Chemotactic Sensitivity Research Articles

Global existence and boundedness in a two‐dimensional parabolic chemotaxis system with competing attraction and repulsion effects

In this paper, we investigate a chemotaxis system under homogeneous Neumann boundary conditions within a bounded domain with a smooth boundary. The system describes the movement of cells in response to two chemical signal substances: one acts as a chemoattractant, while the other serves as a chemorepellent, both produced by the cells. The system takes into account chemotactic sensitivity in the reaction movement when detecting these chemicals. Under certain assumptions, we demonstrate the existence of a unique global bounded classical solution for the proposed problem. To further understand the time evolution of the system's solutions, we conduct numerical experiments and analyze the dynamic properties of the norm of the solutions with respect to variations in chemical production rates.

Boundedness in a Chemotaxis-May-Nowak model for virus dynamics with gradient-dependent flux limitation

This paper investigates an extension of the May-Nowak ODE model for virus dynamics with gradient-dependent flux limitation of cross diffusion. In particular, we consider the associated no-flux initial–boundary value problem (0.1)ut=Δu−∇⋅(uf(|∇v|2)∇v)+κ−u−uw,vt=Δv−v+uw,wt=Δw−w+vin a smoothly bounded domain Ω⊂Rn(n≤3), where the parameter κ≥0. The prototypical chemotactic sensitivity function f∈C2([0,∞)) is given by f(ξ)=(1+ξ)−α,ξ≥0 with some α∈R. It is proved that whenever α∈R,ifn=1,α>n−22(n−1),ifn={2,3},global classical solutions to (0.1) exist and are uniformly bounded. Such result consists with that in [Winkler (2022), Proposition 1.2] when n≤3, which shows that the effect of gradient-dependent flux limitation in weakening the cross-diffusion term remains unchanged even in the context of nonlinear signal production mechanism.

Traveling waves and their spectral instability in volume–filling chemotaxis model

In this paper, I consider a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By the geometric singular perturbation theory together with the center-stable and center unstable manifolds, one gets the existence of a positive traveling wave connecting the two constant steady states (0,0) and (b,αbβ) with a small wave speed ϵc. In addition, the traveling wave is monotone for b≥1 and is not monotone for 0<b<1. Moreover, by the spectral analysis it shows that the above traveling wave is spectrally unstable in some exponentially weighted spaces.

Finite-time blow-up of weak solutions to a chemotaxis system with gradient dependent chemotactic sensitivity

This paper deals with the parabolic–elliptic chemotaxis system with gradient dependent chemotactic sensitivity,{ut=Δu−χ∇⋅(u|∇v|p−2∇v),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0, where Ω:=BR(0)⊂Rn(n≥2) is a ball with some R>0, χ>0, p∈(nn−1,∞) and μ:=1|Ω|∫Ωu0, where u0 is an initial datum of arbitrary size. In the case that p∈(1,nn−1), Negreanu and Tello (2018) [23] established global existence and uniform boundedness of solutions, whereas when p∈(nn−1,2), Tello (2022) [31] showed that solutions blow up in finite time under the condition that μ>6 and χ is large enough. These works imply that the number p=nn−1 certainly plays the role of a critical blow-up exponent, and it is expected that when p>nn−1, for arbitrary μ>0 the system admits at least one solution which blows up in finite time. The purpose of this paper is to prove that this conjecture is true within a framework of weak solutions with a moment inequality.

Generalised solution to a 2D parabolic-parabolic chemotaxis system for urban crime: Global existence and large-time behaviour

AbstractWe consider a parabolic-parabolic chemotaxis system with singular chemotactic sensitivity and source functions, which is originally introduced by Short et al to model the spatio-temporal behaviour of urban criminal activities with the particular value of the chemotactic sensitivity parameter $\chi =2$ . The available analytical findings for this urban crime model including $\chi =2$ are restricted either to one-dimensional setting, or to initial data and source functions with appropriate smallness, or to initial data and source functions with some radial symmetry. In the present work, our first result asserts that for any $\chi \gt 0$ the initial-boundary value problem of this urban crime model possesses a global generalised solution in the two-dimensional setting, without imposing any small or radial conditions on initial data and source functions. Our second result presents the asymptotic behaviour of such solution, under some additional assumptions on source functions.

