This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion \begin{equation}\nonumber
\left\{
\begin{aligned}
&n_t+u\cdot\nabla n=\nabla\cdot\left(|\nabla n|^{p-2}\nabla n\right)-\nabla\cdot(n\nabla c), &x\inΩ,\ t>0,\ \
&c_t+u\cdot\nabla c=Δc-nc,&x\inΩ,\ t>0,\ \
&u_t=Δu+\nabla P+n\nablaϕ,&x\inΩ,\ t>0,\ \
&\nabla\cdot u=0, &x\inΩ,\ t>0,\ \
\end{aligned}
\right. \end{equation} where $ϕ\in W^{2,\infty}(Ω)$ is the gravitational potential. It is proved that global bounded weak solutions exist whenever $p>\frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0\geq 0$ and $c_0\geq 0$.
This work deals with a parabolic chemotaxis model with nonlinear diffusion and nonlocal reaction source. The problem is formulated on the whole space and, depending on a specific interplay between the coefficients associated to such diffusion and reaction, we establish that all given solutions are uniformly bounded in time.
We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant. This paper partially improves some known results in the literature and moreover solves an open question. The research is also complemented with numerical simulations in bi-dimensional domains.
We consider the Neumann problem for a coupled chemotaxis-haptotaxis model of cancer invasion with/without kinetic source in a 2D bounded and smooth domain. For a large class of cell kinetic sources including zero source and sub-logistic sources, we detect an explicit condition involving the chemotactic strength, the asymptotic "damping" rate, and the initial mass of cells to ensure uniform-in-time boundedness for the corresponding Neumann problem. Our finding significantly improves existing 2D global existence and boundedness in related chemotaxis-/haptotaxis systems.
We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann boundary conditions. More precisely, the system considered is \[ \begin{cases}\begin{split} & u_t=Δu - \nabla\cdot(\frac{u}{(1+u)^α}\nabla v) - uw + κ- u, \\ & v_t=Δv + uw - v, \\ & w_t=Δw - w + v, \end{split}\end{cases} \] with $κ\ge 0$, and solvability and boundedness of the solution are shown under the condition that \[ \begin{cases} α> \frac 12 + \frac{n^2…
In this paper, we study chemotaxis effect vs logistic dampening on boundedness for the two-dimensional minimal Keller-Segel model with logistic source in a 2-D smooth and bounded domain. It is well-known that this model allows only for global and uniform-in-time bounded solutions for any chemotactic strength and logistic dampening. Here, we carefully employ a simple and new method to regain its boundedness and, with particular attention to how boundedness depends qualitatively on the coefficient of chemotactic strength and logistic dampening rate. Up to a scaling constant depending only on ini…
In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_t=Δu -χ\nabla \cdot (\frac{u}{v} \nabla v) \\ v_t=Δv-f(u)v \end{cases} \end{equation*} in a smooth and bounded domain $Ω$ of $\mathbb{R}^2$, with $χ>0$ and $f\in C^1(\mathbb{R})$ essentially behaving like $u^β$, $0<β<1$. Precisely for $χ<1$ and any sufficiently regular initial data $u(x,0)\geq 0$ and $v(x,0)>0$ on $\barΩ$, we show the existence of global classical solutions. Moreover, if additionally $m:=\int_Ωu(x,0)$ is sufficiently small, then also their boundedness is achieved.
This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=Δu-χ\nabla\cdot (u\nabla v)+κu-μu^2,\\ v_t=Δv-uv, \end{array} \end{eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $μ$ and prove that for any $μ>0$ there exists a weak solution. Moreover, in the case of $κ>0$ convergence to the constant equilibrium $(\fracκμ,0)$ is shown.
We consider the parabolic chemotaxis model \[ u_t=Δu - χ\nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=Δv - v + u\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $χ\in(0,χ_0)$ for some $χ_0>1$, thereby proving that the value $χ=1$ is not critical in this regard. Our main tool is consideration of the energy functional \[ \mathcal{F}_{a,b}(u,v)=\int_Ωu\ln u - a \int_Ωu\ln v + b \int_Ω|\nabla \sqrt{v}|^2 \] for $a>0$, $b\geq 0$, where using nonzero values of $b$ appears to be new in this context.
