arXiv subjects

Wenxian Shen

Publications and source records attributed to Wenxian Shen.

At least 37 records · Page 2Linked to original sources

arXiv2021

Vanishing-spreading dichotomy in a two-species chemotaxis competition system with a free boundary

Predicting the evolution of expanding population is critical to control biological threats such as invasive species and virus explosion. In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms and a free boundary. Such a model with a free boundary describes the spreading of new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. We first find conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we investigate vanishing-spreading dichotomy scenarios for positive solutions. It is shown that the vanishing-spreading dichotomy in the generalized sense always occurs; that the vanishing spreading dichotomy in the strong sense occurs when the competition between two species is weak-weak competition; and that the vanishing spreading dichotomy in the weak sense occurs when the competition between two species is weak-strong competition.

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arXiv2020

Forced waves of parabolic-elliptic Keller-Segel models in shifting environments

The current paper is concerned with the forced waves of Keller-Segel chemoattraction systems in shifting environments of the form, \begin{equation} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(r(x-ct)-bu),\quad x\in\mathbb{R}\cr 0=v_{xx}- νv+μu,\quad x\in \mathbb{R} \end{cases} (1) \end{equation} where $χ$, $b$, $ν$, and $μ$ are positive constants, $c\in R$, the resource function $r(x)$ is globally Hölder continuous, bounded, $r^*=\sup_{x\in R}r(x)>0$, $r(\pm \infty):=\lim_{x\to \pm\infty}r(x)$ exist, and either $r(-\infty)<0 2χμ$. In the case that $r(-\infty)<0 \frac{χμr^*}{2\sqrt ν(b-χμ)}- 2\sqrt{\frac{r^*(b-2χμ)}{b-χμ}}$. In the case that $r(\pm\infty)<0$, it is shown that (1) has a forced wave solution connecting $(0,0)$ and $(0,0)$ with speed $c$ provided that $χ$ is sufficiently small and $λ_\infty>0$, where $λ_\infty$ is the generalized principal eigenvalue of the operator $u(\cdot)\mapsto u_{xx}(\cdot)+cu_{x}(\cdot)+r(\cdot)u(\cdot)$ on $R$ in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.

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arXiv2020

Non-wandering points for autonomous/periodic parabolic equations on the circle

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle $S^1$, \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 independent of $t$ or $T$-periodic in $t$. Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some $ω$-limit set). More precisely, in the autonomous case, it is proved that any non-wandering point is either a fixed point or generates a rotating wave on the circle. In the periodic case, it is proved that any non-wandering point is a periodic point or generates a rotating wave on a torus. In particular, if $f(t,u,-u_x)=f(t,u,u_x)$, then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case.

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arXiv2020

Almost automorphically forced flows on $S^1$ or $\mathbb{R}$ in one-dimensional almost periodic semilinear heat equations

In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L \end{equation} with periodic boundary condition \begin{equation} \label{bdc1} u(t,0)=u(t,L),\quad u_x(t,0)=u_x(t,L), \end{equation} where $f$ is uniformly almost periodic in $t$. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we refer to it as a minimal set) can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle $S^1=\mathbb{R}/L\mathbb{Z}$. Particularly, if $f(t,u,p)=f(t,u,-p)$, then the flow on a minimal set topologically conjugates to an almost periodically-forced minimal flow on $\mathbb{R}$. Moreover, it is proved that the $ω$-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation. In addition, we further consider the asymptotic dynamics of the skew-product semiflow generated by \eqref{eq0} with Neumann boundary condition \begin{equation*} \label{bcd2} u_x(t,0)=u_x(t,L)=0, \end{equation*} or Dirichlet boundary condition \begin{equation*}\label{bdc3} u(t,0)=u(t,L)=0. \end{equation*} Under certain direct assumptions on $f$, it is proved in this paper that the flow on any minimal set of \eqref{eq0}, with Neumann boundary condition or Dirichlet boundary condition, topologically conjugates to an almost periodically-forced minimal flow on $\mathbb{R}$. Finally, a counterexample is given to show that even for quasi-periodic equations, the results we obtain here cannot be further improved in general.

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arXiv2020

Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments

The current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains. We first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models. Next, under the same conditions on the coefficients, we show that pointwise { persistence} occurs, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough. It should be pointed out that in \cite{TaoWin15}, the authors established the persistence of mass for globally defined positive solutions, which indicates that any extinction phenomenon, if occurring at all, necessarily must be spatially local { in }nature, whereas the population as a whole always persists. The pointwise persistence proved in the current paper implies that not only the population as a whole persists, but also it persists at any location eventually. It also implies the existence of strictly positive entire solutions.

