Keller-Segel Model Research Articles

A posteriori error control for a discontinuous Galerkin approximation of a Keller-Segel model

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional in the sense that an a posteriori computable quantity needs to be small enough—which can be ensured by mesh refinement—and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove the existence of a weak solution up to a certain time based on numerical results.

Optimal critical mass for the multi-species Keller-Segel model with rotational flux terms

Optimal critical mass for the multi-species Keller-Segel model with rotational flux terms

Behavior in time of solutions to a degenerate chemotaxis system with flux limitation

We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system ut=∇⋅(u∇uu2+|∇u|2)−χkf∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0under no flux boundary conditions in a ball B=Ω⊂RN and initial condition u(x,0)=u0(x)>0,χ>0,α>0,kf>0 and μ=1|Ω|∫Ωu0dx. Under suitable conditions on α and u0 it is shown that the solution blows up in L∞-norm at a finite time Tmax and for some p>1 it blows up also in Lp-norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates.

Mathematical analysis and asymptotic predictions of chemical-driven swimming living organisms in weighted networks

This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations of parabolic-parabolic Keller-Segel model coupled with incompressible Navier-Stokes equations must be reformulated in a discrete setting. The starting point is to transpose the coupled system from the Euclidean case to the connected networks with certain network-theoretic simplified approaches, which yields fruitful key results. These results not only enable the construction of global solutions but also serve as a foundation for determining information about the stability and large time behavior of the system. Then, decay rates are well established to predict important features, such as how quickly weak solutions at a given point decrease over time due to dissipative processes. Additionally, the L1- convergence of cell densities towards the self-similar Gaussian solution of the heat equation is well proved by time dependent scaling, which shows that the solution maintains its shape and only scales in time and space as time evolves. Finally, this paper includes many numerical tests through a recent robust numerical scheme to illustrate the theoretical results and to develop computational control and prediction of living organisms' trajectories around central nodes in networked flows.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝ d for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝ d . We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

On the fractional-in-time Keller-Segel model via Sonine kernels

In this paper, we study the existence and asymptotic behavior to a diffusion system which is non-local in time. As consequence of our theorems we deduce new results for the fractional-in-time Keller-Segel model. Our approach is intimately related with the Sonine kernels.

Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption

Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption

Global entropy solutions to a degenerate parabolic–parabolic chemotaxis system for flux-limited dispersal

Existence of global finite-time bounded entropy solutions to a parabolic–parabolic system proposed in Bellomo et al. (2010) is established in bounded domains under no-flux boundary conditions for nonnegative bounded initial data. This modification of the classical Keller–Segel model features degenerate diffusion and chemotaxis that are both subject to flux-saturation. The approach is based on Schauder’s fixed point theorem and calculus of functions of bounded variation.

Existence, Stability and Slow Dynamics of Spikes in a 1D Minimal Keller–Segel Model with Logistic Growth

Existence, Stability and Slow Dynamics of Spikes in a 1D Minimal Keller–Segel Model with Logistic Growth

Blow-up of solutions to the Keller–Segel model with tensorial flux in high dimensions

Over the course of the last decade, there has been a significant level of interest in the analysis of Keller–Segel models incorporating tensorial flux. Despite this interest, the question of whether finite-time blowup solutions exist remains a topic of ongoing research. Our study provides evidence that solutions of this nature are indeed possible in dimensions n≥3, when utilizing a tensorial flux expressed in the form of A∇v, where A denotes a matrix with constant components.

Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D

Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D

Stability of steady‐state solutions of a class of Keller–Segel models with mixed boundary conditions

In this paper, we investigate the existence and stability of non‐trivial steady‐state solutions of a class of chemotaxis models with zero‐flux boundary conditions and Dirichlet boundary conditions on a one‐dimensional bounded interval. By using upper–lower solution and the monotone iteration scheme method, we get the existence of the steady‐state solution of the chemotaxis model. Moreover, by adopting the “inverse derivative” technique and the weighted energy method, we prove the stability of the steady‐state solution of this chemotaxis model.

Analytical Solution of One-Dimensional Keller-Segel Equations via New Homotopy Perturbation Method

For solving a system of nonlinear partial differential equations (PDE) emerging in an attractor one-dimensional chemotaxis model, we used a relatively new analytical method called the new modified homotopy perturbation method (NMHPM). We use NMHPM for solving one-dimensional Keller-Segel models for different types. Some properties show biologically acceptable dependency on parameter values, and numerical solutions are provided. NMHPM’s stability and reduced computing time provide it with a broader range of applications. The algorithm provides analytical approximations for different types of Keller-Segel equations. Some numerical illustrations are given to show the efficiency of the algorithm.

Spinodal decomposition and phase separation in polar active matter.

We develop and study the hydrodynamic theory of flocking with autochemotaxis. This describes large collections of self-propelled entities all spontaneously moving in the same direction, each emitting a substance which attracts the others (e.g., ants). The theory combines features of the Keller-Segel model for autochemotaxis with the Toner-Tu theory of flocking. We find that sufficiently strong autochemotaxis leads to an instability of the uniformly moving state (the "flock"), in which bands of different density form moving parallel to the mean flock velocity with different speeds. This instability is, therefore, completely different from the well-known "banding instability," in which bands form perpendicular to the mean flock velocity. The bands we find, which are reminiscent of ant trails, coarsen over time to reach a phase-separated state, in which one high-density and one low-density band fill the entire system. The same instability, described by the same hydrodynamic theory, can occur in flocks phase separating due to any microscopic mechanism (e.g., sufficiently strong attractive interactions). Although in many ways analogous to equilibrium phase separation via spinodal decomposition, the two steady-state densities here are determined not by a common tangent construction, as in equilibrium, but by an uncommon tangent construction very similar to that found for motility-induced phase separation of disordered active particles. Our analytic theory agrees well with our numerical simulations of our equationsof motion.

