Logistic Source Research Articles

Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source

Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source

Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

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.

An N-dimensional fully parabolic attraction-repulsion chemotaxis system (with logistic source): Global solutions and boundedness

This paper deals with the coupled attraction-repulsion chemotaxis system of aggregation of Microglia in Alzhemer's disease given by(⋆){ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),x∈Ω,t>0,vt=Δv−βv+αu,x∈Ω,t>0,wt=Δw−δw+γu,x∈Ω,t>0, is considered in a bounded Ω⊂RN(N≥1) with smooth boundary, where χ,α,ξ,β as well as γ and δ are positive constants. Here f(u)≡0 or f(u)≤a−bur for all u≥0 with some a∈R,b>0 and r≥1. By virtue of a novel approach on the basis of Maximal Sobolev regularity property for the heat equation, it is showed that when the repulsion cancels the attraction (i.e. χα=ξγ), the solution for an associated initial-boundary problem (⋆) is globally bounded ifN≤3andf(u)≤a−burwith somea∈R,b>0andr≥1 orf(u)≤a−burwith somea∈R,b>0andr>2N+4N+4, which extends (or partly improves) the corresponding results obtained in Tao-Wang (2013) [26] as well as Shi-Liu-Jin (2017) [25] and Wang-Zhuang-Zheng (2018) [31].

Large time behavior of solution to a quasilinear chemotaxis model describing tumor angiogenesis with/without logistic source

In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+μu−μu2,x∈Ω,t>0,vt=Δv+∇⋅(v∇w)−v+u,x∈Ω,t>0,0=Δw−w+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,in a bounded smooth domain Ω⊂Rn(n≤3), where the parameter χ,ξ>0,μ≥0, D(u) is supposed to satisfy the behind property D(u)≥(u+1)αwithα>0.Assume that either μ≥0,α>1 or μ=0,ξ≥λ1∗χ2, where the parameter λ1∗=λ1∗(u0,v0,Ω)>0, then the system admits a global classical solution (u,v,w) by subtle energy estimates. Moreover, when μ=0, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0¯,u0¯,u0¯) provided the initial data u0 is sufficiently small, where u0¯=∫Ωu0|Ω|. Finally, when μ>0, it can be proved that the corresponding solution of the system decays to (1,1,1) exponentially for suitable large μ.

Global classical solutions to an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source

Global classical solutions to an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source

Global solvability and boundedness to a attraction–repulsion model with logistic source

In this paper, we deal with an attraction–repulsion model with a logistic source as follows: {ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+μuq(1−u)in Q,vt=Δv−α1v+β1uin Q,wt=Δw−α2w+β2uin Q,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \ extstyle\\begin{cases} {u_{t}} = \\Delta u - \\chi \ abla \\cdot (u \ abla v) + \\xi \ abla \\cdot (u \ abla w) + \\mu {u^{q}}(1 - u) &\ ext{in } Q , \\\\ {v_{t}} = \\Delta v - {\\alpha _{1}}v + {\\beta _{1}}u &\ ext{in } Q , \\\\ {w_{t}} = \\Delta w - {\\alpha _{2}}w + {\\beta _{2}}u & \ ext{in } Q , \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where Q=Ω×R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q = \\Omega \ imes {\\mathbb{R}^{+} }$\\end{document}, Ω⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset {\\mathbb{R}^{3}}$\\end{document} is a bounded domain. We mainly focus on the influence of logistic damping on the global solvability of this model. In dimension 2, q can be equal to 1 (Math. Methods Appl. Sci. 39(2):289–301, 2016). In dimension 3, we derive that the problem admits a global bounded solution when q>87\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q>\\frac{8}{7}$\\end{document}. In fact, we transfer the difficulty of estimation to the logistic term through iterative methods, thus, compared to the results in (J. Math. Anal. Appl. 2:448 2017; Z. Angew. Math. Phys. 73(2):1–25 2022) in dimension 3, our results do not require any restrictions on the coefficients.

