Keller Segel Equation Research Articles

Travelling waves with continuous profile for hyperbolic Keller-Segel equation

Abstract This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of travelling waves with continuous profiles. This article complements our previous results about sharp travelling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result. An application to wound healing also illustrates the importance of travelling waves with a continuous and discontinuous profile.

Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations

Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations

Asymptotic upper bound life span estimates for L1-solutions of the 2-D Patlak–Keller–Segel equation

We show asymptotic upper bound life span estimates for L1-solutions of the 2-D Patlak–Keller–Segel equation with large initial data. The proof is based on a modification of the monotonicity formula introduced by Wei in [17] recently.

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.

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>

Long-time dynamics of classical Keller–Segel equation

In this article, we obtain global dynamics of 2D Keller–Segel equation (KS for short) in subcritical regime with no additional assumptions. With initial data small in some Triebel-Lizorkin space, we also characterize the long-time asymptotics of global mild solutions to the KS equation in higher dimensions where the spatial dimension n≥3.

Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations

Abstract The Keller–Segel–Navier–Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L 1(Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π may be inherited from the fluid-free Keller–Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.

A simple reaction–diffusion system as a possible model for the origin of chemotaxis

Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis – namely how a directed movement in response to an external chemical signal may occur based on purely reaction–diffusion equations reflecting inner working of the cells. The model is inspired by the well-studied role of the rho-GTPase Cdc42 regulator of cell polarity, in particular in yeast cells. We analyse several versions of the model to better understand its analytic properties and prove global regularity in one and two dimensions. Using computer simulations, we demonstrate that in the framework of this model, at least in certain parameter regimes, the speed of the directed movement appears to be proportional to the size of the gradient of signalling chemical. This coincides with the form of the chemical drift in the most studied mean field model of chemotaxis, the Keller–Segel equation.

An Unconditionally Energy Stable and Positive Upwind DG Scheme for the Keller–Segel Model

The well-suited discretization of the Keller–Segel equations for chemotaxis has become a very challenging problem due to the convective nature inherent to them. This paper aims to introduce a new upwind, mass-conservative, positive and energy-dissipative discontinuous Galerkin scheme for the Keller–Segel model. This approach is based on the gradient-flow structure of the equations. In addition, we show some numerical experiments in accordance with the aforementioned properties of the discretization. The numerical results obtained emphasize the really good behaviour of the approximation in the case of chemotactic collapse, where very steep gradients appear.

Particle approximation of the doubly parabolic Keller-Segel equation in the plane

In this work, we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any N≥2, we show tightness in N of its empirical measure, and that any weak limit point of this empirical measure, as N→∞, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.

Nonlinear Diffusion for Bacterial Traveling Wave Phenomenon.

The bacterial traveling waves observed in experiments are of pulse type which is different from the monotone traveling waves of the Fisher-KPP equation. For this reason, the Keller-Segel equations are widely used for bacterial waves. Note that the Keller-Segel equations do not contain the population dynamics of bacteria, but the population of bacteria multiplies and plays a crucial role in wave propagation. In this paper, we consider the singular limits of a linear system with active and inactive cells together with bacterial population dynamics. Eventually, we see that if there are no chemotactic dynamics in the system, we only obtain a monotone traveling wave. This is evidence that chemotaxis dynamics are needed even if population growth is included in the system.

Singular limit problem for the Keller–Segel system and drift-diffusion system in scaling critical Besov–Morrey spaces

A singular limit problem for the Cauchy problem of the Keller–Segel equation is considered in a critical function space based on Morrey spaces. A solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift-diffusion system of parabolic-elliptic type in the critical space strongly as the relaxation time tends to infinity. The key ingredient is a Banach space-valued Lorentz space in time. By the careful use of the real interpolation, a sharp estimate for the heat semigroup is obtained.

Bound-preserving finite element approximations of the Keller–Segel equations

This paper aims to develop numerical approximations of the Keller–Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a non-negative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear and can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and nonblowup phenomena. In the blowup setting, we identify a locking phenomenon that relates the [Formula: see text]-norm to the [Formula: see text]-norm limiting the growth of the singularity when supported on a macroelement.

Modeling bacterial traveling wave patterns with exact cross-diffusion and population growth

Keller-Segel equations are widely employed to explain chemotaxis-induced bacterial traveling band phenomena. In this system, the dispersal of bacteria is modeled by independently given diffusion and advection terms, and the growth of cell population is neglected. In the paper, we develop a chemotaxis model which consists of cross-diffusion and population growth. In particular, we consider the case that the diffusion and advection terms form an exact cross-diffusion. The developed mathematical models are based on the conversion dynamics between active and inactive cells with different dispersal rates. The process consists of three steps and the performance of each step is complemented by comparing numerical simulations and experimental data.

Transition and bifurcation analysis for chemotactic systems with double eigenvalue crossings

<abstract><p>Our main objective of this research is to study the dynamic transition for diffusive chemotactic systems modeled by Keller-Segel equations in a rectangular domain. The main tool used is the recently developed dynamic transition theory. Through a reduction analysis and focusing on systems with certain symmetry where double eigenvalue crossing occurs during the instability process, it is shown that the chemotactic system can undergo both continuous and jump type transitions from the steady states, depending on non-dimensional parameters $ \alpha $, $ \mu $ and the side length $ L_1 $ and $ L_2 $ of the container. Detailed dynamic structures during transition, including metastable and stable states and orbital connections between them, are rigorously obtained. This result extends the previous work with only one eigenvalue crossing at critical parameters and offers more complex insights given the symmetry of our settings.</p></abstract>

A simple proof of non-explosion for measure solutions of the Keller-Segel equation

<p style='text-indent:20px;'>We give a simple proof, relying on a <i>two-particles</i> moment computation, that there exists a global weak solution to the <inline-formula><tex-math id="M1">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M3">\begin{document}$ f_0( {\mathbb{R}}^2)< 8 \pi $\end{document}</tex-math></inline-formula>.</p>

Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

We consider the singular limit problem for the Cauchy problem of the (Patlak–) Keller–Segel system of parabolic-parabolic type. The problem is considered in the uniformly local Lebesgue spaces and the singular limit problem as the relaxation parameter tau goes to infinity, the solution to the Keller–Segel equation converges to a solution to the drift-diffusion system in the strong uniformly local topology. For the proof, we follow the former result due to Kurokiba–Ogawa [20–22] and we establish maximal regularity for the heat equation over the uniformly local Lebesgue and Morrey spaces which are non-UMD Banach spaces and apply it for the strong convergence of the singular limit problem in the scaling critical local spaces.

On the initial value problem for the hyperbolic Keller-Segel equations in Besov spaces

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u0 is discontinuous at t=0 in the metric of B2,∞s(R), s>32. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces Bp,11+dp(Rd), 1≤p<∞, which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ., 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵΔu as ϵ→0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory.

Analysis of Fractional-Order System of One-Dimensional Keller–Segel Equations: A Modified Analytical Method

In this paper, an analytical method is implemented to solve fractional-order Keller–Segel equations. The Yang transformation along with the Adomian decomposition method is implemented to obtain the solution of the given problems. The present method has an edge over other techniques as it does not need extra calculations and materials. The validity of the suggested technique is verified by considering some numerical problems. The results obtained confirm the better accuracy of the current technique. The suggested technique has a lesser number of calculations and is straightforward to apply and therefore can be applied to other fractional-order partial differential equations.

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.