Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator

TL;DR

This paper studies how the principal eigenvalue of a time-periodic parabolic operator changes with frequency, establishing monotonicity and asymptotic behavior, and confirming a conjecture from prior mathematical biology research.

Contribution

It proves the monotonicity and asymptotic properties of the principal eigenvalue with respect to frequency, confirming a conjecture in the field.

Findings

Principal eigenvalue's monotonicity with frequency established.

Asymptotic behavior of eigenvalue analyzed.

Conjecture by Hutson et al. confirmed.

Abstract

We investigate the effect of frequency on the principal eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow and Pol\'{a}\v{c}ik [2001 J. Math. Biol.].

Equations

$$\begin{cases}\begin{array}[]{ll}\tau\partial_{t}u-\mathrm{div}\left[A(x,t)\nabla u\right]-\nabla m(x,t)\cdot\nabla u+V(x,t)u=\lambda u,&x\in\Omega,t\in[0,1],\\ bu+(1-b)\left[A\nabla u\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ u(x,0)=u(x,1),&x\in\Omega,\end{array}\end{cases}$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\partial_{s}w=\nabla\cdot\left[A^{*}(x,s)\nabla w\right]+\nabla w\cdot\nabla m^{*}(x,s)+wf(x,s,w),&x\in\Omega,s>0,\\ \vskip 3.0pt plus 1.0pt minus 1.0ptbw+(1-b)\left[A^{*}(x,s)\nabla w\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,t>0,\\ w(x,s)=w(x,s+T),&x\in\Omega,\end{array}\end{cases}$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\partial_{s}\varphi-\nabla\cdot\left[A^{*}(x,s)\nabla\varphi\right]-\nabla\varphi\cdot\nabla m^{*}(x,s)-f(x,s,0)\varphi=\lambda\varphi,&x\in\Omega,s>0,\\ \vskip 3.0pt plus 1.0pt minus 1.0ptb\varphi+(1-b)\left[A^{*}(x,s)\nabla\varphi\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,s>0,\\ \varphi(x,s)=\varphi(x,s+T),&x\in\Omega.\end{array}\end{cases}$$

$$\hat{p}(x):={\int }_0^{1}p(x,s)\mathrm{d}s.$$

$$\begin{cases}\begin{array}[]{ll}-\mathrm{div}\left[A(x,t)\nabla\varphi\right]-\nabla m(x,t)\cdot\nabla\varphi+V(x,t)\varphi=\lambda(t)\varphi,&x\in\Omega,\\ b\varphi+(1-b)\left[A\nabla\varphi\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,\end{array}\end{cases}$$

$$\begin{cases}\begin{array}[]{ll}-\mathrm{div}\left[\hat{A}(x)\nabla\varphi\right]-\nabla\hat{m}(x)\cdot\nabla\varphi+\hat{V}(x)\varphi=\lambda\varphi,&x\in\Omega,\\ b\varphi+(1-b)\left[\hat{A}\nabla\varphi\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,\end{array}\end{cases}$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0ptL_{\tau}u:=\tau\partial_{t}u-\mathrm{div}\left[A(x,t)\nabla u\right]+V(x,t)u=\lambda(\tau)u,&x\in\Omega,t\in[0,1],\\ \vskip 3.0pt plus 1.0pt minus 1.0ptbu+(1-b)\left[A\nabla u\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ u(x,0)=u(x,1),&x\in\Omega,\end{array}\end{cases}$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0ptL^{*}_{\tau}v:=-\tau\partial_{t}v-\mathrm{div}\left[A(x,t)\nabla v\right]+V(x,t)v=\lambda(\tau)v,&x\in\Omega,t\in[0,1],\\ \vskip 3.0pt plus 1.0pt minus 1.0ptbv+(1-b)\left[A\nabla v\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ v(x,0)=v(x,1),&x\in\Omega.\end{array}\end{cases}$$

$$\mathbb{S}_{b}=\left\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\zeta\in C^{2,1}(\mathcal{C})\cap C^{1,1}(\overline{\mathcal{C}}):&\zeta(x,0)=\zeta(x,1)~{}\mathrm{in}~{}\Omega,\\ &b\zeta+(1-b)[A\nabla\zeta]\cdot\mathbf{n}=0~{}\mathrm{on}~{}\partial\Omega\times[0,1]\end{array}\right\}.$$

$$J_{\tau }(\zeta )={\int }_{\mathcal{C}}{u}_{\tau }{v}_{\tau }\left(\frac{L_{\tau }\zeta }{\zeta }\right)\mathrm{d}x\mathrm{d}t,\zeta \in {\mathbb{S}}_b^0,$$

$$\mathbb{S}^{0}_{b}=\begin{cases}\begin{split}&\{\zeta\in\mathbb{S}_{b}:\zeta>0~{}\mathrm{in}~{}\overline{\mathcal{C}}\},&0\leq b<1,\\ &\{\zeta\in\mathbb{S}_{1}:\zeta>0~{}\mathrm{in}~{}\mathcal{C},\nabla\zeta\cdot\mathbf{n}<0~{}\mathrm{on}~{}\partial\Omega\times[0,1]\},&b=1.\end{split}\end{cases}$$

