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Wenxian Shen

Publications and source records attributed to Wenxian Shen.

At least 55 records · Page 3Linked to original sources
arXiv2018

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

The current paper is devoted to the study of two species competition systems of the form \begin{equation*} \begin{cases} u_t(t,x)= \mathcal{A} u+u(a_1(t,x)-b_1(t,x)u-c_1(t,x)v),\quad x\in\RR\cr v_t(t,x)= \mathcal{A} v+ v(a_2(t,x)-b_2(t,x)u-c_2(t,x) v),\quad x\in\RR \end{cases} \end{equation*} where $\mathcal{A}u=u_{xx}$, or $(\mathcal{A}u)(t,x)=\int_{\RR}κ(y-x)u(t,y)dy-u(t,x)$ ($κ(\cdot)$ is a smooth non-negative convolution kernel supported on an interval centered at the origin), $a_i(t+T,x)=a_i(t,x)$, $b_i(t+T,x)=b_i(t,x)$, $c_i(t+T,x)=c_i(t,x)$, and $a_i$, $b_i$, and $c_i$ ($i=1,2$) are spatially homogeneous when $|x|\gg 1$, that is, $a_i(t,x)=a_i^0(t)$, $b_i(t,x)=b_i^0(t)$, $c_i(t,x)=c_i^0(t)$ for some $a_i^0(t)$, $b_i^0(t)$, $c_i^0(t)$, and $|x|\gg 1$. Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. We, in particular, study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds.

math.AP↗
arXiv2017

Almost Automorphically and Almost Periodically Forced Circle Flows of Almost Periodic Parabolic Equations on S^1

We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$. The structure of the minimal set $M$ is thoroughly investigated under the assumption that the center space $V^c(M)$ associated with $M$ is no more than $2$-dimensional. Such situation naturally occurs while, for instance, $M$ is hyperbolic or uniquely ergodic. It is shown in this paper that $M$ is a $1$-cover of the hull $H(f)$ provided that $M$ is hyperbolic (equivalently, ${\rm dim}V^c(M)=0$). If ${\rm dim}V^c(M)=1$ (resp. ${\rm dim}V^c(M)=2$ with ${\rm dim}V^u(M)$ being odd), then either $M$ is an almost $1$-cover of $H(f)$ and topologically conjugate to a minimal flow in $\mathbb{R}\times H(f)$; or $M$ can be (resp. residually) embedded into an almost periodically (resp. almost automorphically) forced circle-flow $S^1\times H(f)$. When $f(t,u,u_x)=f(t,u,-u_x)$ (which includes the case $f=f(t,u)$), it is proved that any minimal set $M$ is an almost $1$-cover of $H(f)$. In particular, any hyperbolic minimal set $M$ is a $1$-cover of $H(f)$. Furthermore, if ${\rm dim}V^c(M)=1$, then $M$ is either a $1$-cover of $H(f)$ or is topologically conjugate to a minimal flow in $\mathbb{R}\times H(f)$. For the general spatially-dependent nonlinearity $f=f(t,x,u,u_{x})$, we show that any stable or linearly stable minimal invariant set $M$ is residually embedded into $\mathbb{R}^2\times H(f)$.

math.DS↗
arXiv2017

Time averaging for nonautonomous/random linear parabolic equations

Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.

math.AP↗
arXiv2017

Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$

In the current paper, we consider the following parabolic-elliptic semilinear Keller-Segel model on $\mathbb{R}^{N}$, \begin{equation*} \begin{cases} u_{t}=\nabla\cdot (\nabla u-χu\nabla v)+a u -b u^2, \quad x\in\mathbb{R}^N,\,\, t>0\cr 0=(Δ- I)v+ u, \quad x\in\mathbb{R}^N,\,\, t>0, \end{cases} \end{equation*} where $ χ>0, \ a\ge 0,\ b> 0$ are constant real numbers and $N$ is a positive integer. We first prove the local existence and uniqueness of classical solutions $(u(x,t;u_0),v(x,t;u_0))$ with $u(x,0;u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the constants $a, b, χ$ and the dimension $N$, we prove the global existence and boundedness of classical solution $(u(x,t;u_0),v(x,t;u_0))$ for given initial functions $u_0(x)$. Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants $a, b, χ$ and the dimension $N$, we show that for every strictly positive initial function $u_0(\cdot)$, $$\lim_{t\to\infty} \sup_{x\in\mathbb{R}^N} \big[|u(x,t;u_0)-\frac{a}{b}|+|v(x,t;u_0)-\frac{a}{b}|\big]=0,$$ and that for every nonnegative initial function $u_0(\cdot)$ with non-empty and compact support ${\rm supp}(u_0)$, there are $0 c_{\rm up}^*(u_0).$$