Blow-up for a stochastic model of chemotaxis driven by conservative noise on mathbb {R}^2

We establish criteria on the chemotactic sensitivity chi for the non-existence of global weak solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous, conservative noise on mathbb {R}^2. We show that if chi is sufficiently large then blow-up occurs with probability 1. In this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity. However, for chi in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that blow-up occurs with positive probability.

Chemotaxis-driven stationary and oscillatory patterns in a diffusive HIV-1 model with CTL immune response and general sensitivity.

In this paper, a reaction-diffusion-chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L∞-boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S0, CTL-inactivated infection steady state S1, and CTL-activated infection steady state S∗. Our analyses indicate that S0 is globally asymptotically stable if the basic reproduction number R0 is less than 1; if R0 is between 1 and a threshold, then S1 is globally asymptotically stable. However, if R0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S∗, and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications.

A mathematical modeling technique to understand the role of decoy receptors in ligand-receptor interaction

The ligand-receptor interaction is fundamental to many cellular processes in eukaryotic cells such as cell migration, proliferation, adhesion, signaling and so on. Cell migration is a central process in the development of organisms. Receptor induced chemo-tactic sensitivity plays an important role in cell migration. However, recently some receptors identified as decoy receptors, obstruct some mechanisms of certain regular receptors. DcR3 is one such important decoy receptor, generally found in glioma cell, RCC cell and many various malignant cells which obstruct some mechanism including apoptosis cell-signaling, cell inflammation, cell migration. The goal of our work is to mathematically formulate ligand-receptor interaction induced cell migration in the presence of decoy receptors. We develop here a novel mathematical model, consisting of four coupled partial differential equations which predict the movement of glioma cells due to the reaction-kinetic mechanism between regular receptors CD95, its ligand CD95L and decoy receptors DcR3 as obtained in experimental results. The aim is to measure the number of cells in the chamber’s filter for different concentrations of ligand in presence of decoy receptors and compute the distance travelled by the cells inside filter due to cell migration. Using experimental results, we validate our model which suggests that the concentration of ligands plays an important role in cell migration.

Global classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions

In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.

On the Cauchy problem for Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces

In this paper, the local well‐posedness (which means that the initial‐to‐solution map is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces with and was established, and the solution of PEKS (Keller‐Segel system with ) converges to the solution of HEKS (Keller‐Segel system with ) as the diffusion coefficient in these Besov spaces was proved. In addition, we show that the solution of this parabolic‐elliptic chemotaxis‐growth system is local well‐posedness in the critical spaces but ill‐posed (not continuous) in Besov spaces . Moreover, we show a further continuity result that the initial‐to‐solution map is Hölder continuous in Besov spaces equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces was also obtained.

Existence and large time behavior for a Keller–Segel model with gradient dependent chemotactic sensitivity

The purpose of this paper is to study the chemotaxis growth system $$ \begin{cases} u_t=\Delta u-\nabla \cdot (u|\nabla v|^{p-2}\nabla v)+au-bu^{\alpha},& x\in \Omega ,\, t \gt 0,\\ v_t=\Delta v-v+w,& x\in \Omega ,\, t \gt 0,\\ w_t=\Delta w-w+u,& x\in \Om

Global solutions for chemotaxis-fluid systems with singular chemotactic sensitivity

