$.

-   For given $\lambda_i\in{\ensuremath{\mathbb{R}}}$, if $\mu_i^n(\lambda_i)$ is not the principal eigenvalue of $\mathcal{L}_i+\lambda_i m_i$, then $$\mu_i^n(\lambda_i)=\widehat h_{i,\max}.$$

\(1) The result that (S1) and (S2) are sufficient conditions for $\mu_i^n(\lambda_i)$ to be the principal eigenvalue follows from the argument in Theorem B in [@RaSh]. (S3) is a sufficient condition for $\mu_i^n(\lambda_i)$ to be the principal eigenvalue follows from [@Cov Theorem 1.1] and Proposition \[spectrum-Hi\].

\(2) It follows from Proposition \[spectrum-Hi\].

The next proposition is essential to prove our main theorems. Each property in the proposition is also of independent interest.

\[property-PSP-prop\] Let $1\le i\le 3$.

-   Assume $m_i, m_{i, k} \in \mathcal X_i$ with $\lambda_i^km_{i,k}\to \lambda_im_i$ as $k\to\infty$ in $\mathcal{X}_i$. Then $\mu_i^n( \mathcal L_i+\lambda_i^k m_{i,k})\to\mu_i^n(\mathcal L_i+\lambda_i m_i)$ as $k\to\infty$.

-   We have $\mu_i^n(0)<0$ for $i=1$, and $\mu_i^n(0)=0$ for $i=2, 3$.

-   If $\lambda_i^1m_{i}^1(t, x)\leq \lambda_i^2m_{i}^2(t, x)$, then $\mu_i^n(\mathcal L_i+\lambda_i^1m_{i}^1)\leq \mu_i^n(\mathcal L_i+\lambda_i^2m_{i}^2)$. If, in addition, $\lambda_i^1 m_i^1(t,x)\not \equiv \lambda_i^2 m_i^2(t,x)$, then $\mu_i^n(\mathcal L_i+\lambda_i^1m_{i}^1)< \mu_i^n(\mathcal L_i+\lambda_i^2m_{i}^2)$ .

-   If $\mathcal P(m_i)=\int_0^T\max_{x\in \bar D_i}m_i(t, x)dt>0$, then $\mu_i^n(\mathcal L_i+\lambda_i m_i)>0$ for $\lambda\gg1$.

-   The mapping $\lambda_i\in \R^+\mapsto \mu_i^n(\mathcal L_i+\lambda_i m_i)\in \R$ is convex.

\(1) follows from [@RaSh Proposition 3.11].

\(2) In the case of $\lambda_i=0$, $\mu_i^n(0)$ is the principal eigenvalue of $\mathcal L_i$, since the condition (S2) holds. And by Proposition \[spectrum-autonomous\] (1), $\mu_i^n(0)$ is the principal eigenvalue of $K_i-B_i$, associated with an eigenfunction $\phi_i\in  X_i^{++}$, such that \[lambda1=0\] \_D (y-x)\_1( y)dy-\_1( x)=\_1\^n(0)\_1( x) in the Dirichlet boundary condition case, \[lambda2=0\] \_D (y-x)\_2( y)dy-\_Dk(y-x)dy\_2(x)=\_2\^n(0)\_2( x) in the Neumann boundary condition case, and \[lambda3=0\] \_[\^N]{} (y-x)\_3( y)dy-\_3(x)=\_3\^n(0)\_3(x) in the periodic boundary condition case. In the Neumann and periodic boundary condition cases, we have that $(\mu_2^n(0), \phi_2)=(0, 1)$ and $(\mu_3^n(0), \phi_3)=(0, 1)$ are eigenpairs, respectively. Hence $\mu_i^n(0)=0$ for $i=2,3$. In the Dirichlet boundary case, we have $$\begin{aligned}
\mu_1^n(0)\int_D\phi_1^2(x)dx&=\int_D\int_D\kappa(y-x)\phi_1(y)\phi_1(x)dydx \phi_1-\int_D \phi_1^2(x)dx\\
&\le \int_D\int_D \kappa(y-x)\phi_1(y)\phi_1(x)dydx-\int_D\int_D \kappa(y-x)\phi_1^2(x)dydx\\
&=\int_D\int_D \kappa(y-x)\Big[\phi_1(y)\phi_1(x)-\frac{\phi_1^2(x)+\phi_1^2(y)}{2}\Big]dydx\\
&=-\frac12\int_D\int_D \kappa(y-x)(\phi_1(x)-\phi_1(y))^2dydx.\end{aligned}$$ This implies that $\mu_1^n(0)<0$. Hence the proof is complete.

\(3) Suppose that $m_{i}^1, m_{i}^2\in \mathcal X_i$, and $\lambda_2^1m_{i}^1\leq\lambda_i^2 m_{i}^2$. By Proposition \[comparison\] (3), for any $u_0\in X_i^+$ and $t\geq s$, $$\Phi_i(t, s; \lambda_i^1m_{i}^1)u_0\leq \Phi_i(t, s; \lambda_i^2m_{i, }^2).$$ This implies that $$r(\Phi_i(t, s; \lambda_i^1m_{i}^1))\leq r(\Phi_i(t, s; \lambda_i^2m_{i}^2)).$$ By Proposition \[spectrum-sol-operator\], we have $$\mu_i^n(\mathcal L_i+\lambda_i^1m_{i}^1)\leq \mu_i^n(\mathcal L_i+\lambda_i^2m_{i}^2).$$

\(4) Assume that $\int_0^T \widetilde m_i(t)dt=\int_0^T \max_{x\in D_i} m_i(t, x)dt>0$, and we need to show that $\mu_i^n(\lambda_i)>0$ when $\lambda_i\gg1$. We will prove the case of $i=1$, since other cases can be proved similarly.