$$J_{\tau }(u_{\tau })-J_{\tau }(\zeta )={\int }_{\mathcal{C}}{u}_{\tau }{v}_{\tau }\left[\nabla \log \left(\frac{\zeta }{u_{\tau }}\right)\right]\cdot \left[A\nabla \log \left(\frac{\zeta }{u_{\tau }}\right)\right].$$

$$\begin{split}J_{\tau}(\zeta)=&\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\left(\frac{\zeta_{t}}{\zeta}\right)-\int_{\mathcal{C}}u_{\tau}v_{\tau}\left[\frac{\mathrm{div}(A\nabla\zeta)}{\zeta}\right]+\int_{\mathcal{C}}u_{\tau}v_{\tau}V\\ =&\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\log\zeta-\int_{0}^{1}\!\!\!\int_{\partial\Omega}u_{\tau}v_{\tau}\left[A\nabla\log\zeta\right]\cdot\mathbf{n}+\int_{\mathcal{C}}\nabla\left(u_{\tau}v_{\tau}\right)\cdot\left[A\nabla\log\zeta\right]\\ &-\int_{\mathcal{C}}u_{\tau}v_{\tau}\Big{[}(\nabla\log\zeta)\cdot\left(A\nabla\log\zeta\right)\Big{]}+\int_{\mathcal{C}}u_{\tau}v_{\tau}V.\\ \end{split}$$

$$\begin{array}[]{l}\mathbf{D}J_{\tau}(u_{\tau})\varphi=0\,\,\text{ for all}\,\,\varphi\in\mathbb{S}_{b},\end{array}$$

$$-{\int }_0^1{\int }_{\partial \Omega }{u}_{\tau }{v}_{\tau }\left[A\nabla \log \zeta \right]\cdot \mathbf{n}=\frac{b}{1-b}{\int }_0^1{\int }_{\partial \Omega }{u}_{\tau }{v}_{\tau },$$

$$\begin{split}\mathbf{D}J_{\tau}(\zeta)\varphi=&\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\left(\frac{\varphi}{\zeta}\right)-2\int_{\mathcal{C}}u_{\tau}v_{\tau}(\nabla\log\zeta)\cdot\left[A\nabla\left(\frac{\varphi}{\zeta}\right)\right]\\ &+\int_{\mathcal{C}}\Big{[}A\nabla(u_{\tau}v_{\tau})\Big{]}\cdot\nabla\left(\frac{\varphi}{\zeta}\right).\end{split}$$

$$\begin{split}&\mathbf{D}J_{\tau}(u_{\tau})\varphi\\ =&\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\left(\frac{\varphi}{u_{\tau}}\right)-2\int_{\mathcal{C}}u_{\tau}v_{\tau}(\nabla\log u_{\tau})\cdot\left[A\nabla\left(\frac{\varphi}{u_{\tau}}\right)\right]+\int_{\mathcal{C}}\Big{[}A\nabla(u_{\tau}v_{\tau})\Big{]}\cdot\nabla\left(\frac{\varphi}{u_{\tau}}\right)\\ =&-\tau\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\partial_{t}(u_{\tau}v_{\tau})-2\int_{0}^{1}\!\!\!\int_{\partial\Omega}\left(\frac{\varphi v_{\tau}}{u_{\tau}}\right)A\nabla u_{\tau}\cdot\mathbf{n}+2\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\nabla\cdot\Big{[}v_{\tau}A\nabla u_{\tau}\Big{]}\\ &+\int_{0}^{1}\!\!\!\int_{\partial\Omega}\left(\frac{\varphi}{u_{\tau}}\right)A\nabla(u_{\tau}v_{\tau})\cdot\mathbf{n}-\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\nabla\cdot\Big{[}A\nabla(u_{\tau}v_{\tau})\Big{]}\\ =&-\tau\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\partial_{t}(u_{\tau}v_{\tau})+2\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\Big{\{}\nabla v_{\tau}\cdot\left[A\nabla u_{\tau}\right]+v_{\tau}\mathrm{div}\left[A\nabla u_{\tau}\right]\Big{\}}\\ &-\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\Big{\{}v_{\tau}\mathrm{div}\left[A\nabla u_{\tau}\right]+2\nabla v_{\tau}\cdot\left[A\nabla u_{\tau}\right]+u_{\tau}\mathrm{div}\left[A\nabla v_{\tau}\right]\Big{\}}\\ =&-\tau\int_{\mathcal{C}}\left(\frac{\varphi}{u_{\tau}}\right)\partial_{t}(u_{\tau}v_{\tau})+\int_{\mathcal{C}}\left(\frac{\varphi v_{\tau}}{u_{\tau}}\right)\mathrm{div}\left[A\nabla u_{\tau}\right]-\int_{\mathcal{C}}\varphi\mathrm{div}\left[A\nabla v_{\tau}\right]\\ =&-\int_{\mathcal{C}}\left(\frac{v_{\tau}}{u_{\tau}}\right)\Big{\{}\tau\partial_{t}u_{\tau}-\mathrm{div}\left[A\nabla u_{\tau}\right]\Big{\}}\varphi+\int_{\mathcal{C}}\Big{\{}-\tau\partial_{t}v_{\tau}-\mathrm{div}\left[A\nabla v_{\tau}\right]\Big{\}}\varphi.\\ \end{split}$$