math.AP↗
arXiv2017

Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$

We consider the following chemotaxis systems $$\begin{cases}u_t=Δu-χ_1\nabla(u\nabla v_1)+χ_2\nabla(u\nabla v_2)+u(a-bu),\ \ x\in\mathbb R^N,t>0,\\0=(Δ-λ_1I)v_1+μ_1u,\ \ x\in\mathbb R^N,t>0,\\0=(Δ-λ_2I)v_2+μ_2u,\ \ \text{in}\ x\in\mathbb R^N,\ t>0,\\u(\cdot,0)=u_0,\ \ x\in\mathbb R^N,\end{cases}$$where $χ_i,\ λ_i,\ μ_i,\ i=1,2$ and $a,\ b$ are positive constant real numbers and $N$ is a positive integer. Under some conditions on the parameters, we prove the global existence and boundedness of classical solutions $(u(x,t;u_0),v_1(x,t;u_0),v_2(x,t;u_0))$ for nonnegative, bounded, and uniformly continuous initials $u_0(x)$. Next, we show that, for every strictly positive initial \,$u_0(x)$,$$\lim_{t\to\infty}\left[\|u(\cdot,t;u_0)-\frac{a}{b}\|_{\infty}+\|λ_1v_1(\cdot,t;u_0)-\frac{a}{b}μ_1\|_{\infty}+\|λ_2v_2(\cdot,t;u_0)-\frac{a}{b}μ_2\|_{\infty}\right]=0.$$ Finally, we explore the spreading properties of the global solutions and prove that there are two positive numbers $0 c^*_+(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)$. Furthermore we show that$$\lim_{(χ_1,χ_2)\to(0,0)}c^*_-(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=\lim_{(χ_1,χ_2)\to(0,0)}c^*_+(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=2\sqrt{a}.$$

math.AP↗
arXiv2017

Existence, uniqueness and stability of transition fronts of nonlocal equations in time heterogeneous bistable media

The present paper is devoted to the study of existence, uniqueness and stability of transition fronts of nonlocal dispersal evolution equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space nonincreasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space nonincreasing transition front (if it exists), which implies the uniqueness, up to space shifts, and monotonicity of transition fronts provided that a space nonincreasing transition front exists. Moreover, we show that a transition front must be a periodic traveling wave in periodic media and asymptotic speeds of transition fronts exist in uniquely ergodic media. Finally, we prove the existence of space nonincreasing transition fronts, whose proof does not need the unbalanced condition.

math.AP↗
arXiv2017

Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in [34, 39, 45, 46, 61] for random dispersal Fisher-KPP equations with time and space periodic dependence, in [41, 42, 43, 51, 52, 53, 58, 63] for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in [17, 48, 56] for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable. Up to the author's knowledge, it is the first time that the stability of transition waves of Fisher-KPP equations with both time and space dependence is studied.

math.AP↗
arXiv2017

Asymptotic behavior of semilinear parabolic equations on the circle with time almost-periodic/recurrent dependence

We study topological structure of the $ω$-limit sets of the skew-product semiflow generated by the following scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f(t,u,u_x)$ is $C^2$-admissible with time-recurrent structure including almost-periodicity and almost-automorphy. Contrary to the time-periodic cases (for which any $ω$-limit set can be imbedded into a periodically forced circle flow), it is shown that one cannot expect that any $ω$-limit set can be imbedded into an almost-periodically forced circle flow even if $f$ is uniformly almost-periodic in $t$. More precisely, we prove that, for a given $ω$-limit set $Ω$, if ${\rm dim}V^c(Ω)\leq 1$ ($V^c(Ω)$ is the center space associated with $Ω$), then $Ω$ is either spatially-homogeneous or spatially-inhomogeneous; and moreover, any spatially-inhomogeneous $Ω$ can be imbedded into a time-recurrently forced circle flow (resp. imbedded into an almost periodically-forced circle flow if $f$ is uniformly almost-periodic in $t$). On the other hand, when ${\rm dim}V^c(Ω>1$, it is pointed out that the above embedding property cannot hold anymore. Furthermore, we also show the new phenomena of the residual imbedding into a time-recurrently forced circle flow (resp. into an almost automorphically-forced circle flow if $f$ is uniformly almost-periodic in $t$) provided that $\dim V^c(Ω)=2$ and $\dim V^u(Ω)$ is odd. All these results reveal that for such system there are essential differences between time-periodic cases and non-periodic cases.