By the continuity of $m_1(t,x)$, there are $\delta>0$, $x_0,x_1,\cdots,x_{n-1}\in D$, $0=t_0<t_1<t_2<\cdots<t_n=T$, $r_0,r_1,\cdots,r_{n-1}\in \R^+$, and $m_0,m_1,\cdots,m_{n-1}\in{\ensuremath{\mathbb{R}}}$ such that $\cup_{i=0}^{i=n-1}B(x_i, r_i)\subset D$, and for $i=0,1,\cdots,n-1$ $$m_i+\delta\le m_1(t,x)\le \widetilde m_1(t) \quad {\rm for}\quad t_{i}\le t\le t_{i+1},\,\, x\in B(x_i,r_i),$$ and $$m_0(t_1-t_0)+m_1(t_2-t_1)+\cdots m_{n-1}(t_n-t_{n-1})>0.$$ Let $\bar u(t,x)$ be the function defined as follows: for $ t_0\le t<t_1$, $$\bar u(t,x)=\begin{cases} e^{\lambda m_0 t}\quad &{\rm for}\,\, x\in B(x_0,r_0),\cr
0\quad &{\rm for}\,\, x\in\bar D\setminus B(x_0,r_0),
\end{cases}$$ for $ t_i\le t<t_{i+1} (i=1,2,\cdots,n-1)$, $$\bar u(t,x)=\begin{cases} e^{\lambda [m_0(t_1-t_0)+m_1(t_2-t_1)+\cdots m_{i-1}(t_i-t_{i-1})]+\lambda m_it}\quad &{\rm for }\,\, x\in B(x_i,r_i),\cr
0\quad &{\rm for } \,\ x\in \bar D\backslash B(x_i,r_i),
\end{cases}$$ and for $0\le t<T$ and $k=1,2,\cdots$, $$\bar u(kT+t,x)=\bar u(kT,x)\bar u(t,x),\quad x\in\bar D.$$ We then have that for any $x\in\Bar D$, $\bar u(t,x)$ is differentiable in $t$ at all $t$ but $t_i+kT$ for $i=0,1,2,\cdots,n-1$ and $k=0,1,2,\cdots$. Moreover, for $\lambda\gg 1$, any $t_i+kT< t<t_{i+1}+kT$, and any $x\in\bar\Omega$, we have $$\begin{aligned}
&\bar u_t(t,x)-\left[\int_D \kappa(y-x)\bar u(t,y)dy-\bar u(t,x)+\lambda_1 m_1(t,x)\bar u(t,x)\right]\\
&\le \bar u(t,x)\lambda_1 m_i-[-\bar u(t,x)+ \lambda_1 m_1(t,x) \bar u(t,x)]\\
&=\bar u(t,x)\lambda_1[m_i-m_1(t,x)]+\bar u(t,x)\\
&=\bar u(t,x)[\lambda(m_i-m(t,x))+1]\\
&\le 0\end{aligned}$$ for $\lambda\gg 1$, where $i=0,1,\cdots,n-1$, $k=0,1,\cdots$. Then by Proposition \[comparison\], for $\lambda\gg 1$, $t\ge 0$ and $x\in\bar D$, we have $$u(t,x;u_0)\ge \bar u(t,x)$$ for any $u_0\in X_1^+$ with $u_0(x)\ge \bar u(0,x)$. For $\lambda\gg 1$, it then follows from Proposition \[spectrum-limit\] that $$\begin{aligned}
\mu_1^n(\lambda_1)&\ge \lim_{t\to\infty}\frac{\ln\|\bar u(t,\cdot)\|_\infty}{t}\\
&=\frac{m_0(t_1-t_0)+m_1(t_2-t_1)+\cdots+m_{n-1}(t_n-t_{n-1})}{T}\\
&>0.\end{aligned}$$ The proof is thus complete.

\(5) It suffices to show that for any $0\le\lambda_i^1<\lambda_i^2$, \[convexity\] \_i\^n().

Fix $0\le\lambda_i^1< \lambda_i^2$. By Lemma \[technical-lm\] and Proposition \[pv-existence\], there are $m_i^{1,k},m_i^{2,k}\in \mathcal{X}_i$ for $k=1,2,\cdots$ such that $$m_i(t,x)\le m_i^{j,k}(t,x)\quad {\rm for}\,\, j=1,2,\,\, k=1,2,3,\cdots,$$ $$m_i^{j,k}\to m_i\quad {\rm as}\,\, k\to\infty$$ in $\mathcal{X}_i$ ($j=1,2$), and $\mu_i^n(\mathcal{L}_i+\lambda_i^j m_i^{j,k})$ is the principal eigenvalue of $\mathcal{L}_i+\lambda_i^jm_i^{j,k}$ for $j=1,2$ and $k=1,2,\cdots$. It then suffices to prove $$\label{aux-convexity}
\mu_i^n
\left(\frac{\lambda_i^1+\lambda_i^2}{2}\right)\le\frac{\mu_i^n(\mathcal{L}_i+\lambda_i^1m_i^{1,k})+\mu_i^n(\mathcal{L}_i+
\lambda_i^2m_i^{2,k})}{2}.$$

