Chemotaxis Model Research Articles

Global boundedness and asymptotic behavior in a fully parabolic attraction-repulsion chemotaxis model with logistic source

In this paper, we consider a fully parabolic attraction-repulsion chemotaxis model with logistic source. First of all, we obtain an explicit formula [Formula: see text] for the logistic damping rate [Formula: see text] such that the model has no blow-up when [Formula: see text]. In addition, the asymptotic behavior of the solutions is studied. Our results partially generalize and improve some results in the literature, and partially results are new.

Dynamics and diffusion limit of traveling waves in a two-species chemotactic model with logarithmic sensitivity

This paper discusses the traveling wave analysis of the two-species chemotaxis model with logarithmic sensitivity, which describes diverse biological phenomena such as the initiation of angiogenesis using reinforced random walks theory and the chemotactic response of two interacting species to a chemical stimulus. The objectives are to quantitatively establish the existence of traveling wave solutions only for the parameters in certain parameter regimes. Moreover, we incorporate recent results and discuss many other aspects of traveling wave solutions such as asymptotic decay rates and convergence as the chemical diffusion coefficient goes to zero. The main techniques are analyzed and the analytical results are reviewed graphically.

Stability and instability in a three-component chemotaxis model for alopecia areata

This paper investigates the stability and instability of a generalized reaction–diffusion–advection system that describes the spatio-temporal dynamics of alopecia areata lesions. Using the spectral analysis, we investigate the effect of chemotactic strengths on the stability and instability of a linearized problem around the positive constant equilibrium. We then extend our analysis to the nonlinear system, utilizing higher-order energy estimates and bootstrap techniques to study its stability and instability based on the linearized results. Additionally, we provide numerical simulations to support our theoretical results.

Properties of given and detected unbounded solutions to a class of chemotaxis models

AbstractThis paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow‐up time are established. Finally, for a simplified version of the model, some blow‐up criteria are proved.More precisely, we analyze a zero‐flux chemotaxis system essentially described as The problem is formulated in a bounded and smooth domain Ω of , with , for some , , , and with . A sufficiently regular initial data is also fixed.Under specific relations involving the above parameters, one of these always requiring some largeness conditions on , we prove that any given solution to (), blowing up at some finite time becomes also unbounded in ‐norm, for all ; we give lower bounds T (depending on ) of for the aforementioned solutions in some ‐norm, being ; whenever , we establish sufficient conditions on the parameters ensuring that for some u0 solutions to () effectively are unbounded at some finite time. Within the context of blow‐up phenomena connected to problem (), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic.

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.

Three-dimensional human neural culture on a chip recapitulating neuroinflammation and neurodegeneration.

Neuroinflammation has either beneficial or detrimental effects, depending on risk factors and neuron-glia interactions in neurological disorders. However, studying neuroinflammation has been challenging due to the complexity of cell-cell interactions and lack of physio-pathologically relevant neuroinflammatory models. Here, we describe our three-dimensional microfluidic multicellular human neural culture model, referred to as a 'brain-on-a-chip' (BoC). This elucidates neuron-glia interactions in a controlled manner and recapitulates pathological signatures of the major neurological disorders: dementia, brain tumor and brain edema. This platform includes a chemotaxis module offering a week-long, stable chemo-gradient compared with the few hours in other chemotaxis models. Additionally, compared with conventional brain models cultured with mixed phenotypes of microglia, our BoC can separate the disease-associated microglia out of heterogeneous population and allow selective neuro-glial engagement in three dimensions. This provides benefits of interpreting the neuro-glia interactions while revealing that the prominent activation of innate immune cells is the risk factor leading to synaptic impairment and neuronal loss, validated in our BoC models of disorders. This protocol describes how to fabricate and implement our human BoC, manipulate in real time and perform end-point analyses. It takes 2 d to set up the device and cell preparations, 1-9 weeks to develop brain models under disease conditions and 2-3 d to carry out analyses. This protocol requires at least 1 month training for researchers with basic molecular biology techniques. Taken together, our human BoCs serve as reliable and valuable platforms to investigate pathological mechanisms involving neuroinflammation and to assess therapeutic strategies modulating neuroinflammation in neurological disorders.

Existence and stability of bifurcating solution of a chemotaxis model

This paper focuses on a chemotaxis model with no-flux boundary conditions. We first discuss the stability of the unique positive equilibrium by treating the chemotaxis coefficient ξ \xi as the Hopf bifurcation and the steady state bifurcation parameter. Hereafter, we perform the existence and stability of the bifurcating solution, which bifurcated from the steady state bifurcation, by using the Crandall-Rabinowitz local bifurcation theory. It is noticed that few existing literatures give a discriminant to determine the stability of the bifurcating solution for the chemotaxis models. To this end, we will fill this gap, and an explicit formula will be presented. This technique can also be applied in other ecological models with chemotaxis.

Analysis of complex chemotaxis models

This preface describes motivational aspects related to a special issue focusing on “analysis of complex chemotaxis models”, and briefly discusses the contributions provided by the six papers contained therein.

Well‐posedness and blow‐up of the fractional Keller–Segel model on domains

AbstractThis work deals with well‐posedness and blow‐up in the setting of Lebesgue and Besov spaces to the time‐fractional Keller–Segel model for chemotaxis under homogeneous Neumann boundary conditions in a smooth domain of . The KS model consists in a coupled system of partial differential equations. In particular, we also treat the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution.

