Repulsion Chemotaxis Research Articles

Dissipative Gradient Nonlinearities Prevent δ$\delta$‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models

ABSTRACTWe study a class of zero‐flux attraction–repulsion chemotaxis models, characterized by nonlinearities laws for the diffusion of the cell density , the chemosensitivities and the production rates of the chemoattractant and the chemorepellent . In addition, a source involving also the gradient of is incorporated. Our overall study touches on different aspects: we address questions connected to local well‐posedness, we derive sufficient conditions to ensure boundedness of solutions, and finally, we develop numerical simulations giving insights into the evolution of the system.

Global boundedness in an attraction–repulsion Chemotaxis system with nonlinear productions and logistic source

This paper deals with the attraction–repulsion chemotaxis system with nonlinear productions and logistic source, ut=∇⋅(D(u)∇u)−∇⋅(Φ(u)∇v)+∇⋅(Ψ(u)∇w)+f(u),vt=Δv+αuk−βv,τwt=Δw+γul−δw,τ∈{0,1},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned}& u_{t} = \ abla \\cdot \\bigl( D(u) \ abla u \\bigr) - \ abla \\cdot \\bigl( \\Phi (u) \ abla v \\bigr) + \ abla \\cdot \\bigl( \\Psi (u) \ abla w \\bigr) + f(u),\\\\& v_{t} = \\Delta v+\\alpha {{u}^{k}}-\\beta v,\\qquad \ au w_{t} = \\Delta w+\\gamma {{u}^{l}}-\\delta w,\\quad \ au \\in \\{0,1 \\}, \\end{aligned}$$ \\end{document} in a bounded domain Ω⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset {{\\mathbb{R}}^{n}}$\\end{document} (n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n \\ge 1 $\\end{document}), subject to the homogeneous Neumann boundary conditions and initial conditions, where D,Φ,Ψ∈C2[0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D,\\Phi ,\\Psi \\in {{C}^{2}}[0,\\infty )$\\end{document} are nonnegative with D(s)≥(s+1)p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D(s)\\ge {{(s+1)}^{p}}$\\end{document} for s≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s\\ge 0$\\end{document}, Φ(s)≤χsq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Phi (s)\\le \\chi {{s}^{q}}$\\end{document}, ξsg≤Ψ(s)≤ζsj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\xi {{s}^{g}}\\le \\Psi (s) \\le \\zeta s^{j}$\\end{document}, s≥s0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s\\ge {{s}_{0}}$\\end{document}, for s0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${{s}_{0}}>1$\\end{document}, the logistic source satisfies f(s)≤s(a−bsd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(s)\\le s(a-b{{s}^{d}})$\\end{document}, s>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s>0$\\end{document}, f(0)≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(0)\\ge 0$\\end{document}, and the nonlinear productions for the attraction and repulsion chemicals are described via αuk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha {{u}^{k}}$\\end{document} and γul\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\gamma {{u}^{l}}$\\end{document}, respectively. When k=l=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k=l=1$\\end{document}, it is known that this system possesses a globally bounded solution in some cases. However, there has been no work in the case k,l>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k,l>0$\\end{document}. This paper develops the global boundedness of the solution to the system in some cases and extends the global boundedness criteria established by Tian, He, and Zheng (2016) for the attraction–repulsion chemotaxis system.

Schoener-Polis-Holt's model of the intraguild predation with predator taxis and repulsive chemotaxis

Schoener-Polis-Holt's model of the intraguild predation with predator taxis and repulsive chemotaxis

Finite-time blow-up and boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear signal productions

This article is focused on the no-flux attraction–repulsion chemotaxis model ut=∇⋅(D(u)∇u−S(u)∇v+T(u)∇w)+h(u),x∈Ω,t>0,0=Δv−α(t)+f(u),x∈Ω,t>0,0=Δw−β(t)+g(u),x∈Ω,t>0defined in a smooth and bounded domain Ω⊂Rn(n≥2) and subjected to homogeneous Neumann boundary conditions. The functions D,S and T suitably generalize the singular prototypes D(s)=(s+1)m1−1,S(s)=s(s+1)m2−1,T(s)=s(s+1)m3−1,s≥0,m1,m2,m3∈R,and f and g extend the prototypes f(s)=(s+1)γ1,g(s)=(s+1)γ2,s≥0,γ1,γ2>0,as well as the logistic dampening satisfies h(s)=ks−μsl with k∈R, μ>0, l≥1. Under alternative conditions, we establish the global existence and boundedness of classical solutions to the above model. However, when m2=m3 and no logistic source is taken into account, the finite time blow-up phenomenon preserves provided that γ1>γ2 and γ1>m1∗−m2+2n with m1∗=max{1,m1}.