Critical mass phenomenon for a parabolic–elliptic multispecies chemotaxis system in a two‐dimensional disk

A parabolic–elliptic Keller–Segel model for multipopulations in a two‐dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy–Littlewood–Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow‐up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system.

Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation

<abstract><p>This study focuses on developing efficient numerical techniques for solving the fractional Keller-Segel (KS) model, which is critical in explaining chemotaxis events. Within the Caputo operator framework, the study applied two unique methodologies: The Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). These approaches were used to find precise solutions to the fractional KS equation, resulting in a better understanding of chemotactic behavior in biological systems. The comparative examination of the ARPSM and ATIM revealed their distinct strengths and applications in solving complicated fractional models. The work advances numerical approaches for fractional differential equations and improves our understanding of chemotaxis dynamics using a precise modeling approach.</p></abstract>

Synchronized rotations of active particles on chemical substrates.

Many microorganisms use chemical 'signaling' - a quintessential self-organizing strategy in non-equilibrium - that can induce spontaneous aggregation and coordinated motion. Using synthetic signaling as a design principle, we construct a minimal model of active Brownian particles (ABPs) having soft repulsive interactions on a chemically quenched patterned substrate. The interplay between chemo-phoretic interactions and activity is numerically investigated for a proposed variant of the Keller-Segel model for chemotaxis. Such competition not only results in a chemo-motility-induced phase-separated state, but also results in a new cohesive clustering phase with synchronized rotations. Our results suggest that rotational order can emerge in systems by virtue of activity and repulsive interactions alone without an explicit alignment interaction. These rotations can also be exploited by designing mechanical devices that can generate reorienting torques using active particles.

Can Dirac-type singularities in Keller-Segel systems be ruled out by power-type singular sensitivities?

This paper deals with the chemotaxis system with power-type singular sensitivities: ut=Δu−χ∇⋅(uvα∇v) and vt=Δv−v+u in a bounded domain Ω⊂R2, with χ>0 and α>0. It is notable that, whenever α=0, this system reduces to the minimal Keller-Segel model; the persistent Dirac-type singularities are known to occur at least in radial parabolic-elliptic setting. In this work, our first result shows that for any χ>0, if eitherα≥14, orα∈(0,14) and ‖u0‖L1 is small appropriately, then the corresponding initial-boundary value problem possesses a global generalized solution (u,v) with the property that u belongs to L1(Ω×(0,T)) for any T>0, which further implies that the general sensitivities 1vα, with α≥14, can rule out the emergence of Dirac-type singularities. Our second result indicates that, if α∈(0,12) and ‖u0‖L1 is small properly, then such global generalized solution becomes bounded and smooth at least eventually, and approaches the spatial equilibria in the large time limit.

Critical mass capacity for two-dimensional Keller–Segel model with nonlocal reaction terms

This paper deals with the parabolic–elliptic Keller–Segel system on , involving a source term of logistic type defined in terms of the mass capacity M 0 and the total mass of the individuals. We exhibit that the qualitative behaviour of solutions is decided by the mass capacity M 0 and the initial mass m 0. For general solutions, the existence of a global weak solution is proved under the assumption that both M 0 and m 0 are less than 8π, whereas there exist solutions blowing up in finite time under the hypotheses of either with any integrable initial data or accompanied with large initial data. Moreover, gives rise to a compromise that solutions exist globally and blow up as time goes to infinity. For radially symmetric solutions, we introduce a strategy of relegating the lack of mass conservation via a transformation to the density and then obtain that there are stationary solutions given by with λ > 0. Subsequently we prove that if the initial data is strictly below for some λ > 0, then the solution vanishes in as . If the initial data is strictly above for some λ > 0, then the solution either blows up in finite time or has a mass concentration at the origin as time goes to infinity. Finally, our results are complemented by numerical simulations that demonstrate the asymptotic behaviour of solutions.

Alignment interaction and band formation in assemblies of autochemorepulsive walkers.

Chemotaxis refers to the motion of an organism induced by chemical stimuli and is a motility mode shared by many living species that has been developed by evolution to optimize certain biological processes such as foraging or immune response. In particular, autochemotaxis refers to chemotaxis mediated by a cue produced by the chemotactic particle itself. Here, we investigate the collective behavior of autochemotactic particles that are repelled by the cue and therefore migrate preferentially towards low-concentration regions. To this end, we introduce a lattice model inspired by the true self-avoiding walk which reduces to the Keller-Segel model in the continuous limit, for which we describe the rich phase behavior. We first rationalize the chemically mediated alignment interaction between walkers in the limit of stationary concentration fields, and then describe the various large-scale structures that can spontaneously form and the conditions for them to emerge, among which we find stable bands traveling at constant speed in the direction transverse to the band.