This paper concerns the chemotaxis-Navier-Stokes system \begin{document}$\begin{equation} \begin{cases} n_t+u\cdot {\nabla } n = \Delta n - {{\nabla }\cdot}(n\chi(c) {\nabla } c),\\ c_t+u\cdot {\nabla } c = \Delta c -nf(c),\\ u_t+\kappa (u\cdot {\nabla }) u = \Delta u + {\nabla } P +n {\nabla }\Phi, \quad {{\nabla }\cdot} u = 0 \end{cases} \end{equation} \;\;\;\;\;\;\;\;\;\;\;\;(\star)$ \end{document} in a smoothly convex bounded domain under Neumann/Neumann/Dirichlet boundary conditions. The existing literature reveals that $ (\star) $ possesses the global solution under some mild assumptions on $ \chi $ and $ f $ [25,28]. It is shown in this paper that even though $ \chi $ admits some singularity (e.g. $ \chi(c) = c^{-\alpha} $ for some $ \alpha&gt;0 $), appropriate use of some quantity involving $ \chi $ and $ f $ guarantees the existence of the global solution to $ (\star) $ for any choice of suitably regular (nonnegative) initial data.

Boundedness in a chemotaxis-May-Nowak model for virus dynamics with mildly saturated chemotactic sensitivity and conversion

This work studies a class of chemotaxis systems given by$ \begin{align*} \begin{cases} u_t = \Delta u-\nabla\cdot (uf(u)\nabla v)-u-g(u)w+\kappa, \\ v_t = \Delta v-v+g(u)w, \\ w_t = \Delta w-w+v, \end{cases} \end{align*} $under homogeneous Neumann boundary conditions in smooth bounded n-dimensional domains, where $ \kappa\geq 0 $, $ f\in C^2([0, \infty)) $ satisfies $ |f(s)|\leq K_f(1+s)^{-\alpha} $ for all $ s\geq 0 $ with some $ K_f&gt;0 $ and $ \alpha\in\mathbb{R} $, and $ g\in C^1([0, \infty)) $ is a nonnegative function satisfying $ g(0) = 0 $ and $ g(s)\leq K_g(1+s^{\beta}) $ for all $ s\geq 1 $ with $ K_g&gt;0 $ and $ \beta\in \mathbb{R} $. Under the assumption that $ \beta&lt;\frac{2}{n} $ $ (n\geq 2) $, it is shown that there exists $ \alpha_0 = \alpha_0(\beta)&gt;0 $ such that if $ \alpha&gt;-\alpha_0 $, then the system has a global classical solution which is uniformly bounded. The case when $ n = 1 $ is also discussed. These improve the known results.

Global solutions to a chemotaxis-growth system with signal-dependent motilities and signal consumption

This paper deals with an initial-boundary value problem about the chemotaxis-growth system(0.1){ut=∇(γ(v)∇u−uϕ(v)∇v)+μu(1−u),x∈Ω,t>0,vt=△v−uv,x∈Ω,t>0 in a bounded domain Ω⊂Rn(n≥2) with no-flux boundary conditions. Here one of the two density-dependent motility functions γ(v) describes the strength of diffusion while the other ϕ(v)=(α−1)γ′(v)(α>0) denotes the chemotactic sensitivity. It is proved that for a class generic motility functions there exists a unique global bounded classical solution to (0.1) with some suitable small initial data and some large μ. Furthermore, it asserts that the obtained global solution stabilizes to the spatially uniform equilibrium (1,0) in the sense that‖u(⋅,t)−1‖L∞(Ω)→0,‖v(⋅,t)‖W1,∞(Ω)→0ast→∞.

Further study on the global existence and boundedness of the weak solution in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

We consider an initial-value problem for the incompressible chemotaxis-Stokes equations generalizing the porous-medium-type diffusion model with tensor-valued sensitivity nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)⋅∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0(CNF) in a bounded domain Ω⊆R3 with a smooth boundary, where ϕ∈W2,∞(Ω) is a given gravitational potential function. Problems of this type have been used to describe the mutual interaction between swimming aerobic bacteria populations and the surrounding fluid. The main feature is that the chemotactic sensitivity S is a given parameter matrix on Ω×[0,∞)2, whose Frobenius norm satisfies |S(x,n,c)|≤S0(c) for all (x,n,c)∈Ω̄×[0,∞)×[0,∞) with S0(c) nondecreasing on [0,∞). Based on a new energy-type argument combined with a new estimation technique, we show that if m>114−3, an associated initial–boundary value problem admits at least one globally bounded weak solution. The above mentioned results significantly improved and extended previous results of several authors.