Fix $k\ge 1$. Let $\mu_i^{n,j,k}=\mu_i^n(
\mathcal{L}_i+\lambda_i^j m_i^{j,k})$ ($j=1,2$). Suppose that $\phi_i^{j,k}$ are positive eigenfunctions of $\mathcal{L}_i+\lambda_i^j m_i^{j,k}$ corresponding to $\mu_i^{n,j,k}$. Let $\phi_i^k=\sqrt {\phi_i^{1,k}\phi_i^{2,k}}$. Note that $\ds u_i^{1,k}(t,x)=e^{\mu_i^{n,j,k} t}\phi_i^{1,k}(t,x)$ is a solution of $$u_t=\int_{D} \kappa(y-x)u(t,y)dy-b_i(x) u(t,x)+\lambda_i^1 m_i^{1,k}(t, x) u(t,x),$$ and $\ds u_i^{2,k}(t,x)=e^{\mu_i^{n,j,k}t}\phi_i^{2,k}(t,x)$ is a solution of $$u_t=\int_{D} \kappa(y-x)u(t,y)dy-b_i(x)u(t,x)+\lambda_i^2 m_i^{2,k}(t,x)u(t,x),$$ where $D={\ensuremath{\mathbb{R}}}^N$ in the periodic boundary condition case. Let $$u_i^{1,2,k}(t,x)=e^{\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}t}\sqrt{\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)}.$$ Then, we have $$\begin{aligned}
&\p_t  u_i^{1,2,k}(t,x)\\
=&\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2} e^{\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}t}\sqrt{\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)}\\
& +e^{\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}t}\frac{\p_t \phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)+\phi_i^{1,k}(t,x)\p_t\phi_i^{2,k}(t,x)}{2\sqrt {\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)}}\\
=&e^{\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}t}\frac{\int_D \kappa(y-x)[ \phi_i^{1,k}(t,y)\phi_i^{2,k}(t,x)+\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,y)]dy}{
2\sqrt{\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)}}\\
& +e^{\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}t}\frac{
-2b_i\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)+(\lambda_i^1 m_i^{1,k} +\lambda_i^{2}m_i^{2,k})\phi_i^{1,k}(t,x)
\phi_i^{2,k}(t,x)}{2\sqrt{\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)}}\\
\ge &e^{\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}t}\frac{\int_D \kappa(y-x)\left(2\sqrt{ \phi_i^{1,k}(t,y)\phi_i^{2,k}(t,x)}\sqrt{\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,y)}\right)
dy}{
2\sqrt{\phi_i^{1,k}(t,x)\phi_i^{2,k}(t,x)}}\\
& -b_i(x)u_i^{1,2,k}(t,x)+\frac{\lambda_i^1+\lambda_i^2}{2} m_i(t,x)u_i^{1,2, k}(t,x)\\
\ge& \int_D \kappa(y-x)u_i^{1,2, k}(t,y)dy-b_i(x)u_i^{1,2, k}(t,x)+\frac{\lambda_i^1+\lambda_i^2}{2}m_i(t,x)u_i^{1,2,k}(t,x).\end{aligned}$$ Therefore, $u_i^{1,2,k}$ is a positive super-solution of $$u_t=\int_D \kappa (y-x)u(t,y)dy-b_i(x)u(t,x)+\frac{\lambda_i^1+\lambda_i^2}{2}m_i(t,x)u(t,x).$$ This together with Propositions \[comparison\] and \[spectrum-limit\] implies that $$\begin{aligned}
\mu_i^n\left(\frac{\lambda_i^1+\lambda_i^2}{2}\right)\le \lim_{t\to\infty} \frac{\ln u_i^{1,2,k}(t,x)}{t}=\frac{\mu_i^{n,1,k}+\mu_i^{n,2,k}}{2}.\end{aligned}$$ This proves . Letting $k\to \infty$. we get .

\[strict-convex\] Assume that $m_i(t,x)\not\equiv m_i(t)$ and $0\le \lambda_1^1<\lambda_i^2$. If $\mu_i^n(\mathcal L_i+\lambda_i^{j} m_i)$ ($j=1,2$) are the principal eigenvalue of $\mathcal L_i+\lambda_i^{j} m_i$, then $$\mu_i^n\left(\frac{\lambda_i^1+\lambda_i^2}{2}\right)<\frac{\mu_i^n(\lambda_i^1)+\mu_i^n(\lambda_i^2)}{2}.$$

We denote the principal eigenvalues of $\mathcal{L}_i+\lambda_i^1 m_i$ and $\mathcal{L}_i+\lambda_i^2m_i$ by $\mu_i^n(\lambda_i^1)$ and $\mu_i^n(\lambda_i^2)$, respectively. Let $\phi_i^j$ be a positive eigenfunction of $\mathcal{L}_i+\lambda_i^j m_i$ ($j=1,2$). By the assumption that $m_i(t,x)\not \equiv m_i(t)$, we have $\frac{\phi_i^1(t,x)}{\phi_i^2(t,x)}\not\equiv$constant. In fact, if $\phi_i^1(t,x)=c\phi_i^2(t,x)$ for some $c>0$, then we have $$\mathcal{L}_i \phi_i^1(t,x)+\lambda_i^1 m_i(t,x)\phi_i^1(t,x)=\mu_i^n(\lambda_i^1)\phi_i^1(t,x)$$ and $$\mathcal{L}_i\phi_i^1(t,x)+\lambda_i^2 m_i(t,x)\phi_i^1(t,x)=\mu_i^n(\lambda_i^2)\phi_i^1(t,x).$$ It then follows that $$\lambda_i^1 m_i(t,x)\phi_i^1(t,x)-\mu_i^n(\lambda_i^1)\phi_i^1(t,x)=
\lambda_i^2 m_i(t,x)\phi_i^1(t,x)-\mu_i^n(\lambda_i^2)\phi_i^1(t,x).$$ This implies that $$m_i(t,x)=\frac{\mu_i^n(\lambda_i^1)-\mu_i^n(\lambda_i^2)}{\lambda_i^1-\lambda_i^2}\equiv {\rm constant}$$ This is a contradiction. Hence $\frac{\phi_i^1(t,x)}{\phi_i^2(t,x)}\not\equiv$constant.

We then have $$\frac{\phi_i^1(t,y)\phi_i^2(t,x)+\phi_i^1(t,y)\phi_i^2(t,x)}{2\sqrt{\phi_i^1(t,x)\phi_i^2(t,x)}}>\sqrt{\phi_i^1(t,y)\phi_i^2(t,y)}.$$ By the arguments of Proposition \[property-PSP-prop\], we have $$\mu_i^n\left(\frac{\lambda_i^1+\lambda_i^2}{2}\right)<\frac{\mu_i^n(\lambda_i^1)+\mu_i^n(\lambda_i^2)}{2}.$$ The corollary is thus proved.