$$\begin{array}{rl}\mathbf{D}{J}_{\tau }(u_{\tau })\phi =& -{\int }_{\mathcal{C}}\left(\frac{v_{\tau }}{u_{\tau }}\right)\left[\lambda (\tau )u_{\tau }-V{u}_{\tau }\right]\phi +{\int }_{\mathcal{C}}\left[\lambda (\tau )v_{\tau }-V{v}_{\tau }\right]\phi =0,\end{array}$$

$$\begin{split}&J_{\tau}(u_{\tau})-J_{\tau}(\zeta)\\ =&-\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\log\left(\frac{\zeta}{u_{\tau}}\right)-\int_{\mathcal{C}}u_{\tau}v_{\tau}(\nabla\log u_{\tau})\cdot\left[A\nabla\log u_{\tau}\right]+\int_{\mathcal{C}}u_{\tau}v_{\tau}(\nabla\log\zeta)\cdot\left[A\nabla\log\zeta\right]\\ &-\int_{\mathcal{C}}\nabla(u_{\tau}v_{\tau})\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\\ =&-\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\log\left(\frac{\zeta}{u_{\tau}}\right)+\int_{\mathcal{C}}u_{\tau}v_{\tau}\Big{[}\nabla\log\left(\zeta u_{\tau}\right)\Big{]}\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\\ &-\int_{\mathcal{C}}\nabla(u_{\tau}v_{\tau})\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\\ =&-\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\log\left(\frac{\zeta}{u_{\tau}}\right)+\int_{\mathcal{C}}u_{\tau}v_{\tau}\left[\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)+2\nabla\log u_{\tau}\right]\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\\ &-\int_{\mathcal{C}}\nabla(u_{\tau}v_{\tau})\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\\ =&-\tau\int_{\mathcal{C}}u_{\tau}v_{\tau}\partial_{t}\log\left(\frac{\zeta}{u_{\tau}}\right)+2\int_{\mathcal{C}}u_{\tau}v_{\tau}(\nabla\log u_{\tau})\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\\ &-\int_{\mathcal{C}}\nabla(u_{\tau}v_{\tau})\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]+\int_{\mathcal{C}}u_{\tau}v_{\tau}\left[\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right]\cdot\left[A\nabla\log\left(\frac{\zeta}{u_{\tau}}\right)\right].\\ \end{split}$$

$$\begin{array}{rl}{J}_{\tau }(u_{\tau })-J_{\tau }(\zeta )& =-\mathbf{D}{J}_{\tau }(u_{\tau })\phi +{\int }_{\mathcal{C}}{u}_{\tau }{v}_{\tau }\left[\nabla \log \left(\frac{\zeta }{u_{\tau }}\right)\right]\cdot \left[A\nabla \log \left(\frac{\zeta }{u_{\tau }}\right)\right]\\ & ={\int }_{\mathcal{C}}{u}_{\tau }{v}_{\tau }\left[\nabla \log \left(\frac{\zeta }{u_{\tau }}\right)\right]\cdot \left[A\nabla \log \left(\frac{\zeta }{u_{\tau }}\right)\right],\end{array}$$

$$\underset{x\to x_0}{\lim }\frac{u_{\tau }{v}_{\tau }}{\zeta }=\underset{x\to x_0}{\lim }\frac{v_{\tau }\nabla u_{\tau }\cdot \mathbf{n}+u_{\tau }\nabla v_{\tau }\cdot \mathbf{n}}{\nabla \zeta \cdot \mathbf{n}}=0,\forall x_0\in \partial \Omega .$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\partial_{t}u_{\tau}+\tau\partial_{t}u^{\prime}_{\tau}-\mathrm{div}\left[A\nabla u^{\prime}_{\tau}\right]+Vu^{\prime}_{\tau}=\lambda(\tau)u^{\prime}_{\tau}+\frac{\partial\lambda}{\partial\tau}u_{\tau},&x\in\Omega,t\in[0,1],\\ \vskip 3.0pt plus 1.0pt minus 1.0ptbu^{\prime}_{\tau}+(1-b)\left[A\nabla u^{\prime}_{\tau}\right]\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ u^{\prime}_{\tau}(x,0)=u^{\prime}_{\tau}(x,1),&x\in\Omega.\end{array}\end{cases}$$

$$\frac{\partial \lambda }{\partial \tau }={\int }_{\mathcal{C}}{v}_{\tau }{\partial }_t{u}_{\tau }.$$

$$\begin{split}\int_{\mathcal{C}}v_{\tau}\partial_{t}(u_{\tau})=&\frac{1}{2\tau}\int_{\mathcal{C}}v_{\tau}(L_{\tau}-L^{*}_{\tau})u_{\tau}\\ =&\frac{1}{2\tau}\left[\int_{\mathcal{C}}v_{\tau}L_{\tau}u_{\tau}-\int_{\mathcal{C}}u_{\tau}L_{\tau}v_{\tau}\right]\\ =&\frac{1}{2\tau}\Big{[}J_{\tau}(u_{\tau})-J_{\tau}(v_{\tau})\Big{]}.\end{split}$$

