arXiv subjects

Wenxian Shen

Publications and source records attributed to Wenxian Shen.

At least 19 recordsLinked to original sources

arXiv2026

Existence, uniqueness, stability, and monotonicity of traveling waves for repulsion/attraction chemotaxis models with logistic type source

This paper is devoted to the study of existence, uniqueness, stability, and monotonicity of traveling wave solutions to the following parabolic-elliptic chemotaxis system with logistic type source \begin{equation}\label{E:main-abstract-eq}\tag{CM} \begin{cases} u_t=u_{xx}-χ(u^m v_x)_x +u(1-u^α),\quad &x\in\mathbb{R}\cr 0=v_{xx}-v+u^γ,\quad&x\in\mathbb{R}, \end{cases} \end{equation} where $m,α,γ\ge 1$ and $χ\in\mathbb{R}$. System (CM) can be used to describe the evolution of a biological species influenced by a chemical substance produced by the species itself. In this context, the function $u$ denotes the population density of the biological species, and $v$ denotes the concentration of the chemical agent. Traveling wave solutions of (CM) connecting the two constant solutions $(1,1)$ and $(0,0)$ are among important types of solutions, which characterize the front propagation phenomena in (CM). The existence of such traveling wave solutions to (CM) with $m=α=γ=1$ has been studied in several papers. However, there is little study on the uniqueness, stability, and monotonicity of traveling wave solutions of (CM) in literature and there is also no study on the existence of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ for general $m,α,γ\ge 1$. In the current paper, we prove the existence of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ for any $χ\le 0$ with speed $c$ large than some number $c^*_{χ,m,γ}$, or for $0<χ<1/2$ with any speed $c>2$. We prove that the traveling wave solutions are monotone when $χ\le 0$. We also prove the uniqueness and stability of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ when the speed $c$ is larger than some number $c^{**}_{χ,m,α, γ}(\ge c^*_{χ,m,γ})$.

math.AP↗

arXiv2026

Chemotaxis models with signal-dependent sensitivity and a logistic-type source, II: Persistence and stabilization

This paper is Part II of a series on global existence and asymptotic behavior of positive solutions to \begin{equation*} \begin{cases} \displaystyle u_t=Δu-χ_0\nabla\cdot\left(\frac{u^m}{(1+v)^β}\nabla v\right)+au-bu^{1+α}, & x\inΩ, \cr \displaystyle 0=Δv-μv+νu^γ, & x\inΩ, \cr \displaystyle \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N$ is a bounded and smooth domain. The parameters $α,γ,m,μ,ν$ are positive, $χ_0$ is real, and $a,b,β$ are nonnegative. In Part I, we established boundedness and global existence. Here, we study persistence and stabilization, quantifying how $β$ and $χ_0$ influence long-time dynamics. First, we prove uniform persistence if $m\ge 1$. Next, for $a,b>0$, the unique positive equilibrium is $(u^*,v^*) = \left((\tfrac{a}{b})^{1/α},(\tfracνμ)(\tfrac{a}{b})^{γ/α}\right)$. We identify a threshold $χ^*(u^*)$: $(u^*,v^*)$ is linearly stable if $χ_0<χ^*(u^*)$, with local exponential decay, unstable if $χ_0>χ^*(u^*)$. We also give conditions ensuring every bounded solution converges exponentially to $(u^*,v^*)$. For $a=b=0$, we study stability of the constant equilibria under mass constraint, obtaining a linear stability threshold and global stabilization. We extend the Lyapunov method from $m=1$ to $m>1$ and the rectangle/ODE method from $β=0$ to $β>0$. For $m\ge 1$, signal saturation (large $β$) or repulsion ($χ_0<0$) prevents aggregation and promotes relaxation. In Part III, we study bifurcation and pattern formation when $χ_0$ passes through critical thresholds.