Proofs of the main results
==========================

In this section, we prove our main results.

Dirichlet boundary condition case
---------------------------------

In this subsection, we prove Theorem \[PSP-diri\] and Corollary \[PSP-diri-coro\].

\[Proof of Theorem \[PSP-diri\]\] (1) We first assume that there is $\lambda_1^p>0$ such that $\mu_1^n(\lambda_1^p)=0$, and prove that (D) holds, that is $\int_0^T\widetilde m_1(t)dt=\int_0^T \max_{x\in \bar {D_1}} m_1(t, x)dt>0$.

Assume that $\int_0^T\widetilde m_1(t)dt\le 0$. For any $\lambda_1>0$, we use $\widetilde \mu_1^n(\lambda_1)$ to denote the principal spectrum point of the following eigenvalue problem $$\begin{cases}
-\p_t u+\int_{D_1} \kappa(y-x)u(t,y)dy-u(t,x)+\lambda_1 \widetilde m_1(t)u=\widetilde\mu_1 u\quad \text{ in }\bar D,\cr
u(t+T,x)=u(t,x).
\end{cases}$$ By Proposition \[spectrum-autonomous\] (2), we have $$\widetilde \mu_1^n(\lambda_1)=\mu_i^n(0)+\frac{\lambda_1}{T}\int_0^T \widetilde m(t)dt,$$ where $\mu_1^n(0)<0$ by Proposition \[property-PSP-prop\](2). And by Proposition \[property-PSP-prop\](3), we have $$\mu_1^n(\lambda_1)\le \widetilde\mu_1^n(\lambda_1),$$ since $m_1(t, x)\leq \widetilde m_1(t)$ and $\lambda_1>0$. Hence $\mu_1^n(\lambda_1)<0$ for any $\lambda\ge 0$. This is a contradiction. Hence $\int_0^T\widetilde m_1(t)dt>0$.

Next, we assume that (D) holds, and prove that there is a unique $\lambda_1^p>0$ such that $\mu_1^n(\lambda_1^p)=0$.

According to Proposition \[property-PSP-prop\](2), $\mu_1^n(\lambda_1)<0$ for $\lambda_1=0$. Meanwhile, from Proposition \[property-PSP-prop\](4), we have $\mu_1^n(\lambda_1)>0$ for $\lambda\gg1$. Thus there is $\lambda_1^p>0$ such that $\mu_1^n(\lambda_1^p)=0$.

\(2) Suppose there exist $0<\lambda_1^{p, 1}\leq \lambda_1^{p, 2}$ such that $\mu_1^n(\lambda_1^{p, 1})=\mu_1^n(\lambda_1^{p, 2})=0$. We need to show that $\lambda_1^{p, 1}=\lambda_1^{p, 2}$. Assume that $\lambda_1^{p,1}<\lambda_1^{p,2}$. By the convexity of $\mu_1^n(\lambda_1)$, $\mu_1^n(\lambda_1)=0$ for $\lambda_1^{p,1}\le \lambda_1\le\lambda_1^{p,2}$.

If $m_1(t,x)\equiv m_1(t)$, we have $\mu_1^n(\lambda_1^{p,1})=\mu_1^n(0)+\lambda_1^{p,1}\widehat m_1=0$ and $\mu_1^n(\lambda_1^{p,2})=\mu_1^n(0)+\lambda_1^{p,2}\widehat m_1=0$. Note that $\mu_1^n(0)<0$. We then must have $\widehat m_1>0$ and then $$0=\mu_1^n(\lambda_1^{p,1})=\mu_1^n(0)+\lambda_1^{p,1}\widehat m_1<
\mu_1^n(\lambda_1^{p,2})=\mu_1^n(0)+\lambda_1^{p,2}\widehat m_1=0.$$ This is a contradiction. Hence $\lambda_1^{p,1}=\lambda_1^{p,2}$.

Suppose that $m_1(t,x)\not\equiv m_1(t)$. Assume that $\mu_1^n(\lambda_1^{p,1})$ is the principal eigenvalue of $\mathcal{L}_1+\lambda_1^{p,1}m_1$. Then for $\lambda_1>\lambda_1^{p,1}$ with $\lambda_1-\lambda_1^{p,1}\ll 1$, $\mu_1^n(\lambda_1)$ is also the principal eigenvalue of $\mathcal{L}_1+\lambda_1 m_1$. By Corollary \[strict-convex\], we have $\mu_1^n(\lambda_1)<0$ for $\lambda_1$ with $\lambda_1-\lambda_1^{p,1}\ll 1$. This is a contradiction. Hence $\mu_1^n(\lambda_1^{p,1})$ is not the principal eigenvalue of $\mathcal{L}_1+\lambda_1^{p,1}m_1$. Similarly, $\mu_1^n(\lambda_1^{p,2})$ is not the principal eigenvalue of $\mathcal{L}_1+\lambda_1^{p,2}m_1$. We then have that $\mu_1^n(\lambda_1^{p, 1})$ and $\mu_1^n( \lambda_1^{p, 2})$ are not eigenvalues of $\mathcal L_1+\lambda_1^{p,1}m_1$ and $\mathcal{L}_1+\lambda_1^{p,2}m_1$, respectively. By Proposition \[pv-existence\] (2), we have $$\mu_1^n(\lambda_1^{p,1})=h_{1,\max}(\lambda_1^{p,1})= -1+\lambda_1^{p, 1}m_{1,\max},$$ and $$\mu_1^n(\lambda_1^{p,2})=h_{1,\max}(\lambda_1^{p,2})=-1+\lambda_1^{p, 2}m_{1,\max}.$$ It thus follows that $\lambda_1^{p, 1}=\lambda_1^{p, 2}$ if $\mu_1^n(\lambda_1^{p, 1})=\mu_1^n(\lambda_1^{p, 2})=0$. This is also a contradiction.

Therefore, $\lambda_1^{p,1}=\lambda_1^{p,2}$.