We consider a chemotaxis-fluid system involving nonlinear cell diffusion of porous medium type, signal consumption by cells, and rather general, possibly matrix-valued, chemotactic sensitivities. It is shown that if the corresponding diffusion exponent $m$ satisfies $m>7/6$, then for all reasonably regaular initial data an associated initial-boundary value problem in smoothly bounded three-dimensional domains possesses a globally defined weak solution which is bounded. Under a mild additional assumption on the signal consumption rate, it is moreover shown that any nontrivial of these solutions…
This paper is concerned with the 3-dimensional two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics under homogeneous Neumann boundary conditions and initial conditions. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier--Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions …
Well-posedness and uniform-in-time boundedness of classical solutions are investigated for a three-component parabolic system which describes the dynamics of a population of cells interacting with a chemoattractant and a nutrient. The former induces a chemotactic bias in the diffusive motion of the cells and is accounted for by a density-suppressed motility. Well-posedness is first established for generic positive and non-increasing motility functions vanishing at infinity. Growth conditions on the motility function guaranteeing the uniform-in-time boundedness of solutions are next identified.…
Global existence is established for classical solutions to a chemotaxis model with signal-dependent motility for a general class of motility functions $γ$ which may in particular decay in an arbitrary way at infinity. Assuming further that $γ$ is non-increasing and decays sufficiently slowly at infinity, in the sense that $γ(s)\sim s^{-k}$ as $s\to\infty$ for some $k\in (0,N/(N-2)_+)$, it is also shown that global solutions are uniformly bounded with respect to time. The admissible decay of $γ$ at infinity here is higher than in previous works.
We consider a parabolic-parabolic Keller-Segel system of chemotaxis model with singular sensitivity $u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)$, $v_t=kΔv-v+u$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\geq2)$, with $χ,k>0$. It is proved that for any $k>0$, the problem admits global classical solutions, whenever $χ\in\big(0,-\frac{k-1}{2}+\frac{1}{2}\sqrt{(k-1)^2+\frac{8k}{n}}\big)$. The global solutions are moreover globally bounded if $n\le 8$. This shows an exact way the size of the diffusion constant $k$ of the chemicals $v$ effects th…
Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
Osaka Metropolitan University（大阪公立大学）；Institut de Mathématiques de Toulouse (IMT) UMR5219, Université Toulouse III - Paul Sabatier（图卢兹第三大学-保罗·萨巴蒂尔大学）；Université Savoie Mont Blanc, CNRS（萨瓦-蒙布朗大学）
This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=Δu-\nabla \cdot (uχ(v)\nabla v) +\nabla \cdot (uξ(w)\nabla w), &x \in Ω,\ t>0,\\[1.05mm] v_t=Δv-v+u, &x \in Ω,\ t>0,\\[1.05mm] w_t=Δw-w+u, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with smooth boundary, $χ, ξ$ are functions satisfying some conditions. Global existence and boundedness of classical solutions to the syst…
We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=Δu-\nabla \cdot(u\nabla v),& x\inΩ, t>0,\\ v_t=Δv+wz,& x\inΩ, t>0,\\ w_t=-wz,& x\inΩ, t>0,\\ z_t=Δz-z+u, & x\inΩ, t>0,\\ \end{cases} \end{equation*} in a smooth bounded domain $Ω\subset \mathbb{R}^n(n\geq 1)$ with homogeneous Neumann boundary and initial conditions. This model was recently proposed by Fujie et al. \cite{FIY14} as a model for tumor invasion with the role of extracellular matrix incorporated, and was analyzed by Fujie et al. \cite{FIWY16}, showing the uniform boundedness and co…
In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller-Segel chemotaxis-only model possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model. For negligibility of haptotax…
A semilinear version of parabolic-elliptic Keller-Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large init…
In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,\infty}(1\leq j\leq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one spec…