Global boundedness in a two-species attraction–repulsion chemotaxis system with two chemicals and nonlinear productions

This paper studies the attraction–repulsion chemotaxis system of two-species with two chemicals ut=Δu−χ1∇⋅(u∇¯v)+f1(u), vt=Δv−v+wγ1, wt=Δw+χ2∇⋅(w∇z)+f2(w), 0=Δz−z+uγ2, subject to the homogeneous Neumann boundary conditions in a bounded domain Ω⊂RN (N≥2) with smooth boundary, where the parameters χi,γi>0(i=1,2), and the logistic sources fi(s)∈C2[0,∞) satisfy fi(s)≤μis(1−sθi) with μi,θi>0, fi(0)≥0 (i=1,2). The interactions among the diffusion, attraction, repulsion, logistic sources, and nonlinear productions in the system determine the behavior of solutions. It is obtained that the solutions would be globally bounded whenever the nonlinear productions are dominated by one of the following three mechanisms: (i) the diffusion with γ1<2N, or γ2<4N with γ2≤1; (ii) the logistic sources with min{θ1,θ2}≥max{γ1,γ2}, and (iii) the cooperation of diffusion and logistic sources with θ1+1>γ2min{1+N2,1+Nγ12θ2}, or θ2+1>γ1min{1+N2,1+max{N4,1}γ2θ1}.

Global existence of solutions to the 4D attraction–repulsion chemotaxis system and applications of Brezis–Merle inequality

We consider the Cauchy problem for an attraction–repulsion chemotaxis system in the four space dimension. One of main topics in the study of such a system is the presence of L 1 threshold. In fact, the critical mass phenomenon called 8π-problem is well-known in the two-dimensional setting. In this paper, we show the global existence of solutions to the Cauchy problem for the four-dimensional subcritical case, that is, the total mass of the initial data is less than the four-dimensional L 1 threshold value . The key ingredients are the four-dimensional Brezis–Merle type inequality of the 4th-order elliptic equation and some inequalities derived from rearrangement arguments.

Speed-up of traveling waves by negative chemotaxis

We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of repulsive chemotaxis and provide an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale of chemotaxis. Our study is based on establishing the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-Keller-Segel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPP-Keller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish a new explicit lower bound on the minimal traveling wave speed and provide a novel construction of traveling waves that extends the known existence range to all parameter values.

Critical mass phenomenon in a chemotaxis fluid system

In this paper, we study the following parabolic–parabolic–elliptic attraction–repulsion chemotaxis fluid system driven by a friction force n∇(χ1c−χ2v): (0.1)nt+u⋅∇n=Δn−∇⋅(χ1n∇c)+∇⋅(χ2n∇v),x∈Ω,t>0,ct=Δc−λ1c+n,x∈Ω,t>0,0=Δv−λ2v+n,x∈Ω,t>0,ut+(u⋅∇)u+∇P=Δu+n∇(χ1c−χ2v),∇⋅u=0,x∈Ω,t>0,∂n∂ν=∂c∂ν=∂v∂ν=0,u=0,x∈∂Ω,t>0,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),x∈Ω,in a bounded domain Ω⊂R2 with smooth boundary. With the help of a free energy functional, it is proved that (0.1) admits a global classical solution with arbitrary large initial data in the repulsive dominant or balance case χ1≤χ2. Moreover, there exists a critical mass phenomenon in the attractive dominant case χ1>χ2. Specifically, when the initial mass (χ1−χ2)∫Ωn0(x)dx≤4π, the solution exists globally; while if the initial mass (χ1−χ2)∫Ωn0(x)dx>4π and (χ1−χ2)∫Ωn0(x)dx∉{4πm:m∈N+}, the solution may blow up. This work extends the conclusions of the critical mass phenomenon in the Patlak–Keller–Segel-Navier–Stokes system (Gong and He, 2021) to the parabolic–parabolic–elliptic attraction–repulsion chemotaxis fluid system.

On a Two-Species Attraction–Repulsion Chemotaxis System with Nonlocal Terms

On a Two-Species Attraction–Repulsion Chemotaxis System with Nonlocal Terms

Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption

This paper is framed in a series of studies on attraction–repulsion chemotaxis models combining different effects: nonlinear diffusion and sensitivities and logistic sources, for the dynamics of the cell density, and consumption and/or production impacts, for those of the chemicals. In particular, herein we focus on the situation where the signal responsible of gathering tendencies for the particles’ distribution is produced, while the opposite counterpart is consumed. In such a sense, this research complements the results in Frassu et al. (Math Methods Appl Sci 45:11067–11078, 2022) and Chiyo et al. (Commun Pure Appl Anal, 2023, doi: 10.3934/cpaa.2023047), where the chemicals evolve according to different laws.

Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) In this problem, Ω is a bounded and smooth domain of , for , , , reasonably regular functions generalizing, respectively, the prototypes and , for some and all . Moreover, and have specific expressions, and . Once for any sufficiently smooth , and the local well-posedness of problem () is ensured, and we establish for the classical solution defined in that the life span is indeed and u, v and w are uniformly bounded in in the following cases: For , , , and , provided (I.1) k<l; (I.2) ; (I.3) k = l and , or and . For , , , and , whenever (II.1) ; (II.2) and , or and ; (II.3) . For and and , under the assumptions k<l or (I.3)). In particular, in this paper we partially improve what derived in Viglialoro [Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system. Math Nachr. 2021;294(12):2441–2454] and solve an open question given in Liu and Li [Finite-time blowup in attraction-repulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61:Paper No. 103305, 21]. Finally, the research is complemented with numerical simulations in bi-dimensional domains.

Spreading speed in a fractional attraction–repulsion chemotaxis system with logistic source

This paper studies an attraction-repulsion chemotaxis system with logistic source and a fractional diffusion of order α∈(0,1) on RN: (0.1)ut=−(−Δ)αu−χ1∇⋅(u∇v1)+χ2∇⋅(u∇v2)+u(a−bu),x∈RN,t>0,0=(Δ−λ1I)v1+μ1u,x∈RN,t>0,0=(Δ−λ2I)v2+μ2u,x∈RN,t>0,where α, χi, μi, λi(i=1,2), a and b are constants. We show that (0.1) has a unique global bounded solution and investigate the asymptotic behavior of the global classical solutions under some assumptions on the parameters. Particularly, we consider the spreading properties of the global classical solutions. Under some conditions for the nonnegative initial function u0, we prove that (0.2)lim inft→∞inf|x|≤ectu(x,t;u0)>0,for all0<c<aN+2αand (0.3)limt→∞sup|x|≥ectu(x,t;u0)=0,for allc>aN+2α.This fact shows that the spreading speed of (0.1) is exponential in time, which is different from the linear spreading speed obtained in Salako and Shen (2019).

Does strong repulsion lead to smooth solutions in a repulsion-attraction chemotaxis system even when starting with highly irregular initial data?

It has been well established that, in attraction-repulsion Keller–Segel systems of the form$ \begin{equation*} \left\{ \begin{aligned} u_t &amp; = \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w), \\ \tau v_t &amp; = \Delta v + \alpha u - \beta v, \\ \tau w_t &amp; = \Delta w + \gamma u - \delta w \end{aligned} \right. \end{equation*} $in a smooth bounded domain $ \Omega \subseteq \mathbb{R}^n $, $ n\in\mathbb{N} $, with Neumann boundary conditions and parameters $ \chi, \xi \geq 0 $, $ \alpha, \beta, \gamma, \delta &gt; 0 $ and $ \tau \in \{0, 1\} $, finite-time blow-up can be ruled out in many scenarios given sufficiently smooth initial data if the repulsive chemotaxis is sufficiently stronger than its attractive counterpart. In this paper, we will go - in a sense - a step further than this by studying the same system with initial data that could already be understood as being in a blown-up state (e.g. a positive Radon measure for the first solution component) and then ask the question whether sufficiently strong repulsion has enough of a regularizing effect to lead to the existence of a smooth solution, which is still connected to said initial data in a sensible fashion. Regarding this, we in fact establish that the construction of such a solution is possible in the two-dimensional parabolic-parabolic system and the two- and three-dimensional parabolic-elliptic system under appropriate assumptions on the interaction of repulsion and attraction as well as the initial data.

Global bounded classical solution for an attraction–repulsion chemotaxis system

In this paper, we deal with the global and boundedness of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source ut=Δu−χ∇⋅(u(u+1)r−1∇v)+ξ∇⋅(u(u+1)r−1∇w)+f(u),0=Δv−βv+αu,0=Δw−δw+γuin a bounded domain Ω⊂RN(N≥1) with smooth boundary, where χ, ξ, α, β, δ, γ, r are positive constants and f(u)≤a−buη for all u≥0 with some a≥0, b>0 and η≥1. Under the Neumann boundary conditions, if χα=ξγ and η>max{r+(N−2)+N,1} or χα<ξγ and η≥1, we prove that there exists a unique global-in-time and bounded classical solution for all appropriately regular nonnegative initial data, which extends the results of Li and Xiang (2016), Xu and Zheng (2018) and Xie and Zheng (2021).