\(1) Assume that $m_1(t,x)\equiv m_1(t)$. Then $ \mathcal{P}(m_1)=\int_0^T m_1(t)dt$ and $$\mu_1^n(\mathcal{L}_1+\lambda_1 m_1)=\mu_1^n(0)+\lambda_1 \widehat m_1.$$ (1) then follows from Theorem \[PSP-diri\].

\(2) Assume that $m_1(t,x)\equiv m_1(x)$. Then $\mathcal{P}(m_1)=T\cdot \max_{x\in\bar D_1}m_1(x)$. Then (2) follows from Theorem \[PSP-diri\] and Proposition \[spectrum-autonomous\].

Neumann boundary condition case
-------------------------------

In this subsection, we prove Theorem \[PSP-neum\] and Corollary \[PSP-neum-coro\].

We first prove (2).

Suppose that there exist $0<\lambda_2^{p, 1}\leq \lambda_2^{p, 2}$ such that $\mu_2^n(\lambda_2^{p, 1})=\mu_2^n(\lambda_2^{p, 2})=0$. We need to show that $\lambda_2^{p, 1}= \lambda_2^{p, 2}$.

Assume that $\lambda_2^{p,1}<\lambda_2^{p,2}$. By the convexity of $\mu_2^n(\lambda_2)$, $\mu_2^n(\lambda_2)=0$ for $\lambda_2^{p,1}\le \lambda_2\le\lambda_2^{p,2}$.

By the similar argument as in the Dirichlet boundary case, $\mu_2^n(\lambda_2^{p,1})$ and $\mu_2^n(\lambda_2^{p,2})$ are not eigenvalues of $\mathcal{L}_2+\lambda_2^{p,1}m_2$ and $\mathcal{L}_2+\lambda_2^{p,2}m_2$, respectively (the assumption $m_2(t,x)\not\equiv m_2(t)$ is used here). By Proposition \[pv-existence\] (2), we have $$\mu_2^n(\lambda)=\widehat h_{2,\max}(\lambda)=\max_{x\in  {\bar D_2}}\left[ -b_2(x)+\lambda\widehat m_{2}(x)\right]=0 \quad \text{for all}\,\, \lambda\in[\lambda_2^{p,1},\lambda_2^{p,2}],$$ where $b_2(x)=\int_D\kappa(y-x)dy$. Let $x_\lambda\in \bar D_2$ be such that \[proof-neum-unique\] \_[x|D\_2]{}=-b\_2(x\_)+m\_2(x\_)=\_2\^n()=0 for $\lambda\in [\lambda_2^{p,1},\lambda_2^{p,2}]$. Note that $-b_2(x)<0$ for any $x\in\bar D_2$. Hence we must have $ \widehat m_2(x_\lambda)>0$. In particular, $\widehat m_2(x_{\lambda_2^{p,1}})>0$. We then have $$-b_2(x_{\lambda_2^{p,1}})+\lambda_2^{p,1}\widehat m_2(x_{\lambda_2^{p,1}})<-b_2(x_{\lambda_2^{p,1}})+\lambda_2^{p,2}
\widehat m_2(x_{\lambda_2^{p,1}})\le -b_2(x_{\lambda_2^{p,2}})+\lambda_2^{p,2}
\widehat m_2(x_{\lambda_2^{p,2}}),$$ which contradicts to .

It thus follows that $\lambda_2^{p, 1}=\lambda_2^{p, 2}$ if $\mu_2^n(\lambda_2^{p, 1})=\mu_2^n(\lambda_2^{p, 2})=0$.

Next, we prove (1).

First suppose that there is $\lambda_2^p>0$ such that $\mu_2^n(\lambda_2^p)=0$. We prove that $\int_0^T \widetilde m_2(t)dt>0$ and $\int_D\widehat m_2(x)dx<0$ with $\widetilde m_2(t)=\max_{x\in \bar D_2}m_2(t, x)$ and $\widehat m_2(x)=\frac1T\int_0^Tm_2(t, x)dt$.

Assume that $\int_0^T \widetilde m_2(t)dt\le 0$. Note that $m_2(t,x)\le \widetilde m_2(t)$ and $m_2(t,x)\not\equiv \tilde m_2(t)$. Hence, by Proposition \[spectrum-autonomous\](2) and Proposition \[property-PSP-prop\](3), we have $$\mu_2^n(\lambda_2)< \lambda_2\frac{\int_0^T\widetilde m_2(t)dt}{T}\le 0$$ for $\lambda_2> 0$. This is a contradiction. Therefore $\int_0^T\widetilde m_2(t)dt>0$.

Proposition \[property-PSP-prop\], together with $\mu_2^n(\lambda_2^p)=0$ and (2), leads to $\mu_2^n(\lambda_2)<0$ for $0<\lambda_2\ll 1$. Note that for $|\lambda_2|\ll 1$, $\mu_2^n(\lambda_2)$ is a principal eigenvalue, associated with a positive eigenfunction, denoted by $\phi_2^{\lambda_2}(t,x)$. Then $$-\frac{\p_t\phi_2^{\lambda_2}(t,x)}{\phi_2^{\lambda_2}(t,x)}+\int_{D_2} \kappa(y-x)\frac{\phi_2^{\lambda_2}(t,y)-\phi_2^{\lambda_2}(t,x)}{\phi_2^{\lambda_2}(t,x)}dy
+\lambda_2 m_2(t,x)=\mu_2^n(\lambda_2).$$ Hence for $0<\lambda_2\ll 1$, $$\begin{aligned}
&\mu_2^n(\lambda_2)\cdot T\cdot |D_2| -\lambda_2 T \int_{D_2} \widehat m_2(x)dx\\
&=
\int_0^T \int_{D_2}\int_{D_2} \kappa(y-x)\frac{\phi_2^{\lambda_2}(t,y)-\phi_2^{\lambda_2}(t,x)}{\phi_2^{\lambda_2}(t,x)}dydxdt\\
&=\frac{1}{2}\int_0^T \int_{D_2}\int_{D_2} \kappa(y-x)\Big[\frac{\phi_2^{\lambda_2}(t,y)-\phi_2^{\lambda_2}(t,x)}{\phi_2^{\lambda_2}(t,x)}
+\frac{\phi_2^{\lambda_2}(t,x)-\phi_2^{\lambda_2}(t,y)}{\phi_2^{\lambda_2}(t,y)}\Big]dydxdt\\
&=\frac{1}{2}\int_0^T\int_{D_2}\int_{D_2} \kappa(y-x)\frac{(\phi_2^{\lambda_2}(t,y)-\phi_2^{\lambda_2}(t,x))^2}{\phi_2^{\lambda_2}(t,x)\phi_2^{\lambda_2}(t,y)}dydxdt.\end{aligned}$$ Since $m_2(t,x)\not \equiv m_2(t)$, $\phi_2^{\lambda_2}(t,y)\not\equiv \phi_2^{\lambda_2}(t,x)$. We then have $$0>\mu_2^n(\lambda_2)\cdot T\cdot |D_2|> \lambda_2 T\int_{D_2}\widehat m_2(x)dx \quad \text{ for } 0<\lambda_2\ll 1,$$ and then $\int_{D_2} \widehat m_2(x) dx<0$.