Bacterial chemotaxis to saccharides is governed by a trade-off between sensing and uptake

To swim up gradients of nutrients, E. coli senses nutrient concentrations within its periplasm. For small nutrient molecules, periplasmic concentrations typically match extracellular concentrations. However, this is not necessarily the case for saccharides, such as maltose, which are transported into the periplasm via a specific porin. Previous observations have shown that, under various conditions, E. coli limits maltoporin abundance so that, for extracellular micromolar concentrations of maltose, there are predicted to be only nanomolar concentrations of free maltose in the periplasm. Thus, in the micromolar regime, the total uptake of maltose from the external environment into the cytoplasm is limited not by the abundance of cytoplasmic transport proteins but by the abundance of maltoporins. Here, we present results from experiments and modeling suggesting that this porin-limited transport enables E. coli to sense micromolar gradients of maltose despite having a high-affinity ABC transport system that is saturated at these micromolar levels. We used microfluidic assays to study chemotaxis of E. coli in various gradients of maltose and methyl-aspartate and leveraged our experimental observations to develop a mechanistic transport-and-sensing chemotaxis model. Incorporating this model into agent-based simulations, we discover a trade-off between uptake and sensing: although high-affinity transport enables higher uptake rates at low nutrient concentrations, it severely limits the range of dynamic sensing. We thus propose that E. coli may limit periplasmic uptake to increase its chemotactic sensitivity, enabling it to use maltose as an environmental cue.

Boundedness and asymptotic behavior in a Keller-Segel(-Navier)-Stokes system with indirect signal production

This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production(⋆){nt+u⋅∇n=Δn−∇⋅(n∇v)+rn−μn2,vt+u⋅∇v=Δv−v+w,wt+u⋅∇w=Δw−w+n,ut+κ(u⋅∇)u=Δu−∇P+n∇Φ,∇⋅u=0 in a bounded and smooth domain Ω⊂RN(N=2,3) with no-flux boundary for n, v, w and no-slip boundary for u, where r∈R, μ≥0, κ∈{0,1} and Φ∈W2,∞(Ω). In the case without logistic source (r=μ=0), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (⋆) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (r∈R, μ>0), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (⋆) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases r<0, r=0 and r>0. We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for μ≥23 (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for r=0 and μ≥23. Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system.

The one-dimensional stochastic Keller–Segel model with time-homogeneous spatial Wiener processes

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak–Keller–Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak–Keller–Segel system perturbed by random processes.We study a stochastic version of the classical Patlak–Keller–Segel system under homogeneous Neumann boundary conditions on an interval O=[0,1]. In particular, let W1, W2 be two time-homogeneous spatial Wiener processes over a filtered probability space A. Let u and v denote the cell density and concentration of the chemical signal. We investigate the coupled system{du−(ruΔu−χdiv(u∇v))dt=u∘dW1,dv−(rvΔv−αv)dt=βudt+v∘dW2,with initial conditions (u(0),v(0))=(u0,v0). The positive terms ru and rv are the diffusivity of the cells and chemoattractant, respectively, the positive value χ is the chemotactic sensitivity, α≥0 is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given T>0, we will prove the existence of a martingale solution on [0,T].

Global well-posedness of 2D chemotaxis Euler fluid systems

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.

Global solvability and asymptotic stabilization in a three-dimensional Keller–Segel–Navier–Stokes system with indirect signal production

This paper deals with the Keller–Segel–Navier–Stokes model with indirect signal production in a three-dimensional (3D) bounded domain with smooth boundary. When the logistic-type degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the associated no-flux/no-flux/no-flux/Dirichlet problem possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in [Formula: see text] with any [Formula: see text]. Moreover, under an explicit condition on the chemotactic sensitivity, these solutions are shown to stabilize toward the corresponding spatially homogeneous state in the sense of some suitable norms. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if the degradation is quadratic. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the 3D Keller–Segel–Navier–Stokes system.