math.DS↗
arXiv2017

Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

We study traveling wave solutions of the following chemotaxis systems,$$\begin{cases}u_t=Δu-χ_1\nabla(u\nabla v_1)+χ_2\nabla(u\nabla v_2)+u(a-bu),\ x\in\mathbb{R}^N\\ 0=Δv_1-λ_1v_1+μ_1u,\ x\in\mathbb{R}^N,\\ 0=Δv_2-λ_2v_2+μ_2u,\ x\in\mathbb{R}^N,\end{cases}$$where $u(x,t), v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\geq0$ such that if $b+χ_2μ_2>χ_1μ_1+K$, then the steady solution $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ is asymptotically stable with respect to positive perturbations. In this paper, we prove that if $b+χ_2μ_2>χ_1μ_1+K$, then there exist a number $c^*(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)\geq 2\sqrt a$ such that for every $c\in ( c^*(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2) , \infty)$ and $ξ\in S^{N-1}$, the system has a traveling wave solution $(u,v_1,v_2)=(U(x\cdotξ-ct),V_1(x\cdotξ-ct),V_2(x\cdotξ-ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ and $(0,0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover we prove that$$\lim_{(χ_1,χ_2)\to(0^+,0^+)}c^{*}(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=\begin{cases}2\sqrt a\ \text{if}\ a\leq \min\{λ_1, λ_2\}\\ \frac{a+λ_1}{\sqrt{λ_1}}\ \text{if}\ λ_1\leq \min\{a, λ_2\}\\ \frac{a+λ_2}{\sqrt{λ_2}}\ \text{if}\ λ_2\leq \min\{a, λ_1\}\end{cases},\forall λ_1,λ_2,μ_1,μ_2>0,$$and$$\lim_{x\to\infty}\frac{U(x)}{e^{-\sqrt{a}μx}}=1,$$ where $μ$ solves $\sqrt a(μ+\frac{1}μ)=c$ in the interval $(0 , \min\{1,\sqrt{\frac{λ_1}{a}},\sqrt{\frac{λ_2}{a}}\})$.

math.AP↗
arXiv2017

Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources

This paper considers the dynamics of the following chemotaxis system $$ \begin{cases} u_t=Δu-χ\nabla (u\cdot \nabla v)+u\left(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_Ωu\right),\quad x\in Ω\cr 0=Δv+ u-v,\quad x\in Ω\quad \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad x\in\partialΩ, \end{cases} $$ where $Ω\subset \mathbb{R}^n(n\geq 1)$ is a bounded domain with smooth boundary $\partialΩ$ and $a_i(t,x)$ ($i=0,1,2$) are locally Hölder continuous in $t\in\mathbb{R}$ uniformly with respect to $x\in\barΩ$ and continuous in $x\in\barΩ$. We first prove the local existence and uniqueness of classical solutions $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the coefficients $a_1(t,x)$, $a_2(t,x)$, $χ$ and $n$, we prove the global existence and boundedness of classical solutions $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with given nonnegative initial function $u(x,t_0;t_0,u_0)=u_0(x)$. Then, under the same conditions for the global existence, we show that the system has an entire positive classical solution $(u^*(x,t),v^*(x,t))$. Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic in $t$ with period $T$ or are independent of $t$, then the system has a time periodic positive solution $(u^*(x,t),v^*(x,t))$ with periodic $T$ or a steady state positive solution $(u^*(x),v^*(x))$. If $a_i(t,x)$ $(i=0,1,2)$ are independent of $x$ , then the system has a spatially homogeneous entire positive solution $(u^*(t),v^*(t))$. Finally, under some further assumptions, we prove that the system has a unique entire positive solution $(u^*(x,t),v^*(x,t))$ which is globally stable . Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic or almost periodic in $t$, then $(u^*(x,t),v^*(x,t))$ is also periodic or almost periodic in $t$.

math.AP↗
arXiv2017

Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media

The current paper is devoted to the study of spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. We first prove the existence, uniqueness, and stability of spatially homogeneous entire positive solutions. Next, we establish lower and upper bounds of the (generalized) spreading speed intervals. Then, by constructing appropriate sub-solutions and super-solutions, we show the existence and continuity of transition fronts with given front position functions. Also, we prove the existence of some kind of critical front.