Next, suppose that $\int_0^T\widetilde m_2(t)dt>0$ and $\int_{D_2} \widehat m_2(x)dx<0$. We prove that there is $\lambda_2^p>0$ such that $\mu_2^n(\lambda_2^p)=0$.

By Proposition \[property-PSP-prop\] (4), $\mu_2^n(\lambda_2)>0$ for $\lambda_2\gg 1$. Note that $\mu_2(0)=0$ and for $0<\lambda_2\ll 1$, $\mu_2^n(\lambda_2)$ is a principal eigenvalue of . Suppose that $\phi_2^{\lambda_2}(t,x)$ is a positive principal eigenfunction with $\|\phi_2^{\lambda_2}(\cdot,\cdot)\|_\infty=1$. Note that $\phi_2^{\lambda_2}(t,x)$ is differentiable in $\lambda_2$ and $\phi_2^0(t,x)\equiv 1$. Hence $$\phi_2^{\lambda_2}(t,y)-\phi_2^{\lambda_2}(t,x)=\phi_2^{\lambda_2}(t,y)-1-(\phi_2^{\lambda_2}(t,x)-1)=O(\lambda_2).$$ Then by the above arguments, $$\begin{aligned}
&\mu_2^n(\lambda_2)\cdot T\cdot|D_2|\\
=&\lambda_2\int_{D_2} \widehat m_2(x)dx+\frac{1}{2}\int_0^T\int_{D_2}\int_{D_2} \kappa(y-x)\frac{(\phi_2^{\lambda_2}(t,y)-\phi_2^{\lambda_2}(t,x))^2}{\phi_2^{\lambda_2}(t,x)\phi_2^{\lambda_2}(t,y)}dydxdt\\
=&\lambda_2\int_{D_2}\widehat m_2(x)dx+O(\lambda_2^2).\end{aligned}$$ It then follows that $\mu_2^n(\lambda_2)<0$ for $0<\lambda_2\ll 1$. Then there is $\lambda_2^p>0$ such that $\mu_2^n(\lambda_2^p)=0$.

\(1) Assume that $m_2(t,x)\equiv m_2(t)$. Then by Proposition \[spectrum-autonomous\] (2) and \[property-PSP-prop\] (2), we have $$\mu_2^n(\mathcal{L}_2+\lambda_2 m_2)=\mu_2^n(0)+\lambda_2 \widehat m_2=\lambda_2\widehat m_2.$$ It then follows that if $\widehat m_2\not =0$, then there is no positive principal spectrum point of $\mathcal{L}_2+\lambda_2 m_2$. If $\widehat m_2=0$, then every positive $\lambda$ is a principal spectrum point of $\mathcal{L}_2+\lambda_2m_2$.

\(2) Assume that $m_2(t,x)\equiv m_2(x)$. Then $\mathcal{P}(m_2)=T\cdot \max_{x\in\bar D_2}m_2(x)$. (2) then follows from Theorem \[PSP-neum\] and Proposition \[spectrum-autonomous\].

Periodic boundary condition case
--------------------------------

In this subsection, we prove Theorem \[PSP-peri\] and Corollary \[PSP-peri-coro\]

First, (2) can proved by the similar arguments as those in Theorem \[PSP-neum\](2).

We prove (1) in the following.

Suppose that there is $\lambda_3^p>0$ such that $\mu_3^n(\lambda_3^p)=0$. We prove that $\int_0^T \widetilde m_3(t)dt>0$ and $\int_{D_3}\widehat m_3(x)dx<0$.

Assume that $\int_0^T \widetilde m_3(t)dt\le 0$. Note that $m_3(t,x)\le \widetilde m_3(t)$ and $m_3(t,x)\not\equiv \widetilde m_3(t)$. Hence $$\mu_3^n(\lambda_3)<\frac{\lambda_3}T\int_0^T\widetilde m_3(t)dt\le 0$$ for $\lambda_3> 0$. This is a contradiction. Therefore $\int_0^T\widetilde m_3(t)dt>0$.

