Keller-Segel Model Research Articles

Global Well-Posedness and Boundary Layer Effects of Radially Symmetric Solutions for the Singular Keller–Segel Model

Global Well-Posedness and Boundary Layer Effects of Radially Symmetric Solutions for the Singular Keller–Segel Model

Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller—Segel Model with Signal-Dependent Motilities for Local Sensing Chemotaxis

In this paper we consider the initial Neumann boundary value problem for a degenerate Keller—Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in the boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos—Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in [1] and the exponential stabilization is obtained for the first time.

Global existence of weak solutions to a Keller-Segel model with $$L^1$$ initial data

Global existence of weak solutions to a Keller-Segel model with $$L^1$$ initial data

On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

In this paper, we study the following fractional semilinear Neumann problem arising from Keller–Segel model: where is a smooth bounded domain and is the outward unit normal to . First, under the superlinear and subcritical growth assumptions on , we prove that there exists at least one positive nonconstant solution for small and the family of solutions is uniformly bounded. Moreover, we build a Pohozaev‐type identity for the ( ‐) Neumann harmonic extension of the following problem: where is a function such that . As a direct application of this identity, when satisfies , where , we deduce the nonexistence of weak bounded solution in star‐shaped domains.

Asymptotic stability of contraction-driven cell motion.

We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two- dimensional free-boundary model that generalizes a previous one-dimensional model [P. Recho, T. Putelat, and L. Truskinovsky, Phys. Rev. Lett. 111, 108102 (2013)10.1103/PhysRevLett.111.108102] by combining a Keller-Segel model, a Hele-Shaw boundary condition, and the Young-Laplace law with a regularizing term which precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We find a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is nonlinear asymptotic stability of traveling solutions that model observable steady cell motion. We derive an explicit asymptotic formula for the stability-determining eigenvalue via asymptotic expansions in small speed. This formula greatly simplifies computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability.

Finite element analysis of a Keller-Segel model with additional cross-diffusion and logistic source. Part II: Time convergence and numerical simulation

In [1], a finite element method for the (Patlak-) Keller-Segel equations, with additional cross-diffusion and logistic source terms in the elliptic equation, has been introduced. Some a priori estimates of the regularized solutions have been derived, independently of the regularization parameters. Moreover, the existence of the finite element solutions has been shown by using the fixed point theorem and some stability bounds on the fully discrete approximations are obtained. The convergence of the approximate solutions in space has been shown. In this article, we have completed what has been achieved in [1], where we have completed the convergence of time, where this article starts from where the study in [1] ended. The current study aims to prove the existence of weak solutions to this model, which includes passing the limit to the remaining regulation parameter and achieving the convergence in time. We firstly show that the solutions, independent of regulation parameter, can be bounded. Then these bounds are used to derive bounds on the time-derivatives which are also independent of regulation parameter. In addition, compactness arguments have been used to discuss the convergence of the finite element the approximate problem. We demonstrated the existence of a weak solution for the model under study. Finally, we use the implicit scheme to conduct simulations in one and two space dimensions.

Degenerate Cahn–Hilliard and incompressible limit of a Keller–Segel model

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy's law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard system.

Traveling waves in a Keller–Segel model with logistic growth

Traveling waves in a Keller–Segel model with logistic growth

THE ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO THE CHEMOTAXIS MODEL IN THE CRITICAL FRAMEWORK

<abstract> The Keller-Segel model is an effective mathematical model (derived by Keller and Segel), which is used to describe the phenomenon of chemotaxis in biological sciences. The purpose of this paper is to investigate the asymptotic behavior of solutions in the $ L^p $ framework by developing the pure energy approach (independent of spectral analysis). Precisely, a new low-frequency regularity of initial data is posted, which enables us to establish the Lyapunov-type inequality in time for energy norms. As a result, the large-time behavior of strong solutions near constant equilibrium can be obtained. The proof crucially depends on non standard product estimates and interpolations. It's worth noting that the smallness requirement of low frequencies is no longer needed. </abstract>

A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion

&lt;p style='text-indent:20px;'&gt;Many problems arising in biology display a complex system dynamics at different scales of space and time. For this reason, multiscale mathematical models have attracted a great attention as they enable to take into account phenomena evolving at several characteristic lengths. However, they require advanced model reduction techniques to reduce the computational cost of solving all the scales.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;In this work, we present a novel version of the Keller-Segel model of chemotaxis on embedded multiscale geometries, &lt;i&gt;i.e.&lt;/i&gt;, one-dimensional networks embedded in three-dimensional bulk domains. Applying a model reduction technique based on spatial averaging for geometrical order reduction, we reduce a fully three-dimensional Keller-Segel system to a coupled 3D-1D multiscale model. In the reduced model, the dynamics of the cellular population evolves on a one-dimensional network and its migration is influenced by a three-dimensional chemical signal evolving in the bulk domain. We propose the multiscale version of the Keller-Segel model as a realistic approach to describe the invasion of malignant cancer cells along the collagen fibers that constitute the extracellular matrix. Performing several numerical simulations, we investigate how the invasive abilities of the cells are affected by the topology of the network (&lt;i&gt;i.e.&lt;/i&gt;, matrix fibers orientation and alignment) as well as by three-dimensional spatial effects. We discuss these results in light of biological evidences.&lt;/p&gt;