$$\begin{array}{rl}\frac{\partial \lambda }{\partial \tau }& =\frac{1}{2\tau }{\int }_{\mathcal{C}}{u}_{\tau }{v}_{\tau }\left[\nabla \log \left(\frac{v_{\tau }}{u_{\tau }}\right)\right]\cdot \left[A\nabla \log \left(\frac{v_{\tau }}{u_{\tau }}\right)\right],\end{array}$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\tau\partial_{t}U_{\tau}-\mathrm{div}[\hat{A}\nabla U_{\tau}]+\hat{V}(x)U_{\tau}=\lambda(\tau)U_{\tau},&x\in\Omega,t\in[0,1],\\ \vskip 3.0pt plus 1.0pt minus 1.0ptbU_{\tau}+(1-b)[\hat{A}\nabla U_{\tau}]\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ U_{\tau}(x,0)=U_{\tau}(x,1),&x\in\Omega.\end{array}\end{cases}$$

$$c^{\prime }(t)v_{{\tau }_0}+2c(t){\partial }_t{v}_{{\tau }_0}=0.$$

$$-{\tau }_0\frac{T_{{\tau }_0}^{\prime }(t)}{T_{{\tau }_0}(t)}-\frac{\mathrm{div}[\hat{A}{X}_{{\tau }_0}](x)}{X_{{\tau }_0}(x)}+V(x,t)=\lambda ({\tau }_0)\mathrm{in}\mathcal{C}.$$

$$\begin{cases}\begin{array}[]{ll}\tau\partial_{t}u-D\Delta u-\nabla\hat{m}(x)\cdot\nabla u+V(x,t)u=\lambda(\tau)u,&x\in\Omega,t\in[0,1],\\ bu+(1-b)D\nabla u\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ u(x,0)=u(x,1),&x\in\Omega.\end{array}\end{cases}$$

$$\begin{cases}\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\mathcal{L}_{\tau}\varphi:=\tau\partial_{t}\varphi-D\Delta\varphi+h(x,t)\varphi=\lambda(\tau)\varphi,&x\in\Omega,t\in[0,1],\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\left(b-\frac{1-b}{2}\nabla\hat{m}\cdot\mathbf{n}\right)\varphi+(1-b)D\nabla\varphi\cdot\mathbf{n}=0,&x\in\partial\Omega,t\in[0,1],\\ \varphi(x,0)=\varphi(x,1),&x\in\Omega,\end{array}\end{cases}$$

Full text

S. Liu: Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, PRC. Email: liushuangnqkg@ruc.edu.cn

Y. Lou: Department of Mathematics, Ohio State University, Columbus, OH 43210, USA. Email: lou@math.ohio-state.edu

R. Peng: School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PRC. Email: pengrui $\b{}$ seu@163.com

M. Zhou: Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW 2341, Australia. Email: zhouutokyo@gmail.com

1. Introduction

We consider the linear time-periodic parabolic eigenvalue problem

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial\Omega$, and $\mathbf{n}$ denotes the unit outward normal vector on $\partial\Omega$. The positive constant $\tau$ is referred as the frequency, and the constant $b\in[0,1]$. Denote by $\mathcal{C}=\Omega\times(0,1)$ the periodicity cell. The matrix function $A\in C^{1+\sigma,1}(\overline{\mathcal{C}})$, and functions $m\in C^{1+\sigma,1}(\overline{\mathcal{C}})$ and $V\in C^{\sigma,1}(\overline{\mathcal{C}})$, with $\sigma\in(0,1)$, are assumed to be periodic in $t$ with unit period. Furthermore, $A$ is a symmetric and uniformly elliptic matrix field, i.e. there exist positive constants $\gamma_{1}$ and $\gamma_{2}$ such that $\gamma_{1}|\xi|^{2}\leq\xi^{\mathrm{T}}A(x,t)\xi\leq\gamma_{2}|\xi|^{2}$ holds for all $(x,t)\in\mathcal{C}$ and $\xi\in\mathbb{R}^{N}$.

Proposition 14.4 in [5] guarantees the existence and uniqueness of the principle eigenvalue, denoted by $\lambda(\tau)$, of problem (1), which is real, simple and its corresponding eigenfunction can be chosen positive in $\Omega\times[0,1]$. Furthermore, $\lambda(\tau)<Re(\lambda)$ for any other eigenvalue $\lambda$ of (1).

Problem (1) arises in connection with the nonlinear reaction-diffusion equation

[TABLE]

which models various ecological and evolutionary processes in spatio-temporally varying environments [2, 3, 4, 10, 11, 13]. The matrix function $A^{*}$ and functions $m^{*}$, $f$ are periodic in $s$ with a common period $T>0$. It is natural to inquire how the temporal variability of the environment affects the population dynamics of (2). The persistence of populations is closely associated with the stability of the steady state $w=0$ for (2), which in turn is determined by the sign of the principal eigenvalue, denoted by $\lambda^{*}(T)$, of the linear eigenvalue problem

[TABLE]

Set $\tau=1/T,t=s/T,A(x,t)=A^{*}(x,s),m(x,t)=m^{*}(x,s),V(x,t)=-f(x,s,0),$ and $u(x,t)=\varphi(x,s).$ Then $u(x,t)$ satisfies problem (1), and $\lambda(\tau)=\lambda^{*}(T)$. Hence, determining the stability of $w=0$ is reduced to understanding the sign of $\lambda(\tau)$. The goal of this paper is to study the dependence of $\lambda(\tau)$ on the frequency $\tau$.