Proposition \[property-PSP-prop\], together with $\mu_3^n(\lambda_3^p)=0$, leads to $\mu_3^n(\lambda_3)<0$ for $0<\lambda_3\ll 1$. Note that for $|\lambda_3|\ll 1$, $\mu_3^n(\lambda_3)$ is a principal eigenvalue. Suppose that $\phi_3^{\lambda_3}(t,x)$ is a positive principal eigenfunction. Then $$-\frac{\p_t\phi_3^{\lambda_3}(t,x)}{\phi_3^{\lambda_3}(t,x)}+\int_{{\ensuremath{\mathbb{R}}}^N} \kappa(y-x)\frac{\phi_3^{\lambda_3}(t,y)-\phi_3^{\lambda_3}(t,x)}{\phi_3^{\lambda_3}(t,x)}dy
+\lambda_3 m_3(t,x)=\mu_3^n(\lambda_3).$$ Let $$\widetilde \kappa(z)=\sum_{(k_1,k_2,\cdots,k_N)\in{\ensuremath{\mathbb{Z}}}^N}\kappa(z+(k_1p_1,k_2p_2,\cdots,k_Np_N)).$$ Then $\widetilde \kappa(-z)=\widetilde \kappa(z)$ and $$\int_{{\ensuremath{\mathbb{R}}}^N}\kappa(y-x)\phi_3^{\lambda_3}(t,y)dy=\int_{D_3}\widetilde \kappa(y-x) \phi_3^{\lambda_3}(t,y)dy.$$ Hence for $0<\lambda_3\ll 1$, $$\begin{aligned}
&\mu_3^n(\lambda_3)\cdot T\cdot |D_3| -\lambda_3T \int_{D_3} \widehat m_3(x)dx\\
&=
\int_0^T \int_{D_3}\int_{D_3}  \widetilde\kappa(y-x)\frac{\phi_3^{\lambda_3}(t,y)-\phi_3^{\lambda_3}(t,x)}{\phi_3^{\lambda_3}(t,x)}dydxdt\\
&=\frac{1}{2}\int_0^T \int_{D_3}\int_{D_3}\widetilde \kappa(y-x)\Big[\frac{\phi_3^{\lambda_3}(t,y)-\phi_3^{\lambda_3}(t,x)}{\phi_3^{\lambda_3}(t,x)}
+\frac{\phi_3^{\lambda_3}(t,x)-\phi_3^{\lambda_3}(t,y)}{\phi_3^{\lambda_3}(t,y)}\Big]dydxdt\\
&=\frac{1}{2}\int_0^T\int_{D_3}\int_{D_3}\widetilde \kappa(y-x)\frac{(\phi_3^{\lambda_3}(t,y)-\phi_3^{\lambda_3}(t,x))^2}{\phi_3^{\lambda_3}(t,x)\phi_3^{\lambda_3}(t,y)}dydxdt.\end{aligned}$$ Since $m_3(t,x)\not \equiv m_3(t)$, $\phi_3^{\lambda_3}(t,y)\not\equiv \phi_3^{\lambda_3}(t,x)$. Hence for $0<\lambda_3\ll 1$, $$0>\mu_3^n(\lambda_3)\cdot T\cdot |D_3|> \lambda_3 T\int_{D_3}\widehat m_3(x)dx$$ and then $\int_{D_3} \widehat m_3(x) dx<0$.

Next, suppose that $\int_0^T\widetilde m_3(t)dt>0$ and $\int_{D_3} \widehat m_3(x)dx<0$. We prove that there is $\lambda_3^p>0$ such that $\mu_3^n(\lambda_3^p)=0$.

By Proposition \[property-PSP-prop\] (4), $\mu_3^n(\lambda_3)>0$ for $\lambda_3\gg 1$. Note that $\mu_3^n(0)=0$ and for $0<\lambda_3\ll 1$, $\mu_3^n(\lambda_3)$ is a principal eigenvalue of . Suppose that $\phi_3^{\lambda_3}(t,x)$ is a positive principal eigenfunction with $\|\phi_3^{\lambda_3}(\cdot,\cdot)\|_\infty=1$. Note that $\phi_3^{\lambda_3}(t,x)$ is differentiable in $\lambda_3$ and $\phi_3^0(t,x)\equiv 1$. Hence $$\phi_3^{\lambda_3}(t,y)-\phi_3^{\lambda_3}(t,x)=\phi_3^{\lambda_3}(t,y)-1-(\phi_3^{\lambda_3}(t,x)-1)=O(\lambda_3).$$ Then by the above arguments, $$\begin{aligned}
&\mu_3^n(\lambda_3)\cdot T\cdot|D_3|\\
=&\lambda_3\int_{D_3} \widehat m_3(x)dx+\frac{1}{2}\int_0^T\int_{D_3}\int_{D_3} \widetilde\kappa(y-x)\frac{(\phi_3^{\lambda_3}(t,y)-\phi_3^{\lambda_3}(t,x))^2}{\phi_3^{\lambda_3}(t,x)\phi_3^{\lambda_3}(t,y)}dydxdt\\
=&\lambda_3\int_{D_3}\widehat m_3(x)dx+O(\lambda_3^2).\end{aligned}$$ It then follows that $\mu_3^n(\lambda_3)<0$ for $0<\lambda_3\ll 1$. Then there is $\lambda_3^p>0$ such that $\mu_3^n(\lambda_3^p)=0$.

It can be proved by a similar argument as in the proof of Corollary \[PSP-neum-coro\].

Upper bounds and the existence of principal eigenvalues
-------------------------------------------------------

In this subsection, we prove Theorem \[upper-bounds\].

\[Proof of Theorem \[upper-bounds\]\] (1) First, we prove the upper bound of in the case of $i=1$. If has a unique positive principal spectrum point $\lambda_1^p(\widehat m_1)$, we need to show that also has a unique positive principal spectrum point $\lambda_1^p(m_1)$, and $\lambda_1^p(m_1)\leq \lambda_1^p(\widehat m_1)$.

By Corollary \[PSP-diri-coro\] (2), has a unique positive principal spectrum point $ \lambda_1^p(\widehat m_1)$ if and only if \[upper-bound-diri\] m\_1(x\_0)&gt;0 x\_0D\_1, where $\widehat m_1(x_0)=\frac1T\int_0^Tm_1(t, x_0)dt$. Since $\int_0^Tm_1(t, x_0)dt\leq \int_0^T\max_{x\in \bar D_1}m_1(t, x)dt$, we know that $m_1(t, x)$ satisfies (D), that is $$\mathcal P(m_1)=\int_0^T\max_{x\in \bar D_1}m_1(t, x)dt>0.$$ By Theorem \[PSP-diri\], we know that also has a unique positive principal spectrum point $\lambda_1^p(m_1)$.