In this paper we consider the initial Neumann boundary value problem for a degenerate Keller--Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos--Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the moti…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nabla u)-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+μu(1- u-w), x\in Ω, t>0,\\ τv_t=Δv- v +u,\quad x\in Ω, t>0,\\ w_t=- vw,\quad x\in Ω, t>0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, where $τ\in\{0,1\}$ and $χ$, $ξ$ and $μ$ are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result…
In present paper, we consider a chemotaxis consumption system with density-signal governed sensitivity and logistic source: $u_t=Δu-\nabla\cdot(\frac{S(u)}{v}\nabla v)+ru-μu^2$, $v_t=Δv-uv$ in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\ge2)$, where parameters $r,μ>0$ and density governed sensitivity fulfills $ S(u) \simeq u(u+1)^{β-1}$ for all $u\ge0$ with $β\in \mathbb{R}$. It is proved that for any $r,μ>0$, there exists a global classical solution if $β<1$ and $n\ge2$. Moreover, the global boundedness and the asymptotic behavior of the classical solution are determined for the case $…
This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, \begin{equation*} \begin{cases} u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v) + λu - μu^κ, & x\inΩ,\ t>0, \\[1mm] 0=Δv - \overline{M_f}(t) + f(u), & x\inΩ,\ t>0, \end{cases} \end{equation*} where $λ>0$, $μ>0$, $κ>1$ and $\overline{M_f}(t):=\frac{1}{|Ω|}\int_Ω f(u(x,t))\,dx$, and $D$, $S$ and $f$ are functions generalizing the prototypes \begin{align*} D(u)=(u+1)^{m-1},\quad S(u)=u(u+1)^{α-1}\quad\mbox{and}\quad f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion \begin{align*} u_t=&\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w),\\ v_t=&Δv-v+u,\\ w_t=&-vw\end{align*} under homogeneous Neumann boundary conditions in a bounded smooth domain $Ω\subset\mathbb{R}^n$, $n=2, 3, 4$, where $χ, ξ$ and $μ$ are given nonnegative parameters. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geqδu^{m-1}$ for all $u>0$ with some $δ>0$. It is proved that for sufficiently regular initial data global bounde…
This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left\{ \begin{array}{llc} u_t=Δu-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w), &(x,t)\in Ω\times (0,T),\\ v_t=Δv-v+u, &(x,t)\inΩ\times (0,T),\\ w_t=-vw,&(x,t)\in Ω\times (0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann boundary conditions. Here $Ω\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and the parameters $ξ,χ,μ>0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $χ/μ$ is sufficiently small, ($\star$) possesses a global class…
This paper is concerned with the Neumann initial-boundary value problem for the two-species chemotaxis system with consumption of chemoattractant \begin{equation*} u_t=Δu-χ_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*} v_t=Δv-χ_2\nabla\cdot(v\nabla w), \end{equation*} \begin{equation*} w_t=Δw-(αu+βv)w \end{equation*} in a smooth bounded domain $Ω\subset\mathbb{R}^n$ ($n\geq2$), where the parameters $χ_1$, $χ_2$, $α$ and $β$ are positive. It is proved that if \begin{equation*} \max\{χ_1,χ_2\}\|w(x,0)\|_{L^{\infty}(Ω)}<\sqrt{\frac{2}{n}}π\end{equation*} the problem possesses a uniqu…
In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)+ru-μu^k$, $0=Δv-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $Ω\subset\mathbb{R}^n$, $χ,r,μ>0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{χ^2}{4}$ for $0<χ\le 2$, or $r> χ-1$ for $χ>2$. In addition, under the same condition for $r,χ$, the system admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3…
This work studies the chemotaxis-haptotaxis system $$\left\{ \begin{array}{ll} u_t= Δu - χ\nabla \cdot (u\nabla v) - ξ\nabla \cdot (u\nabla w) + μu(1-u-w), &\qquad x\in Ω, \, t>0, \\[1mm] v_t=Δv-v+u, &\qquad x\in Ω, \, t>0, \\[1mm] w_t=-vw, &\qquad x\in Ω, \, t>0, \end{array} \right. $$ in a bounded smooth domain $Ω\subset\mathbb{R}^2$ with zero-flux boundary conditions, where the parameters $χ, ξ$ and $μ$ are assumed to be positive. It is shown that under appropriate regularity assumption on the initial data $(u_0, v_0, w_0)$, the corresponding initial-boundary problem possesses a unique clas…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partialΩ$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0…