math.DS↗
arXiv2016

Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems

The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= Δu-χ\nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N τv_t=Δv-v+u, \quad x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ represents the population density of a mobile species and $v(x,t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We prove that for every $τ>0$, there is $0<χ_τ^*<\frac{b}{2}$ such that for every $0<χ<χ_τ^*$, there exist two positive numbers $2\sqrt a \le c^{*}(χ,τ)<c^{**}(χ,τ)$ satisfying that for every $ c\in [ c^{*}(χ,τ)\,\ c^{**}(χ,τ))$ and $ξ\in S^{N-1}$, the system has a traveling wave solution $(u(x,t),v(x,t))=(U(x\cdotξ-ct;τ),V(x\cdotξ-ct;τ))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{a}{b})$ and $(0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover, $$ \lim_{χ\to 0^+}c^{**}(χ,τ)=\infty,$$ $$ \lim_{χ\to 0^+}c^{*}(χ,τ)=\begin{cases} 2\sqrt{a}\qquad \qquad \qquad\ \text{if} \quad 0<a\leq \frac{1+τa}{(1-τ)_+}\cr \frac{1+τa}{(1-τ)_{+}}+\frac{a(1-τ)_{+}}{1+τa}\quad \text{if} \quad a\geq \frac{1+τa}{(1-τ)_+}, \end{cases} $$ and $$ \lim_{x\to -\infty}\frac{U(x;τ)}{e^{-μx}}=1, $$ where $μ$ is the only solution of the equation $μ+\frac{a}μ=c$ in the interval $(0, \min\{\sqrt a, \sqrt{\frac{1+τa}{(1-τ)_+}}\})$. Furthermore, it hods that $\lim_{τ\to 0^+}χ_τ^*=\frac{b}{2}$.

math.AP↗
arXiv2016

Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on R^N

In this paper, we study the spreading speeds and traveling wave solutions of the PDE $$ \begin{cases} u_{t}= Δu-χ\nabla \cdot (u \nabla v) + u(1-u),\ \ x\in\mathbb{R}^N 0=Δv-v+u, \ \ x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ and $v(x,t)$ represent the population and the chemoattractant densities, respectively, and $χ$ is the chemotaxis sensitivity. It has been shown in an earlier work by the authors of the current paper that, when $0<χ<1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x,t;u_0),v(x,t;u_0))$ with initial condition $u(x,0;u_0)=u_0(x)$. Furthermore, if $0<χ<\frac{1}{2}$, then the constant steady-state solution $(1,1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0<χ<1$, then there are nonnegative constants $c_{-}^*(χ)\leq c_+^*(χ)$ such that for every nonnegative initial function $u_0(\cdot)$ with nonempty compact support $$ \lim_{t\to\infty} \sup_{|x|\le ct} \big[|u(x,t;u_0)-1|+|v(x,t;u_0)-1|\big]=0\ \ \forall\ 0 c_{+}^*(χ).$$ We also show that if $0<χ<\frac{1}{2}$, there is a positive constant $c^*(χ)$ such that for every $c\ge c^*(χ)$, the system has a traveling wave solution $(u(x,t),v(x,t))$ with speed $c$ and connecting $(1,1)$ and $(0,0)$, that is, $(u(x,t),v(x,t))=(U(x-ct),V(x-ct))$ for some functions $U(\cdot)$ and $V(\cdot)$ satisfying $(U(-\infty),V(-\infty))=(1,1)$ and $(U(\infty),V(\infty))=(0,0)$. Moreover, $$ \lim_{χ\to 0}c^*(χ)=\lim_{χ\to 0}c_+^*(χ)=\lim_{χ\to 0}c_-^*(χ)=2. $$ We first give a detailed study in the case $N=1$ and next we extend these results to the case $N\ge 2$.

math.DS↗
arXiv2016

Structure of $ω$-limit Sets for Almost-periodic Parabolic Equations on $S^1$ with Reflection Symmetry

The structure of the $ω$-limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$ and satisfies $f(t,u,u_x)=f(t,u,-u_x)$. We show that any $ω$-limit set $Ω$ contains at most two minimal sets. Moreover, any hyperbolic $ω$-limit set $Ω$ is a spatially-homogeneous $1$-cover of hull $H(f)$. When $\dim V^c(Ω)=1$ ($V^c(Ω)$ is the center space associated with $Ω$), it is proved that either $Ω$ is a spatially-homogeneous, or $Ω$ is a spatially-inhomogeneous $1$-cover of $H(f)$.

math.DS↗
arXiv2016

Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc. 365 (2013), pp. 5329-5365, to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.

math.DS↗
arXiv2016

Criteria for the Existence of Principal Eigenvalues of Time Periodic Cooperative Linear Systems with Nonlocal Dispersal

The current paper {establishes} criteria for the existence of principal eigenvalues of time periodic cooperative linear nonlocal dispersal systems with Direchlet type, Neumann type or periodic type boundary conditions. It is shown that such a nonlocal dispersal system has a principal eigenvalue in the following cases: the nonlocal dispersal distance is sufficiently small; the spatial inhomogeneity satisfies a so called vanishing condition; or the spatial inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the principal eigenvalue of a time periodic cooperative linear nonlocal dispersal system (if it exists) is algebraically simple. A linear nonlocal dispersal system may not have a principal eigenvalue. The results established in the current paper extend those in literature for time independent or periodic nonlocal dispersal equations to time periodic cooperative nonlocal dispersal systems and {will} serve as {a} basic tool for the study of cooperative nonlinear systems with nonlocal dispersal.

math.AP↗
arXiv2016

Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications

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.

math.DS↗