A quasilinear attraction–repulsion chemotaxis system with logistic source

This paper is concerned with the quasilinear attraction–repulsion chemotaxis system with logistic source ut=∇⋅(D(u)∇u)−∇⋅(Φ(u)∇v)+∇⋅(Ψ(u)∇w)+f(u),(x,t)∈Ω×(0,∞),0=Δv+αu−βv,(x,t)∈Ω×(0,∞),0=Δw+γu−δw,(x,t)∈Ω×(0,∞)under Neumann boundary conditions in a bounded domain Ω⊂RN(N≥1), where D,Φ,Ψ∈C2[0,∞) are nonnegative functions with D(s)≥(s+1)p for s≥0, Φ(s)≤χsq,ξsr≤Ψ(s)≤ζsr for s>1, and the smooth function f satisfies f(s)≤μs(1−sk) for s>0,f(0)≥0. Tian et al. (2016) proved that when q=max{r,k} and q−p≥2N, if one of the following assumptions holds: (i) q=r=k,μ>(αχ−γξ)(1−2N(q−p))/(1+2(q−1)N(q−p)); (ii) q=r>k,αχ−γξ<0; (iii) q=k>r,μ>αχ(1−2N(p−q))/(1+2(q−1)N(p−q)), then the solution of the equations is globally bounded. The present work further shows that the same conclusion still holds for the critical cases: (a) q=r=k,μ=(αχ−γξ)(1−2N(q−p))/(1+2(q−1)N(q−p)); (b) q=r>k,αχ−γξ=0 with q−p<1Nmin{4(N+1)N+2,N+2}; (c) q=k>r,μ=αχ(1−2N(p−q))/(1+2(q−1)N(p−q)).

Global boundedness and asymptotic behavior in an attraction–repulsion chemotaxis system with nonlocal terms

Global boundedness and asymptotic behavior in an attraction–repulsion chemotaxis system with nonlocal terms

Boundedness and finite-time blow-up in a quasilinear parabolic–elliptic–elliptic attraction–repulsion chemotaxis system

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 -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha u-\beta v, \\[1.05mm] 0=\Delta w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partial\Omega$, where $m, p, q \in \mathbb{R}$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$ 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 $m>0$ and $f$ is a function of logistic type such as $f(u)=\lambda u-\mu u^{\kappa}$ with $\lambda \ge 0$, $\mu>0$ and $\kappa>1$ sufficiently close to~$1$, in the radially symmetric setting. In the case that $\xi=0$ and $f(u) \equiv 0$, global existence and boundedness have been proved under the condition $p<m+\frac2n$. Also, in the case that $m=1$, $p=q=2$ and $f$ is a function of logistic type, finite-time blow-up has been established by assuming $\chi\alpha-\xi\gamma>0$. This paper classifies boundedness and blow-up into the cases $p<q$ and $p>q$ without any condition for the sign of $\chi\alpha-\xi\gamma$ and the case $p=q$ with $\chi\alpha-\xi\gamma<0$ or $\chi\alpha-\xi\gamma>0$.

Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system ut=∇⋅(D(u)∇u)−∇⋅(G(u)χ(v)∇v)+∇⋅(H(u)ξ(w)∇w),x∈Ω,t>0,vt=d1Δv+αu−βv,x∈Ω,t>0,wt=d2Δw+γu−δw,x∈Ω,t>0,under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary, d1,d2,α,β,γ,δ>0 are constants. Also, D,G,H∈C2([0,∞)) fulfill that a0(s+1)m−1≤D(s)≤a1(s+1)m−1 with a0,a1>0 and m∈R; G(0)=0, 0≤G(s)≤b0(s+1)q−1 with b0>0 and q<min{2,m+1}; H(0)=0, 0≤H(s)≤c0(s+1)r−1 with c0>0 and r<min{2,m+1}, and χ,ξ satisfy that 0≤χ(s)≤χ0sk1 with χ0>0 and k1>1; 0≤ξ(s)≤ξ0sk2 with ξ0>0 and k2>1. Global existence and boundedness in the case that w=0 were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.

Boundedness and stabilization in the 3D minimal attraction–repulsion chemotaxis model with logistic source

Boundedness and stabilization in the 3D minimal attraction–repulsion chemotaxis model with logistic source

Two-species competition model with chemotaxis: well-posedness, stability and dynamics

We study a system of PDEs modelling the population dynamics of two competitive species whose spatial movements are governed by both diffusion and mutually repulsive chemotaxis effects. We prove that solutions to this system are globally well-posed, without any smallness assumptions on the chemotactic coefficients. Moreover, in the weak competition regime, we prove that neither species can be driven to extinction as the time goes to infinity, regardless of how strong the chemotaxis coefficients are. Finally, long-time behaviours of the system are studied both analytically in the weakly nonlinear regime, and numerically in the fully nonlinear regime.