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arXiv2025

Chemotaxis models with signal-dependent sensitivity and a logistic-type source, I: Boundedness and global existence

We study, in Part I of this series, boundedness and global existence of positive classical solutions to a parabolic-elliptic chemotaxis system with signal-dependent sensitivity and a logistic-type source on a bounded smooth domain $Ω\subset\mathbb{R}^N$: \begin{equation*} \begin{cases} \displaystyle u_t=Δu-χ_0\nabla\cdot\left(\frac{u^m}{(1+v)^β}\nabla v\right)+au-bu^{1+α}, & x\inΩ, \cr \displaystyle 0=Δv-μv+νu^γ, & x\inΩ, \cr \displaystyle \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ. \end{cases} \end{equation*} Here, $u$ denotes the population density and $v$ the chemical concentration. The parameters $α,γ,m,μ,ν$ are positive, $χ_0$ is real, and $a,b,β$ are nonnegative. We analyze boundedness from three viewpoints: negative chemotaxis ($χ_0<0$), the strength of the nonlinear cross diffusion rate $\frac{u^m}{(1+v)^β}$, and the strength of the logistic-type damping $u(a-bu^α)$. Under explicit conditions reflecting these mechanisms, all positive classical solutions remain bounded. Moreover, when $m\ge 1$, boundedness implies global existence. Although the decay of $χ(v) = \dfrac{χ_0}{(1+v)^β}$ for large $v$ has a damping effect, it also introduces new analytical difficulties; our techniques yield, for example, global existence for $m=1$ provided that \begin{equation*} β>\max\left\{1,\frac12+\frac{χ_0}{4}\max\{2,γN\}\right\}. \end{equation*} Several known results for special cases are recovered. Part II is devoted to the asymptotic behavior of globally defined solutions, including uniform persistence as well as stability and bifurcation of positive constant equilibria.

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arXiv2025

The spreading of global solutions of chemotaxis systems with logistic source and consumption on $\mathbb{R}^{N}$

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=Δu - χ\nabla\cdot(u\nabla v)+ u(a-bu),\quad &(t,x)\in [0,\infty)\times\mathbb{R}^N, \\ &{τ\partial_tv}=Δv-uv,\quad & (t,x)\in [0,\infty)\times\mathbb{R}^N, \end{aligned} \right. \] where $u(t,x)$ represents the population density of a biological species, and $v(t,x)$ denotes the density of a chemical substance. Key findings of this study include: (i) the species spreads at least at the speed $c^*=2\sqrt a$ (equalling the speed when $v\equiv 0$), suggesting that the chemical substance does not hinder the spreading; (ii) the chemical substance does not induce infinitely fast spreading of $u$; (iii) the spreading speed remains unaffected under conditions that $v(0,\cdot)$ decays spatially or $0<-χ\ll 1$ and $τ=1$. Additionally, our numerical simulations reveal a noteworthy phase transition in $χ$: for $v(0, \cdot)$ uniformly distributed across space, the spreading speed accelerates only when $χ$ surpasses a critical positive value.

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arXiv2025

Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on $\mathbb{R}^N$: I. Global Solvability and Boundedness

This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption: \begin{equation*} \begin{cases} u_t = m\nabla\cdot \left((\eps+u)^{m-1}\nabla u\right) - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{ in } (0,\infty)\times\mathbb{R}^N, \\ v_t = Δv - uv, & \text{ in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where $m > 1$ and $\eps \geq 0$. The present paper focuses on the global solvability and boundedness of weak solutions. For general bounded initial data, which may be non-integrable, we prove the existence of global weak solutions that remain uniformly bounded for all times. The proof relies on deriving local $L^p$ estimates that are uniform in time via a new continuity-type argument and obtaining $L^\infty$ bounds using Moser's iteration; all of these estimates are uniform as $\eps\to0$. In part II, we will study the regularity and uniqueness of weak solutions.

math.AP↗

arXiv2024

Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics

The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i,a_i, b_i, c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. In [25], we proved that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$, and $u_0+v_0$ is not small in the case that $(χ_1-χ_2)^2\le \max\{4χ_1,4χ_2\}$ and $u_0+v_0$ is neither small nor big in the case that $(χ_1-χ_2)^2>\max\{4χ_1,4χ_2\}$. In this paper, we proved that (0.1) has a unique positive constant solution $(u^*,v^*,w^*)$, where $$ u^*=\frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2},\quad v^*=\frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^*=\fracνμu^*+\fracλμ v^*. $$ We obtain some explicit conditions on $χ_1,χ_2$ which ensure that the positive constant solution $(u^*,v^*,w^*)$ is globally stable in the sense that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0\not \equiv 0$ and $v_0\not \equiv 0$, $$ \lim_{t\to\infty}\Big(\|u(t,\cdot;u_0,v_0)-u^*\|_\infty +\|v(t,\cdot;u_0,v_0)-v^*\|_\infty+\|w(t,\cdot;u_0,v_0)-w^*\|_\infty\Big)=0. $$

math.AP↗

arXiv2024

Well-posedness of Keller-Segel systems on compact metric graphs

Chemotaxis phenomena govern the directed movement of micro-organisms in response to chemical stimuli. In this paper, we investigate two Keller--Segel systems of reaction-advection-diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic type Keller--Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions.