Given any function $p(x,t)$, which is $1$-periodic in time, set

[TABLE]

For $A(x,t)\equiv\hat{A}(x)$ and $\nabla m(x,t)\equiv\nabla\hat{m}(x)$, it was shown in [6] that $\lambda(\tau)\leq\lim\limits_{\tau\rightarrow\infty}\lambda(\tau)$ for all $\tau>0$, i.e. $\lambda(\tau)$ attains its maximum at $\tau=\infty$. It is natural to ask whether $\lambda(\tau)$ is increasing in $\tau$. The monotonicity of $\lambda(\tau)$ seems to be open [9], even for the case $\nabla m=0$ as conjectured by Hutson et. al [7]. We now answer this conjecture positively as follows:

Theorem 1.1**.**

Assume $\nabla m=0$. Then $\lambda(\tau)$ is non-decreasing in $\tau>0$. If further assume $A(x,t)\equiv\hat{A}(x)$, then the following assertions hold:

• (i)

If $V=\hat{V}(x)+g(t)$ for some $1$-periodic function $g(t)$, then $\lambda(\tau)$ is constant for $\tau>0$;

• (ii)

Otherwise $\frac{\partial\lambda}{\partial\tau}(\tau)>0$ for every $\tau>0$.

The monotonicity of $\lambda(\tau)$ in Theorem 1.1 may fail if $\nabla m\neq 0$; See Example 3.1 in Sect. 3. Our next result concerns the case $A=I_{N\times N}$ and $\nabla m(x,t)=\nabla\hat{m}(x)$.

Theorem 1.2**.**

Assume $A(x,t)=DI_{N\times N}$ for some constant $D>0$ and $\nabla m(x,t)=\nabla\hat{m}(x)$. Then $\lambda(\tau)$ is non-decreasing in $\tau>0$ and the following assertions hold:

• (i)

If $V=\hat{V}(x)+g(t)$ for some $1$-periodic function $g(t)$, then $\lambda(\tau)$ is constant for $\tau>0$;

• (ii)

Otherwise $\frac{\partial\lambda}{\partial\tau}(\tau)>0$ for every $\tau>0$.

Interpreting them in the context of spatio-temporal variation of environmental resources, Theorems 1.1 and 1.2 suggest that if the species disperse by random diffusion and/or by advection along some gradient $\nabla m$, increasing the temporal variation of the resources tends to favor the persistence of populations. The condition $\nabla m(x,t)=\nabla\hat{m}(x)$ in Theorem 1.2 is necessary; See Example 3.2 in Sect. 3.

We next state the asymptotic behaviors of $\lambda(\tau)$ for sufficiently small or large $\tau$:

Theorem 1.3**.**

The following assertions hold:

• (i)

For each fixed $t\in[0,1]$, denote by $\lambda^{0}(t)$ the principal eigenvalue of the linear problem

[TABLE]

then $\lim_{\tau\rightarrow 0}\lambda(\tau)=\int_{0}^{1}\lambda^{0}(s)\mathrm{d}s;$

• (ii)

Denote by $\lambda^{\infty}$ the principal eigenvalue of the linear problem

[TABLE]

then $\lim_{\tau\rightarrow\infty}\lambda(\tau)=\lambda^{\infty}.$

Part (i) in Theorem 1.3 appears to be new, while part (ii) is due to Nadin [9] for the space-time periodic case. Theorems 1.1, 1.2 and 1.3 imply that if $\nabla m(x,t)=\nabla\hat{m}(x)$, $\int_{0}^{1}\lambda^{0}(s)\mathrm{d}s\leq\lambda(\tau)\leq\lambda^{\infty}$ for any $\tau$, and the equality holds if and only if $V$ has the form of $\hat{V}(x)+g(t)$. Here the estimate $\lambda(\tau)\leq\lambda^{\infty}$ agrees with the result in [6], while $\lambda(\tau)\geq\int_{0}^{1}\lambda^{0}(s)\mathrm{d}s$ is new, even in the simplest scenario when $A=I_{N\times N}$ and $\nabla m=0$; See [7] for numerical results for this case.

The rest of the paper is organized as follows: In Sect. 2, Theorems 1.1 and 1.2 are proved. Two examples are presented in Sect. 3 to complement Theorems 1.1 and 1.2. Sect. 4 is devoted to the proof of Theorem 1.3.

2. Monotonicity of $\lambda(\tau)$

This section is devoted to the proofs of Theorems 1.1 and 1.2. The proofs are based upon some functional, which was first introduced in [8] for an elliptic eigenvalue problem.

2.1. The case $\nabla m=0$

In this subsection, we consider the scenario when $\nabla m=0$ and prove Theorem 1.1. In this case, problem (1) becomes

[TABLE]

where we denote $u_{\tau}$ as a positive eigenfunction associated with $\lambda(\tau)$. Furthermore, consider the adjoint problem of (5), i.e.

[TABLE]

Let $v_{\tau}$ be a positive eigenfunction of (6) corresponding to $\lambda(\tau)$. We normalize $u_{\tau}$ and $v_{\tau}$ such that $\int_{\mathcal{C}}u^{2}_{\tau}=\int_{\mathcal{C}}u_{\tau}v_{\tau}=1$ for any $\tau>0$.