Front stability of infinitely steep travelling waves in population biology

Reaction–diffusion models are often used to describe biological invasion, where populations of individuals that undergo random motility and proliferation lead to moving fronts. Many reaction–diffusion models of biological invasion are extensions of the well–known Fisher–Kolmogorov–Petrovskii–Piskunov model that describes the spatiotemporal evolution of a 1D population density, u(x,t) , as a result of linear diffusion with flux J=−∂u/∂x , and logistic growth source term, S=u(1−u) . In 2020 Fadai introduced a new reaction–diffusion model of biological invasion with a nonlinear degenerate diffusive flux, J=−u∂u/∂x , and the model was formulated as a moving boundary problem on 0<x<L(t) , with u(L(t),t)=0 and dL(t)/dt=−κu∂u/∂x at x=L(t) (Fadai and Simpson 2020 J. Phys. A: Math. Theor. 53 095601). Fadai’s model leads to travelling wave solutions with infinitely steep, well–defined fronts at the moving boundary, and the model has the mathematical advantage of being analytically tractable in certain parameter limits. In this work we consider the stability of the travelling wave solutions presented by Fadai. We provide general insight by first presenting two key extensions of Fadai’s model by considering: (i) generalised nonlinear degenerate diffusion with flux J=−um∂u/∂x for some constant m > 0; and, (ii) solutions describing both biological invasion with dL(t)/dt>0 , and biological recession with dL(t)/dt<0 . After establishing the existence of travelling wave solutions for these two extensions, our main contribution is to consider stability of the travelling wave solutions by introducing a lateral perturbation of the travelling wavefront. Full 2D time–dependent level–set numerical solutions indicate that invasive travelling waves are stable to small amplitude lateral perturbations, whereas receding travelling waves are unstable. These preliminary numerical observations are corroborated through a linear stability analysis that gives more formal insight into short time growth/decay of wavefront perturbation amplitude. Julia–based software, including level–set algorithms, is available on Github to replicate all results in this study.

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

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].

The convergence rate of solutions in chemotaxis models with density-suppressed motility and logistic source

The convergence rate of solutions in chemotaxis models with density-suppressed motility and logistic source

On the global existence of solutions to a chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source

This paper is devoted to a chemotaxis system with signal-dependent motility under homogeneous Neumann boundary conditions in a bounded domain. We prove that this problem possesses a global classical solution which is uniformly bounded under weaker conditions than that obtained by Lv and Wang (2020). The findings of our study demonstrate the presence of a consistent decay rate, effectively ruling out the occurrence of blow-up phenomena in the system across all spatial dimensions.

Boundedness and Finite-Time Blow-up in a Chemotaxis System with Flux Limitation and Logistic Source

Boundedness and Finite-Time Blow-up in a Chemotaxis System with Flux Limitation and Logistic Source

On an attraction-repulsion chemotaxis model involving logistic source

This paper is concerned with the attraction-repulsion chemotaxis system involving logistic source: u_{t}=Δu-χ∇⋅(u∇υ)+ξ∇⋅(u∇ω)+f(u), ρυ_{t}=Δυ-α₁υ+β₁u, ρω_{t}=Δω-α₂ω+β₂u under homogeneous Neumann boundary conditions with nonnegative initial data (u₀,υ₀,ω₀)∈ (W^{1,∞}(Ω))³, the parameters χ, ξ, α₁, α₂, β₁, β₂&gt;0, ρ≥0 subject to the non-flux boundary conditions in a bounded domain Ω⊂ℝ^{N}(N≥3) with smooth boundary and f(u)≤au-μu² with f(0)≥0 and a≥0, μ&gt;0 for all u&gt;0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a globally bounded classical solution provided that χ+ξ0 is sufficiently small for all β₁, β₂

Global solvability of a two-species chemotaxis–fluid system with Lotka–Volterra type competitive kinetics

In this paper, we study a two-species chemotaxis–fluid system with Lotka–Volterra type competitive kinetics in a bounded and smooth domain [Formula: see text] with no-flux/Dirichlet boundary conditions. We present the global existence of weak energy solution to a two-species chemotaxis Navier–Stokes system, and then the global weak energy solution which coincides with a smooth function throughout [Formula: see text], where [Formula: see text] represents a countable union of open intervals which is such that [Formula: see text]. In such two-species chemotaxis–fluid setting, our existence improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up. This finding significantly extends previously known ones.

Global Existence of Chemotaxis-Navier–Stokes System with Logistic Source on the Whole Space R2

In this article, we study the Cauchy problem of the chemotaxis-Navier–Stokes system with the consumption and production of chemosignals with a logistic source. The parameters χ≠0, ξ≠0, λ&gt;0 and μ&gt;0. The system is a model that involves double chemosignals; one is an attractant consumed by the cells themselves, and the other is an attractant or a repellent produced by the cells themselves. We prove the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space R2.

Global weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

Global weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

Solving a fractional chemotaxis system with logistic source using a meshless method

We study the numerical solution of the fractional Keller–Segel system with logistic source. We derive the discretization of the fractional Laplacian and integer derivatives using a meshless method. A condition for convergence is given and several examples illustrating the dynamics of both fully parabolic and parabolic–elliptic systems on irregular meshes are provided.

Global existence in a two-species chemotaxis system with signal-dependent sensitivity and logistic source

Global existence in a two-species chemotaxis system with signal-dependent sensitivity and logistic source

具有Logistic源的三维趋化模型的适定性研究

具有Logistic源的三维趋化模型的适定性研究

Global boundedness in a chemotaxis system with signal-dependent motility and indirect signal consumption and logistic source

Global boundedness in a chemotaxis system with signal-dependent motility and indirect signal consumption and logistic source