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arXiv2020

Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments

The current paper is concerned with the persistence and spreading speeds of the following Keller-Segel chemoattraction system in shifting environments, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(r(x-ct)-bu),\quad x\in\R\cr 0=v_{xx}- νv+μu,\quad x\in\R, \end{cases} \end{equation} where $χ$, $b$, $ν$, and $μ$ are positive constants, { $c\in\R$ }, $r(x)$ is Hölder continuous, bounded, $r^*=\sup_{x\in\R}r(x)>0$, $r(\pm \infty):=\lim_{x\to \pm\infty}r(x)$ exist, and $r(x)$ satisfies either $r(-\infty)<0 χμ$ and $b\ge \big(1+\frac{1}{2}\frac{(\sqrt{r^*}-\sqrtν)_+}{(\sqrt{r^*}+\sqrtν)}\big)χμ$. In the case that $r(-\infty)<0 c^*:=2\sqrt{r^*}$, then the species becomes extinct in the habitat. If the moving speed $ -c^*\leq c c^*$, then the species will become extinct in the habitat. If $λ_{\infty}$, defined to be the generalized principle eigenvalue of the operator $u\to u_{xx}+cu_{x}+r(x)u$, is negative and the degradation rate $ν$ of the chemo-attractant is grater than or equal to some number $ν^*$, then the species will also become extinct in the habitat. If $λ_{\infty}>0$, then the species will persist surrounding the good habitat.

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arXiv2020

Long time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(θ_tω)u(1-u),\quad x\in\mathbb{R} $$ where $ω\inΩ$, $(Ω, \mathcal{F},\mathbb{P})$ is a given probability space, $θ_t$ is an ergodic metric dynamical system on $Ω$, and $a(ω)>0$ for every $ω\inΩ$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\mathbb{R}$$ where $a_0(t)$ is a positive locally Hölder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $a(ω)$, we show that the constant solution $u=1$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $[2\sqrt{\underline a}, 2\sqrt{\bar a}]$, where $\underline{a}$ and $\bar a$ are the least and the greatest means of $a(\cdot)$, respectively, and hence the spreading speed interval is linearly determinant. It is shown that the solution of (1) with the initial function which is bounded away from $0$ for $x\ll -1$ and is $0$ for $x\gg 1$ propagates at the speed $2\sqrt {\hat a}$, where $\hat a$ is the average of $a(\cdot)$. Under some assumption on $a_0(\cdot)$, we also show that the constant solution $u=1$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $a(ω)$ and $a_0(t)$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher-KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).

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arXiv2020

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II. Spreading-vanishing dichotomy in a domain with a free boundary

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first of the series, we investigated the dynamical behaviors of logistic type chemotaxis models on the half line $\mathbb{R}^+$, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

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arXiv2019

Persistence and Extinction of Nonlocal Dispersal Evolution Equations in Moving Habitats

This paper is devoted to the study of persistence and extinction of a species modeled by nonlocal dispersal evolution equations in moving habitats with moving speed $c$. It is shown that the species becomes extinct if the moving speed $c$ is larger than the so called spreading speed $c^*$, where $c^*$ is determined by the maximum linearized growth rate function. If the moving speed $c$ is smaller than $c^*$, it is shown that the persistence of the species depends on the patch size of the habitat, namely, the species persists if the patch size is greater than some number $L^*$ and in this case, there is a traveling wave solution with speed $c$, and it becomes extinct if the patch size is {smaller} than $L^*$.

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arXiv2019

Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(a-bu),\quad x\in\R\cr 0=v_{xx}- λv+μu,\quad x\in\R, \end{cases} \end{equation} where $χ$, $a$, $b$, $λ$, and $μ$ are positive constants. Assume $b>χμ$. Then if in addition, $\big(1+\frac{1}{2}\frac{(\sqrt{a}-\sqrtλ)_+}{(\sqrt{a}+\sqrt{\la})}\big)χμ{ \leq} b$ holds, it is proved that $c_0^*=2\sqrt a$ is the spreading speed of the solutions of \eqref{abstract-eq1} with nonnegative continuous initial function $u_0$ with nonempty compact support, that is, $$ \limsup_{|x|\ge ct, t\to\infty}u(t,x;u_0)=0\quad \forall\, c>c_0^* $$ and $$ \liminf_{|x|\le ct,t\to\infty} u(t,x;u_0)>0\quad \forall \, 0 2χμ$ and $λ\geq a$ holds, then $c_0^*=2\sqrt a$ is the minimal speed of the traveling wave solutions of \eqref{abstract-eq1} connecting $(0,0)$ and $(\frac{a}{b},\fracμλ\frac{a}{b})$, that is, for any $c\ge c_0^*$, \eqref{abstract-eq1} has a traveling wave solution connecting $(0,0)$ and $(\frac{a}{b},\fracμλ\frac{a}{b})$ with speed $c$, and \eqref{abstract-eq1} has no such traveling wave solutions with speed less than $c_0^*$. Note that $c_0^*=2\sqrt a$ is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, \begin{equation} \label{abstract-eq2} u_t=u_{xx}+u(a-bu),\quad x\in\R. \end{equation}

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arXiv2019

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line $\mathbb{R}^+$, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

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arXiv2019

Stability and uniqueness of generalized traveling waves of lattice Fisher-KPP equations in heterogeneous media

In this paper, we investigate the stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations with general time and space dependence. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability and uniqueness of generalized traveling waves connecting the unique strictly positive entire solution and the trivial solution zero. Applying the general stability and uniqueness theorem, we then prove the existence, stability and uniqueness of periodic traveling wave solutions of lattice Fisher-KPP equations in time and space periodic media, and the existence, stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations in time heterogeneous media. The general stability result established in this paper implies that the generalized traveling waves obtained in many cases are asymptotically stable under well-fitted perturbation.