For $b\in[0,1]$, define set $\mathbb{S}_{b}$ by

[TABLE]

Define functional $J_{\tau}$ by

[TABLE]

where

[TABLE]

Clearly, for any $\tau>0$, $u_{\tau},v_{\tau}\in\mathbb{S}^{0}_{b}$ and $J_{\tau}$ is well defined on the cone $\mathbb{S}^{0}_{b}$. The following property of $J_{\tau}$ turns out to be crucial in establishing Theorem 1.1.

Lemma 2.1**.**

For any $\zeta\in\mathbb{S}^{0}_{b}$, we have

[TABLE]

Proof.

By the definition of $J_{\tau}$, we observe that, for every $\zeta\in\mathbb{S}^{0}_{b}$,

[TABLE]

In what follows, we need to distinguish two cases: $0\leq b<1$ and $b=1$.

Case 1: $0\leq b<1$. First, we claim that $u_{\tau}$ is a critical point of $J_{\tau}$ in the sense that

[TABLE]

where $\mathbf{D}J_{\tau}(u_{\tau})$ is the Fréchet derivative of $J_{\tau}$ at the point $u_{\tau}\in\mathbb{S}^{0}_{b}$.

To prove (10), we first observe that, by $\zeta\in\mathbb{S}^{0}_{b}$,

[TABLE]

which is independent of $\zeta$ in the current case. Making use of this fact and (9), for any $\varphi\in\mathbb{S}_{b}$, we have

[TABLE]

Through straightforward calculations, we further have

[TABLE]

In the above, the boundary integrals vanish due to the boundary conditions of $u_{\tau}$ and $v_{\tau}$.

By $L_{\tau}u_{\tau}=\lambda(\tau)u_{\tau}$ and $L^{*}_{\tau}v_{\tau}=\lambda(\tau)v_{\tau}$, we obtain

[TABLE]

and (10) thus follows.

We now proceed to prove formula (8) through some tedious manipulations. With the help of (9), direct calculation shows

[TABLE]

As $u_{\tau}\log\left(\frac{\zeta}{u_{\tau}}\right)\in\mathbb{S}_{b}$, we choose $\varphi=u_{\tau}\log\left(\frac{\zeta}{u_{\tau}}\right)$ in (11). By $\mathbf{D}J_{\tau}(u_{\tau})\varphi=0$, we have

[TABLE]

as desired.

Case 2: $b=1$. The Hopf Boundary Lemma implies that $\nabla u_{\tau}\cdot\mathbf{n}<0$ and $\nabla v_{\tau}\cdot\mathbf{n}<0$ on $\partial\Omega\times[0,1]$, and thus $u_{\tau},v_{\tau}\in\mathbb{S}^{0}_{1}$, so that $J_{\tau}(u_{\tau})$ and $J_{\tau}(v_{\tau})$ are well defined. Fix $t\in[0,1]$. For any $\zeta\in\mathbb{S}^{0}_{1}$, noting that $\nabla\zeta\cdot\mathbf{n}<0$ on $\partial\Omega$, we have

[TABLE]

Hence, it is easy to see that $\int_{\partial\Omega}u_{\tau}v_{\tau}\left[A\nabla\log\zeta\right]\cdot\mathbf{n}=0$ in (9).

Similarly as in Case 1, we can show that the principal eigenfunction $u_{\tau}$ is still a critical point of $J_{\tau}$, i.e., $\mathbf{D}J(u_{\tau})\varphi=0$ for all $\varphi\in\mathbb{S}_{1}$. Based on this fact, formula (8) can be proved by a similar argument as in Case 1. The proof of Lemma 2.1 is thus complete. ∎

With the help of Lemma 2.1, we are in a position to prove Theorem 1.1.

$\mathbf{Proof~{}of~{}Theorem~{}\ref{FR_thm 1}}$: We substitute $u=u_{\tau}$ into (5) and differentiate the resulting equation with respect to $\tau$. Denoting $\frac{\partial u_{\tau}}{\partial\tau}=u^{\prime}_{\tau}$ for brevity, we obtain

[TABLE]

We multiply the above equation by $v_{\tau}$ and integrate the result over $\mathcal{C}$. Together with the facts of $L^{*}_{\tau}v_{\tau}=\lambda(\tau)v_{\tau}$ and the normalization $\int_{\mathcal{C}}u_{\tau}v_{\tau}=1$, we find that

[TABLE]

Recalling the definitions of $L_{\tau}$, $L^{*}_{\tau}$ and $J_{\tau}$, we further derive

[TABLE]

In view of $v_{\tau}\in\mathbb{S}^{0}_{b}$, Lemma 2.1 implies

[TABLE]

which shows that $\frac{\partial\lambda}{\partial\tau}\geq 0$ for all $\tau>0$.

It remains to prove Parts (i) and (ii) in Theorem 1.1. In what follows we assume that $A(x,t)=\hat{A}(x)$. When $V(x,t)=\hat{V}(x)+g(t)$ for some $1$-periodic function $g(t)$, we set $U_{\tau}(x,t)=e^{\left(\frac{1}{\tau}\int_{0}^{1}g(s)\mathrm{d}s\right)}u_{\tau}$. Then $U_{\tau}$ is $1$-periodic and solves

[TABLE]

Observe that $\hat{A}$ and $\hat{V}$ are independent of $t$. By the uniqueness of principal eigenfunction (up to multiplication by a constant), it is clear that $\lambda(\tau)$ is constant for $\tau>0$. This proves Part (i) in Theorem 1.1.