math.AP↗

arXiv2024

Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i$, $a_i$, $b_i$, $ c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$ and $u_0+v_0$ is not small. Moreover, under the same condition, we prove that \begin{equation*} \limsup_{t\to\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\le M^*, \end{equation*} and \begin{equation*} \liminf_{t\to\infty} \inf_{x\inΩ}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*, \end{equation*} for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.

math.AP↗

arXiv2024

Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

In the current paper, we study stability, bifurcation, and spikes of positive stationary solutions of the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{cases} u_t=u_{xx}-χ(\frac{u}{v} v_x)_x+u(a-b u), & 0 0,\cr 0=v_{xx}- μv+ νu, & 0 0 \cr u_x(t,0)=u_x(t,L)=v_x(t,0)=v_x(t,L)=0, & t>0, \tag{1} \end{cases} where $χ$, $a$, $b$, $μ$, $ν$ are positive constants. Among others, we prove there are $χ^*>0$ and $\{χ_k^*\}\subset [χ^*,\infty)$ ($χ^*\in\{χ_k^*\}$) such that the constant solution $(\frac{a}{b},\fracνμ\frac{a}{b})$ of (1) is locally stable when $0<χ<χ^*$ and is unstable when $χ>χ^*$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ_k^*$, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions $\{(u(\cdot;χ_n),v(\cdot;χ_n))\}$ of (1) with $χ=χ_n(\to \infty)$ develops spikes at any $x^*$ satisfying $\liminf_{n\to\infty} u(x^*;χ_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ^*$ can be extended to $χ=\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $χ\gg 1$ and develop spikes as $χ\to\infty$.

math.AP↗

arXiv2024

Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions

This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(t,x)-b(t,x) u), & x\in Ω,\cr 0=Δv- μv+ νu, & x\in Ω, \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $a(t,x)$ and $b(t,x)$ are positive smooth functions, and $χ$, $μ$ and $ν$ are positive constants. In the very recent paper [25], we proved that for given nonnegative initial function $0\not\equiv u_0\in C^0(\bar Ω)$ and $s\in\mathbb{R}$, (0.1) has a unique globally defined classical solution $(u(t,x;s,u_0),v(t,x;s,u_0))$ with $u(s,x;s,u_0)=u_0(x)$, provided that $a_{\inf}=\inf_{t\in\mathbb{R},x\inΩ}a(t,x)$ is large relative to $χ$ and $u_0$ is not small. In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption that $a_{\inf}$ is large relative to $χ$ and $u_0$ is not small. Among others, we provide some concrete estimates for $\int_Ωu^{-p}$ and $\int_Ωu^q$ for some $p>0$ and $q>\max\{2,N\}$ and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a ``rectangular'' type bounded invariant set (in $L^q$) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution $(u^*(t,x),v^*(t,x))$, which is periodic in $t$ if $a(t,x)$ and $b(t,x)$ are periodic in $t$ and is independent of $t$ if $a(t,x)$ and $b(t,x)$ are independent of $t$.

math.AP↗

arXiv2024

Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current paper is to study the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u^{1+σ}),\quad &x\in Ω\cr 0=Δv-μv+νu,\quad &x\in Ω\quad \cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partialΩ$, $χ$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $μ,ν$ are positive constants, and $σ\ge 0$. When $σ>0$, we prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar Ω)$, (0.1) has a unique globally defined classical solution $(u_σ(x,t;u_0),v_σ(x,t;u_0))$ with $u_σ(x,0;u_0)=u_0(x)$, which shows that, in any space dimensional setting, strong logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $χ$. In addition, the solutions are shown to be uniformly bounded under the conditions \begin{equation*} a_{\inf}> \begin{cases} \frac{μχ^2}{4}, &\text{if $0< χ\leq 2,$}\\ μ(χ-1), &\text{if $χ>2$.}\\ \end{cases} \end{equation*} When $σ=0$, we show that the classical solution $(u(x,t;u_0,0),v(x,t;u_0,0))$ exists globally and stays bounded provided that both $a(x,t)$ and $u_0(x)$ are not small.

math.AP↗

arXiv2023

Global existence of classical solutions of chemotaxis systems with logistic source and consumption or linear signal production on $\mathbb{R}^{n}$