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arXiv2018

Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media

In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions, and then obtain some sufficient conditions ensuring the uniform persistence. In particular, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.

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arXiv2018

Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments

The current paper is concerned with the asymptotic dynamics of two species competition systems with/without chemotaxis in heterogeneous media. In the previous work \cite{ITBWS17a}, we find conditions on the parameters in such systems for the persistence of the two species and the existence of positive coexistence states. In this paper, we find conditions on the parameters for the uniqueness and stability of positive coexistence states of such systems. The established results are new even for the two species competition systems without chemotaxis but with space dependent coefficients.

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arXiv2018

Long time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(θ_tω)u(1-u),\quad x\in\R, \eqno(1) $$ where $ω\inΩ$, $(Ω, \mathcal{F},\mathbb{P})$ is a given probability space, $θ_t$ is an ergodic metric dynamical system on $Ω$, and $a(ω)>0$ for every $ω\inΩ$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\R, \eqno(2) $$ where $a_0(t)$ is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1) and (2). In this second part of the series, we investigate the existence and stability of transition fronts of (1) and (2). We first study the transition fronts of (1). Under some proper assumption on $a(ω)$, we show the existence of random transition fronts of (1) with least mean speed greater than or equal to some constant $\underline{c}^*$ and the nonexistence of ranndom transition fronts of (1) with least mean speed less than $\underline{c}^*$. We prove the stability of random transition fronts of (1) with least mean speed greater than $\underline{c}^*$. Note that it is proved in the first part that $\underline{c}^*$ is the infimum of the spreading speeds of (1). We next study the existence and stability of transition fronts of (2). It is not assumed that $a(ω)$ and $a_0(t)$ are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.

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arXiv2018

Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments

In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms in heterogeneous media. We first find various conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we find further conditions on the parameters which establish the persistence of the two species. Then, under the same set of conditions for the persistence of two species, we prove the existence of coexistence states. Finally we prove the extinction phenomena in the sense that one of the species dies out asymptotically and the other reaches its carrying capacity as time goes to infinity. The persistence in general two species chemotaxis systems is studied for the first time. Several important techniques are developed to study the persistence and coexistence of the two species chemotaxis systems. Many existing results on the persistence, coexistence, and extinction on two species competition systems without chemotaxis are recovered.

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arXiv2018

Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading

The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$\partial_tu=Δu-χ\nabla(u\nabla v)+u(a(x,t)-ub(x,t))\ ,\ 0=Δv-λv+μu \ \ (1)$$where $χ, λ, μ$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0\in R$ and every $u_0\in C^{b}_{\rm unif}(R^N)$, $u_0\geq 0$. Next we show the pointwise persistence phenomena in the sense that, for any solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ of (1) with strictly positive initial function $u_0$, then$$0<\inf_{t_0\in R, t\geq 0}u(x,t+t_0;t_0,u_0)\le\sup_{t_0\in R, t\geq 0} u(x,t+t_0;t_0,u_0)<\infty$$and show the uniform persistence phenomena in the sense that there are $0 0$ such that$$m\le u(x,t+t_0;t_0,u_0)\le M\ \forall\,t\ge T(u_0),\ x\in R^N.$$We then discuss the spreading properties of solutions to (1) with compactly supported initial and prove that there are positive constants $0 c_+^*,$$and$$\liminf_{t\to\infty}\sup_{|x|\le ct}u(x,t+t_0;t_0,u_0)>0, \ \forall 0<c<c_-^*.$$We also discuss the spreading properties of solutions to (1) with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions and the existence of transition fronts, respectively.

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arXiv2018

Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. II. Existence, uniqueness, and stability of strictly positive entire solutions

This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$$\partial_tu=Δu-χ\nabla\cdot(u\nabla v)+u(a(x,t)-ub(x,t)),\quad 0=Δv-λv+μu ,\ x\in\mathbb{R}^N, \ (1) $$where $N\ge1$ is a positive integer, $χ,λ,μ>0$, and the functions $a(x,t), b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In this direction, we prove that, if $0\leμχ 0$, where $(u_χ(x,t+t_0;t_0,u_0),v_χ(x,t+t_0;t_0,u_0))$ denotes the unique classical solution of (1) with $u_χ(x,t_0;t_0,u_0)=u_0(x)$, for every $0\leχ<b_\inf$.

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