A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller–Segel systems

&lt;abstract&gt;&lt;p&gt;It was shown that unbounded solutions of the Neumann initial-boundary value problem to the two-dimensional Keller–Segel system can be induced by initial data having large negative energy if the total mass $ \Lambda \in (4\pi, \infty)\setminus 4\pi \cdot \mathbb{N} $ and an example of such an initial datum was given for some transformed system and its associated energy in Horstmann–Wang (2001). In this work, we provide an alternative construction of nonnegative nonradially symmetric initial data enforcing unbounded solutions to the original Keller–Segel model.&lt;/p&gt;&lt;/abstract&gt;

Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

&lt;p style='text-indent:20px;'&gt;One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.&lt;/p&gt;

Existence and uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations

Abstract Keller-Segel chemotaxis model is described by a system of nonlinear partial differential equations: a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller-Segel model coupled with Boussinesq equations. The main objective of this work is to study the global existence and uniqueness and boundedness of the weak solution for the problem, which is carried out by the Galerkin method.

Radial spiky steady states of a flux‐limited Keller–Segel model: Existence, asymptotics, and stability

AbstractThis paper is concerned with the radial stationary problem of a flux‐limited Keller–Segel model derived in a multidimensional bounded domain with Neumann boundary conditions. With the global bifurcation theory and Helly compactness theorem by treating the chemotactic coefficient as a bifurcation parameter, we establish the existence of nonconstant monotone radial stationary solutions and further show that the radial stationary solution will tend to a Dirac delta mass as the chemotactic coefficient tends to infinity. By using the stability criterion of Crandall and Rabinnowitz, we prove the linearized stability of bifurcating stationary solutions near the bifurcation points.

Maximal regularity and a singular limit problem for the Patlak\u2013Keller\u2013Segel system in the scaling critical space involving BMO

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter tau rightarrow infty . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces dot{B}^s_{q,sigma }({mathbb {R}}^n) and dot{F}^s_{q,sigma }({mathbb {R}}^n) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

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].

Weak entropy solution for a Keller-Segel type fluid model

In this paper, we consider a Keller-Segel type fluid model, which is a kind of Euler-Poisson system with a self-gravitational force. We show that similar to the parabolic case, there is a critical mass $8\pi$ such that if the initial total mass $M$ is supercritical, i.e., $M> 8\pi$, then any weak entropy solution with the same mass $M$ must blow up in finite time. The a priori estimates of weak entropy solutions for critical mass $M=8\pi$ and subcritical mass $M<8\pi$ are also obtained.

Spectral stability of bacteria pulses for a Keller-Segel chemotactic model

This paper is concerned with the stability of traveling waves for a Keller-Segel model with singular chemotactic term and zero chemo-attractant diffusion, where the model and the waves are established to explain the propagation of bacteria pulses along a capillary tube observed in Adler's experiment [1]. By applying the detailed spectral analysis, Evans function method and special transformations and combining with some numerical simulations, in some range of the parameters all the waves are shown to be spectrally stable in some exponentially weighted spaces, and in other range of parameters all the waves are shown to be unstable in any exponentially weighted spaces. The local well-posedness of the classical positive solution to the Cauchy problem of the model is also obtained by applying semigroup argument and some special transformations.

Blow-up criteria for the classical Keller-Segel model of chemotaxis in higher dimensions

We study the simplest parabolic-elliptic model of chemotaxis in space dimensions N ≥ 3 , and show the optimal conditions on the initial data for the finite time blow-up and the global existence of solutions in terms of stationary solutions. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution.

Finite element approximation of a Keller–Segel model with additional self- and cross-diffusion terms and a logistic source

A fully finite element approximation of the Keller–Segel model, including self- and cross-diffusion terms and a logistic source, has been considered. The existence of a fully finite element approximation of the Keller–Segel model, some stability bounds of the solution and the convergence of the approximated solution have been shown. As a result, the existence of a solution to the nonlinear Keller–Segel model in Rd, d≤3 has been demonstrated. Moreover, an iterative approach to solving the resulted nonlinear discrete system has been introduced. Finally, some numerical results have been shown.