Boundedness and weak stabilization in a degenerate chemotaxis model arising from tumor invasion

This paper is concerned with the degenerate system{ut=∇⋅(f(u,w)∇u−g(u)∇v),x∈Ω,t>0,vt=Δv+wz,x∈Ω,t>0,wt=−wz,x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0 in a bounded domain Ω⊂RN (N≥2) under the no-flux boundary condition for u and the homogeneous Neumann boundary condition for v,z with non-negative initial data u0,v0,w0,z0. Here, the diffusivity f and the sensitivity g are assumed to fulfill f(u,w)≥um−1 (m>1), 0≤g(u)≤uα(α∈R). It is shown that if α+1<m+4N(N≥2) or α+1=m+4N(N≥3) with small mass of u0, then the system possesses a global bounded weak solution which converges to the constant equilibrium in the weak⁎ topology in L∞(Ω) as t→∞.

Cross-diffusion models in complex frameworks from microscopic to macroscopic

This paper deals with the micro–macro derivation of models from the underlying description provided by methods of the kinetic theory for active particles. We consider the so-called exotic models according to the definition proposed in [ N. Bellomo, N. Outada, J. Soler, Y. Tao and M. Winkler, Chemotaxis and cross diffusion models in complex environments: Modeling towards a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. We refer to a selection of case studies, in detail, the transport of virus models, social dynamics, and Keller–Segel in a fluid. The second part shows how a Hilbert-type approach can be developed to derive models at the macroscale from the underlying description provided by the kinetic theory of active particles. The third part deals with the derivation of macroscopic models corresponding to the selected case studies. Finally, a forward look into the future research perspectives is proposed.

Free Boundaries Problem for a Class of Parabolic Type Chemotaxis Model

Free Boundaries Problem for a Class of Parabolic Type Chemotaxis Model

Nonlinear stability of strong traveling waves for a chemotaxis model with logarithmic sensitivity and periodic perturbations

In this paper, we are concerned with the nonlinear stability of traveling wave solutions for a conservation laws arising from the well‐known chemotaxis model with logarithmic sensitivity. By constructing appropriate “ansatz,” we shall prove the nonlinear stability of traveling waves even though the perturbations oscillate at the far fields. More precisely, it is shown that as time tends to infinite, the solutions converge to arbitrarily large amplitude viscous shock wave solutions with a shift, which are partially determined by the periodic oscillations.

On the Global Well-Posedness for a Hyperbolic Model Arising from Chemotaxis Model with Fractional Laplacian Operator

Several cells and microorganisms, such as bacteria and somatic, have many essential features, one of which can be modeled by the chemotaxis system, which we consider to be our main interest in this article. More precisely, we studied the hyperbolic system derived from the chemotaxis model with fractional dissipation, which is a generalization for the hyperbolic system with classical dissipation. The results of this article are divided into two parts. In the first part, we used energy methods to obtain the existence of small solutions in the Besov spaces. The second one deals with the optimal decay of perturbed solutions using a refined time-weighted energy combined with the Littlewood-Paley decomposition technique. To the authors’ best knowledge, this type of system (with fractional dissipation) has not been studied in the literature.

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.

A probe into research of complex chemotaxis models

A probe into research of complex chemotaxis models

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

Blow-up solutions of a chemotaxis model with nonlocal effects

In this paper, we investigate the radially symmetric solutions to the following chemotaxis system with nonlocal term (0.1)ut=△u−∇⋅(u∇v)+μu1−∫Ωuκdx,x∈Ω,t>0,0=△v−m(t)+u,x∈Ω,t>0,(u,v)(x,0)=(u0(x),v0(x)),x∈Ω,where Ω≔BR(0) is a ball whose center is the origin, n≥3, μ>0 and m(t)=1|Ω|∫Ωu(x,t)dx. We show that for any m0>0, there exists a nonnegative initial data u0∈C1(Ω̄) with 1|Ω|∫Ωu0dx=m0 such that the corresponding solution of (0.1) blows up in finite time under the assumption 0<κ<min{2,n2}.

Competition between chemoattractants causes unexpected complexity and can explain negative chemotaxis

Negative chemotaxis, where eukaryotic cells migrate away from repellents, is important throughout biology, for example, in nervous system patterning and resolution of inflammation. However, the mechanisms by which molecules repel migrating cells are unknown. Here, we use predictive modeling and experiments with Dictyostelium cells to show that competition between different ligands that bind to the same receptor leads to effective chemorepulsion. 8-CPT-cAMP, widely described as a simple chemorepellent, is inactive on its own and only repels cells when it acts in combination with the attractant cAMP. If cells degrade either competing ligand, the pattern of migration becomes more complex; cells may be repelled in one part of a gradient but attracted elsewhere, leading to populations moving in different directions in the same assay or converging in an arbitrary place. More counterintuitively still, two chemicals that normally attract cells can become repellent when combined. Computational models of chemotaxis are now accurate enough to predict phenomena that have not been anticipated by experiments. We have used them to identify new mechanisms that drive reverse chemotaxis, which we have confirmed through experiments with real cells. These findings are important whenever multiple ligands compete for the same receptors.

New RK type time-integration methods for stiff convection–diffusion–reaction systems

Convection–diffusion–reaction equations (CDREs) are extensively used to model the various physical processes, which are generally stiff in both reaction and diffusion terms. This paper discusses a new class of implicit–explicit Runge–Kutta (IMEX RK) type methods for numerical simulations of the convection–diffusion–reaction systems. The developed approach does not need to invert the coefficient matrix, despite being implicit in nature. The Fourier stability analysis is performed to gauge the properties of the developed methods using one- and two-dimensional scalar CDRE as model systems. Additionally, numerical simulation of the one-dimensional inhomogeneous linear CDRE, contaminant transport with kinetic Langmuir sorption model, and two-dimensional chemotaxis model that incorporates volume-filling effect and nonlinear diffusion are used to validate the effectiveness and robustness of the developed methods. The numerical solutions match the exact solutions discussed in the literature very well.