To prove the upper bound of $\lambda_1^p(m_1)$, denote the principal spectrum point of by $\mu_1^n(\lambda_1,\widehat m_1)$, and the principal spectrum point of by $\mu_1^n(\lambda_1,m_1)$. We have $$\mu_1^n(\lambda_1^p(\widehat m_1),\widehat m_1)\leq \mu_1^n(\lambda_1^p(\widehat m_1),m_1))$$ by Proposition \[property-PSP-prop\] (3). By Definition \[PSP-def\], we have $$\mu_1^n(\lambda_1^p(\widehat m_1),\widehat m_1)=\mu_1^n(\lambda_1^p(m_1),m_1)=0.$$ Hence, $$\mu_1^n(\lambda_1^p(m_1),m_1)\leq \mu_1^n( \lambda_1^p(\widehat m_1),m_1).$$ By Proposition \[property-PSP-prop\] (3), we have $$\lambda_1^p(m_i)\leq \lambda_1^p(\widehat m_i).$$ The proof in the case of $i=2$ or $i=3$ is similar to the case of $i=1$. So we omit it.

\(2) (i) It follows from the fact that the conditions in (i) imply that (S1) holds.

\(ii) It follows from (1) and Proposition \[pv-existence\].

Applications
------------

In this section, we apply Theorem \[PSP-diri\], \[PSP-neum\] and \[PSP-peri\] to study the existence of positive time-periodic solution of the KPP type equations , , and .

Recall the KPP type equation ( or ) with $f_i(t, x, u) (i=1, 2, 3)$ satisfying (F). The eigenvalue problems of , , and linearized at $u=0$ are $$\label{KPP-diri-L}
\begin{cases}
 -\p_t u(t, x)+\left[\int_D \kappa(y-x)u(t, y)dy-u(t, x)\right]+\lambda_1 m_1(t,x)u(t, x)=\mu_1  u,\\
u(t+T, x)=u(t, x).\\
u(t, x)\geq0,
\end{cases}$$ for the Dirichlet boundary condition, $$\label{KPP-neum-L}
\begin{cases}
 -\p_t u(t, x)+\int_D \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_2 m_2(t,x)u(t, x)=\mu_2  u,\\
u(t+T, x)=u(t, x),\\
u(t, x)\geq 0
\end{cases}$$ for the Neumann boundary condition, and $$\label{KPP-peri-L}
\begin{cases}
 -\p_t u(t, x)+\int_{\R^N} \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_3 m_3(t,x)u(t, x)=\mu_3  u,\\
u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x),\\
u(t, x)\geq 0
\end{cases}$$ for the periodic boundary condition, where $m_i(t, x)= f_i(t, x, 0)$. It is assumed that $m_1$, $m_2$, and $m_3$ satisfies (D), (N), and (P), respectively.

Denote the principal spectrum point of , , and by $\mu_1^n(\lambda_1)$, $\mu_2^n(\lambda_2)$, and $\mu_3^n(\lambda_3)$, respectively. Suppose $\lambda_i^p$ is such that $\mu_i^n(\lambda_i^p)=0$, we need to show that ( or ) admits a unique positive time-periodic solution $u^*(t, x)$ if and only if $\lambda_i>\lambda_i^p$.

We only prove the case of $i=1$, and other cases ($i=2$ or $i=3$) can be shown similarly.

First, we show if $\lambda_1>\lambda_1^p$, then admits a unique positive time-periodic solution $u^*(t, x)$. By [@RaSh Theorem E], the KPP type equation has a unique positive time periodic solution, if $f_1$ satisfies the following monostable assumptions:

[**(H1)**]{} [*$f_1$ is $C^1$ in $t\in{\ensuremath{\mathbb{R}}}$ and $C^3$ in $(x,u)\in {\ensuremath{\mathbb{R}}}^N\times{\ensuremath{\mathbb{R}}}$; $f_1(t, x, u)<0$ for $u\gg 1$ and $\partial _u f_1(t, x, u)<0$ for $u\geq0$*]{}; $f_1(t+T,x,u)=f_1(t,x,u)$.

[**(H2)**]{} [*$\mu_1^n(\lambda_1)>0$, where $\mu_1^n(\lambda_1)$ is the principle spectrum point of .* ]{}

The condition (H1) is exactly (F), so we only need to show that (H2) is satisfied if $\lambda_1>\lambda_1^p$. Since $m_1=f_1(t, x, 0)$ satisfies (D), we have $\mu_1^n(\lambda_1^p)=0$ by Theorem \[PSP-diri\]. Since $\lambda_1>\lambda_1^p$, we know that $\mu_1^n(\lambda_1)>\mu_1^n(\lambda_1^p)$ by Theorem \[PSP-diri\] and \[P\]roposition \[property-PSP-prop\] (3). It follows that $\mu_1^n(\lambda_1)>0$. Hence (H2) is satisfied.

Second, consider , and we need to show that if has a solution, then $\lambda_1>\lambda_1^p$, where $\lambda_1^p$ is the unique positive number such that $\mu_1^n(\lambda_1^p)=0$.

Suppose that admits a positive solution $u^*(t, x)$. Then $$-\p_t u^*(t,x)+\int_D\kappa(y-x)u^*(t,y)dy-u^*(t,x)+\lambda_1 f_1(t,x,u^*(t,x)) u^*(t,x)=0.$$ This implies that $\mu_1^n(\mathcal{L}_1+\lambda_1 m^*_1)=0$, where $m^*_1(t,x)=f_1(t,x,u^*(t,x))$. Note that $f_1(t,x,0)>f_1(t,x,u^*(t,x))$. Hence $\mu_1^n(\lambda_1)>0$. Therefore, $\lambda_1^p<\lambda_1.$

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