Finally, we show $\frac{\partial\lambda}{\partial\tau}>0$ for all $\tau>0$ if $V$ does not take the form of $V=\hat{V}(x)+g(t)$. Suppose to the contrary that there exists some $\tau_{0}>0$ such that $\frac{\partial\lambda}{\partial\tau}(\tau_{0})=0$. According to (12), we have $u_{\tau_{0}}=c(t)v_{\tau_{0}}$ for some $1$-periodic function $c(t)>0$. Substituting $u_{\tau_{0}}=c(t)v_{\tau_{0}}$ into $L_{\tau_{0}}u_{\tau_{0}}=\lambda(\tau_{0})u_{\tau_{0}}$ and using $L^{*}_{\tau_{0}}v_{\tau_{0}}=\lambda(\tau_{0})v_{\tau_{0}}$, we can deduce

[TABLE]

It then follows $\partial_{t}\log v_{\tau_{0}}=-\frac{c^{\prime}(t)}{2c(t)}$ in $\mathcal{C}$, which depends only on $t$. Hence, $v_{\tau_{0}}$ is of the form $v_{\tau_{0}}=X_{\tau_{0}}(x)T_{\tau_{0}}(t)$ with some $1$-periodic function $T_{\tau_{0}}(t)>0$ in $[0,T]$ and function $X_{\tau_{0}}(x)>0$ in $\Omega$. Again using $L^{*}_{\tau}v_{\tau_{0}}=\lambda(\tau_{0})v_{\tau_{0}}$, we arrive at

[TABLE]

Thus, it is necessary that $V$ has the form of $V=\hat{V}(x)+g(t)$, contradicting the previous assumption. The proof of Theorem 1.1 is now complete. ∎

2.2. The case $A(x,t)=DI_{N\times N}$ and $\nabla m(x,t)=\nabla\hat{m}(x)$

We now prove Theorem 1.2. Under our assumption, problem (1) reduces to the following:

[TABLE]

$\mathbf{Proof~{}of~{}Theorem~{}\ref{FR_thm 1.1}}$: The argument is similar to that of Theorem 1.1 and hence we only give a sketch here. By the transformation $\varphi=e^{\frac{\hat{m}}{2D}}u$, problem (13) can be rewritten as

[TABLE]

with $h=\frac{\Delta\hat{m}}{2}+\frac{|\nabla\hat{m}|^{2}}{4D}+V(x,t)$. Denote by $\widetilde{u}_{\tau}>0$ the eigenfunction of problem (13) corresponding to $\lambda(\tau)$. Then $\varphi_{\tau}=e^{\frac{\hat{m}}{2D}}\widetilde{u}_{\tau}$ (normalized by $\int_{\mathcal{C}}\varphi^{2}_{\tau}=1$) solves (14). We also consider the adjoint problem to (14):

[TABLE]

Choose $\psi_{\tau}>0$ to be the principal eigenfunction of (15), normalized by $\int_{\mathcal{C}}\varphi_{\tau}\psi_{\tau}=1$. Similarly as in Subsection 2.1, we introduce the functional

[TABLE]

which is well defined on the cone

[TABLE]

Here $\widetilde{\mathbb{S}}_{b}$ is defined by

[TABLE]

Proceeding similarly as in the proof of Lemma 2.1, one can check that $\widetilde{J}_{\tau}$ satisfies the property:

[TABLE]

Then the same analysis as in Theorem 1.1 enables one to conclude Theorem 1.2. ∎

3. Non-monotonicity of $\lambda(\tau)$ in the general case

In this section, we will construct two examples to show that the monotonicity of $\lambda(\tau)$ stated in Theorems 1.1 and 1.2 may not hold for a general $m(x,t)$.

Example 3.1** (A counterexample for Theorem 1.1 when $\nabla m\neq 0$).**

Let $m(x,t)=x_{1}$, $A(x,t)=a(t)$, $V(x,t)=\frac{-x_{1}a^{\prime}(t)}{2a^{2}(t)}$, and $b=0$ in problem (1). Here $a(t)\in C^{1}([0,1])$ is some $1$-periodic positive function satisfying $a^{\prime}(t)\not\equiv 0$. Now, consider the eigenvalue problem

[TABLE]

Since $\hat{V}(x)=0$, Part (ii) in Theorem 1.3 concludes that

[TABLE]

Let $u_{\tau}>0$ be the principal eigenfunction of (16) and introduce $\varphi_{\tau}=e^{\frac{x_{1}}{2a(t)}}u_{\tau}$. By direct calculation, we have

[TABLE]

Taking $\tau=1$, we obtain

[TABLE]

We multiply the above equation by $\varphi_{1}$ and integrate the resulting equation over $\mathcal{C}=\Omega\times(0,1)$. Then it follows from the boundary condition of $\varphi_{1}$ that

[TABLE]

If $\lambda(1)=0$, it is easily seen that $\partial_{x_{1}}\varphi_{1}=\frac{\varphi_{1}}{2a(t)}$ and $\partial_{x_{i}}\varphi_{1}=0$ for all $2\leq i\leq N$, whence $\varphi_{1}=Ce^{\frac{x_{1}}{2a(t)}}$ with some constant $C>0$. In view of $\varphi_{1}=e^{\frac{x_{1}}{2a(t)}}u_{1}$, we arrive at $u_{1}=C$. Substituting $u_{1}=C$ in problem $(\ref{fr15})$ gives a contradiction to $a^{\prime}(t)\not\equiv 0$. Thus $\lambda(1)>0=\lambda(\infty)$ and in turn $\lambda(\tau)$ is not non-decreasing in $\tau$.