For the classical zero-flux chemotaxis-consumption model \begin{equation*} u_t= Δu - χ\nabla \cdot (u \nabla v) \quad \textrm{and}\quad v_t=Δv- uv, \quad \text{ with } (x,t)\in Ω\times (0,T_{max}), \end{equation*} $Ω$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 3$, $χ$ some positive number and $T_{max} \in (0,\infty]$, the following was established in a paper by Tao: for every sufficiently regular initial data $u(x,0)=u_0(x)\geq 0$ and $v(x,0)=v_0(x) \geq 0$, there is $χ(\lVert v_0 \rVert_{L^\infty(Ω)})$ such that for all $0<χ\leq χ(\lVert v_0 \rVert_{L^\infty(Ω)})$, the initia…
This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (G(u)χ(v)\nabla v) +\nabla\cdot(H(u)ξ(w)\nabla w), \quad v_t=d_1Δv+αu-βv, \quad w_t=d_2Δw+γu-δw, \quad x \in Ω,\ t>0, \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, α, β, γ, δ>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \…
We study the system \begin{align*} \label{prob:star} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - u - f(u) w + κ, \\ v_t = Δv - v + f(u) w, \\ w_t = Δw - w + v, \end{cases} \end{align*} which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain $Ω\subset \mathbb R^n, n \in \mathbb N,$ for a parameter $κ\ge 0$ and a given function $f \in C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^α$ for all $s \ge 1$, some $K_f \gt 0$ and $α\in \mathbb R$. We prove that whenever \begin{align*} α\lt \frac2n, \end{align*} so…
In this work, we study global existence, eventual smoothness and asymptotical behavior of positive solutions for a two-species chemotaxis consumption model in a bounded smooth but not necessarily convex domain $Ω\subset \mathbb{R}^n (n=2,3,4,5)$ with nonnegative initial data and homogeneous Neumann boundary data Under a smallness condition, boundedness of classical solutions and stabilization to constant equilibrium is known. Here, without any smallness condition, we show global existence and uniform-in-time boundedness of classical solutions in 2D and global existence, eventual smoothness and…
In this paper, the three-dimensional chemotaxis-stokes system \begin{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla n=Δn^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\inΩ,\ \ t>0, \medskip c_t+u\cdot\nabla c=Δc-nf(c),&x\inΩ,\ \ t>0, \medskip u_t+\nabla P=Δu +n\nablaϕ,&x\inΩ,\ \ t>0, \nabla\cdot u=0, &x\inΩ,\ \ t>0,, \end{array}\right. \end{eqnarray*} posed in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary is considered under the no-flux boundary condition for $n$, $c$ and the Dirichlect boundary condition for $u$ under the assumption that the Frobenius norm of the t…
This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{equation*} u_t=Δu - \nabla \cdot (u χ(v)\nabla v), \quad v_t=Δv + u - v, \quad x\inΩ,\ t>0, \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$; $χ$ is a function satisfying $χ(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established global existence of bounded solutions under the condition $K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664--1673) asserted glob…
This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=Δ(γ(v) u ) &\mathrm{in}\ Ω\times(0,\infty), \\[1mm] v_t=Δv - v + u &\mathrm{in}\ Ω\times(0,\infty), \\[1mm] \displaystyle \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 &\mathrm{on}\ \partialΩ\times (0,\infty), \\[1mm] u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 &\mathrm{in}\ Ω, \end{cases} \end{align*} where $Ω$ is a smooth bounded domain in $\mathbf{R}^n$($n \geq 3$), $γ(v)=v^{-k}$ ($k>0$) and the initial data $(u_0,v_0)$ is positive and regular. This system has striking featu…
Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.
This paper deals with the two-species chemotaxis-competition system $u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u(1 - u - a_1 v)$, $v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v(1 - a_2 u - v)$, $0 = d_3 Δw + αu + βv - γw$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $χ_i$ and $μ_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{χ_i}{μ_i}$ are small. However, the conditions in the above…