While much literature on chemotaxis systems focuses on bounded domains, this paper emphasizes the global existence of classical solutions for three primary chemotaxis systems with a logistic source on $\mathbb{R}^n$. We present a unified proof demonstrating global existence of solutions can be deduced from their locally uniform boundedness in $L^p(\mathbb{R}^n)$ for some $p>\max\{1,\frac{n}{2}\}$. We then provide sufficient conditions for the global existence and boundedness of classical solutions. Notably, our findings even improve several existing results for bounded domains.

math.AP↗

arXiv2023

Stability and bifurcation for logistic Keller--Segel models on compact graphs

This paper concerns asymptotic stability, instability, and bifurcation of constant steady state solutions of the parabolic-parabolic and parabolic-elliptic chemotaxis models on metric graphs. We determine a threshold value $χ^*>0$ of the chemotaxis sensitivity parameter that separates the regimes of local asymptotic stability and instability, and, in addition, determine the parameter intervals that facilitate global asymptotic convergence of solutions with positive initial data to constant steady states. Moreover, we provide a sequence of bifurcation points for the chemotaxis sensitivity parameter that yields non-constant steady state solutions. In particular, we show that the first bifurcation point coincides with threshold value $χ^*$ for a generic compact metric graph. Finally, we supply numerical computation of bifurcation points for several graphs.

math.AP↗

arXiv2022

Non-local dispersal equations with almost periodic dependence. II. Asymptotic dynamics of Fisher-KPP equations

This series of two papers is devoted to the study of the principal spectral theory of nonlocal dispersal operators with almost periodic dependence and the study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In the first part of the series, we investigated the principal spectral theory of nonlocal dispersal operators from two aspects: top Lyapunov exponents and generalized principal eigenvalues. Among others, we provided various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, established the relations between them, and studied the effect of time and space variations on them. In this second part of the series, we study the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence applying the principal spectral theory developed in the first part. In particular, we study the existence, uniqueness, and stability of almost periodic solutions of Fisher KPP equations with nonlocal dispersal and almost periodic dependence. By the properties of the asymptotic dynamics of nonlocal dispersal Fisher-KPP equations, we also establish a new property of the generalized principal eigenvalues of nonlocal dispersal operators in this second part of the series.

math.AP↗

arXiv2021

Nonlocal dispersal equations with almost periodic dependence. I. Principal spectral theory

math.AP↗

arXiv2021

Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N},\cr {v_t}=Δv -λv+μu,\quad x\in \mathbb{R}^{N}. \end{cases}(1) \end{equation} where $χ, \ a,\ b,\ λ,\ μ$ are positive constants. Assume $b>\frac{Nμχ}{4}$. Among others, it is proved that $2\sqrt{a}$ is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is, $$ \lim_{t\to\infty}\sup_{|x|\geq ct}u(x,t;u_0,v_0)=0\quad \forall\,\, c>2\sqrt{a} $$ and $$ \liminf_{t\to\infty}\inf_{|x|\leq ct}u(x,t;u_0,v_0)>0 \quad \forall\,\, 0 \frac{Nμχ}{4}$, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.

math.AP↗

arXiv2021

Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N}\,\,\, t>0\cr {v_t}=Δv -λv+μu,\quad x\in \mathbb{R}^{N}\,\,\, t>0 \end{cases}(1) \end{equation} where $χ, \ a,\ b,\ λ,\ μ$ are positive constants and $N$ is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption $b>\frac{Nχμ}{4}$, the global existence of a unique classical solution $(u(x,t;u_0, v_0),v(x,t;u_0, v_0))$ of (1) with $u(x,0;u_0, v_0)=u_0(x)$ and $v(x,0;u_0, v_0)=v_0(x)$ for every nonnegative, bounded, and uniformly continuous function $u_0(x)$, and every nonnegative, bounded, uniformly continuous, and differentiable function $v_0(x)$. Next, under the same assumption $b>\frac{Nχμ}{4}$, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function $u_0$ is bounded below by a positive constant independent of $(u_0, v_0)$ when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function $u_0$. We show that there is $K=K(a,λ,N)>\frac{N}{4}$ such that if $b>K χμ$ and $λ\geq \frac{a}{2}$, then for every strictly positive initial function $u_0(\cdot)$, it holds that $$\lim_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\fracμλ\frac{a}{b}\|_{\infty}\big]=0.$$

math.AP↗

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.

math.AP↗