Example 3.2** (A counterexample for Theorem 1.2 when $A=I_{N\times N}$ but $\nabla m(x,t)\neq\nabla\hat{m}(x)$).**

Let $\Omega=(0,2\pi)$, $m(x,t)=\cos x\sin t$, $D=1$, $V(x,t)=\frac{1}{2}\cos x(\sin t+\cos t)$, and $b=0$ in problem (13). Thus we are led to the following problem:

[TABLE]

Clearly, $m(x,t)$ does not satisfy the assumption in Theorem 1.2. Let $\lambda(\tau)$ be the principal eigenvalue of (17) and $u_{\tau}>0$ be the corresponding eigenfunction. By Part (ii) in Theorem 1.3, we infer that

[TABLE]

Set $\varphi_{\tau}=e^{\frac{1}{2}\cos x\sin t}u_{\tau}$ and $\tau=1$. Then $(\lambda(1),\varphi_{1})$ solves

[TABLE]

Multiplying the above equation by $\varphi_{1}$ and integrating the resulting equation over $\mathcal{C}=(0,2\pi)\times(0,2\pi)$, we arrive at

[TABLE]

which implies immediately that $\lambda(1)>0=\lambda(\infty)$. Hence $\lambda(\tau)$ is not non-decreasing in $\tau$.

4. Asymptotic behaviors of $\lambda(\tau)$

The goal of this section is to prove Theorem 1.3.

$\mathbf{Proof~{}of~{}Theorem~{}\ref{FR_thm 2}}$. As already noted, the proof of Part (ii) in Theorem 1.3 has been carried out by [9] for the space-time periodic case, which can be easily adapted to the present setting. We thus omit the details here and refer the interested reader to Lemma 3.10 in [9].

To prove Part (i) in Theorem 1.3, denote by $u_{0}(x,t)$ the principal eigenfunction of (3) corresponding to $\lambda^{0}(t)$ for fixed $t$. For any $x\in\overline{\Omega}$, it is easy to see that $u_{0}(x,\cdot),\nabla u_{0}(x,\cdot)\in C^{1}([0,1])$ and $u_{0}(x,t+1)=u_{0}(x,t)$.

Define $\underline{u}=\rho(t)u_{0}$ for some $1$-periodic function $\rho(t)>0$. Given arbitrary $\epsilon>0$, it is desirable to specify $\rho(t)$ such that

[TABLE]

provided that $\tau>0$ is sufficiently small. Then a direct application of Proposition 2.1 of [12] gives

[TABLE]

To verify (18), substitute $\underline{u}=\rho(t)u_{0}$ into (18). Direct calculation shows that $\rho(t)$ should satisfy

[TABLE]

With this in mind, we now claim that $\frac{\partial_{t}u_{0}}{u_{0}}$ is bounded in $\overline{\Omega}\times[0,1]$. Indeed, the claim is obvious for $0\leq b<1$ because of $u_{0}>0$ in $\overline{\mathcal{C}}$. For $b=1$ (i.e., zero Dirichlet boundary condition), it suffices to prove $\lim\limits_{x\rightarrow\partial\Omega}\frac{\partial_{t}u_{0}}{u_{0}}$ is bounded. The well-known Hopf Boundary Lemma for parabolic equations (see e.g. Proposition 13.3 of [5]) implies that $\nabla u_{0}\cdot\mathbf{n}<0$ on $\partial\Omega\times[0,1]$. Fix any $x_{0}\in\partial\Omega$. Thanks to $u_{0}(x_{0},t)=\partial_{t}u_{0}(x_{0},t)=0$, the claim follows from

[TABLE]

Consequently, we can choose $\tau$ so small that $\tau|\partial_{t}u_{0}|\leq\epsilon u_{0}$. Then define $\rho(t)$ by

[TABLE]

i.e., $\rho(t)=\exp\left[\frac{1}{\tau}\left(t\int_{0}^{1}\lambda^{0}(s)\mathrm{d}s-\int_{0}^{t}\lambda^{0}(s)\mathrm{d}s\right)\right]$. We can check readily that $\rho(t)=\rho(t+1)$ and $\underline{u}$ verifies (18), whence (19) holds.

In a similar manner, we may construct $\overline{u}$ such that

[TABLE]

for sufficiently small $\tau>0$, which combined with Proposition 2.1 of [12] leads to

[TABLE]

Part (i) in Theorem 1.3 follows from (19) and (20). ∎

Acknowledgments. SL was partially supported by the NSFC grant No. 11571364. YL was partially supported by the NSF grant DMS-1411176. RP was partially supported by the National Science Foundation (NSF) of China (grant nos. 11671175 and 11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (grant no. PPZY2015A013), and the Qing Lan Project of Jiangsu Province. MZ was partially supported by the Australian Research Council (grant no. DE170101410).