Keller-Segel Chemotaxis Model Research Articles

Spatial heterogeneity localizes Turing patterns in reaction-cross-diffusion systems

Motivated by bacterial chemotaxis and multi-species ecological interactions in heterogeneous environments, we study a general one-dimensional reaction-cross-diffusion system in the presence of spatial heterogeneity in both transport and reaction terms. Under a suitable asymptotic assumption that the transport is slow over the domain, while gradients in the reaction heterogeneity are not too sharp, we study the stability of a heterogeneous steady state approximated by the system in the absence of transport. Using a WKB ansatz, we find that this steady state can undergo a Turing-type instability in subsets of the domain, leading to the formation of localized patterns. The boundaries of the pattern-forming regions are given asymptotically by 'local' Turing conditions corresponding to a spatially homogeneous analysis parameterized by the spatial variable. We developed a general open-source code which is freely available, and show numerical examples of this localized pattern formation in a Schnakenberg cross-diffusion system, a Keller-Segel chemotaxis model, and the Shigesada-Kawasaki-Teramoto model with heterogeneous parameters. We numerically show that the patterns may undergo secondary instabilities leading to spatiotemporal movement of spikes, though these remain approximately within the asymptotically predicted localized regions. This theory can elegantly differentiate between spatial structure due to background heterogeneity, from spatial patterns emergent from Turing-type instabilities.

High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

<abstract><p>In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.</p></abstract>

Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model.

The paper is concerned with development of an accurate and effective positivity-preserving high-order compact difference method for solving the Keller-Segel chemotaxis model, which is a kind of nonlinear parabolic-parabolic system in mathematical biology. Firstly, a stiffly-stable five-step fourth-order fully implicit compact difference scheme is proposed. The new scheme not only has fourth-order accuracy in the spatial direction, but also has fourth-order accuracy in the temporal direction, and the computational strategy for the nonlinear chemotaxis term is provided. Then, a positivity-preserving numerical algorithm is presented, which ensures the non-negativity of cell density at all time without accuracy loss. And a time advancement algorithm is established. Finally, the proposed method is applied to the numerical simulation for chemotaxis phenomena, and the accuracy, stability and positivity-preserving of the new scheme are validated with several numerical examples.

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

<p style='text-indent:20px;'>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.</p>

Application of weak Galerkin finite element method for nonlinear chemotaxis and haptotaxis models

In this paper, we introduce a weak Galerkin finite element method for 2D Keller-Segel chemotaxis models and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue. The chemotaxis and haptotaxis models consist of a system of time-dependent nonlinear reaction-diffusion-taxis partial differential equations. It is well known that solutions of chemotaxis systems may blow-up in finite time in the cell density. This blow-up is a mathematical description of a cell concentration phenomenon, which mathematically results in rapidly growing solutions in small neighborhoods of concentration points or curves. Capturing such singular solutions numerically is a challenging problem. We applied the weak Galerkin finite element method (WGFEM) to approximate the spatial variables. Weak Galerkin finite element method can be considered as an extension of the standard finite element method where classical derivatives are replaced in the variational equation by the weakly defined derivatives on discontinuous weak functions. The weak derivatives of weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the WGFEM is impacted by the selection of such polynomials. We show that the proposed method produced numerical results that are nonnegative, oscillation-free and suitable for solving models that have a blow-up of the cell density. Error bounds for cell density and chemoattractant equation are obtained. The proposed method is applied to several Keller-Segel chemotaxis models and extended to the haptotaxis model for simulating the behavior of cancer cell invasion of surrounding healthy tissue. Numerical results demonstrate the accuracy and efficiency of the proposed scheme.

Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs

In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times. As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold.

Lower Bounds of Finite-Time Blow-Up of Solutions to a Two-Species Keller-Segel Chemotaxis Model

Lower Bounds of Finite-Time Blow-Up of Solutions to a Two-Species Keller-Segel Chemotaxis Model

Spatial movement with distributed memory

Diffusion has been widely applied to model animal movement that follows Brownian motion. However, animals typically move in non-Brownian ways due to their perceptual judgment. Spatial memory and cognition recently have received much attention in characterizing complicated animal movement behaviours. Explicit spatial memory is modeled via a distributed delayed diffusion term in this paper. The distributed time represents the memory growth and decay over time, and the spatial nonlocality reflects the dependence of spatial memory on location. When the temporal delay kernel is weak under the assumption that animals can immediately acquire knowledge and memory decays over time, the equation is equivalent to a Keller-Segel chemotaxis model. For the strong kernel with learning and memory decay stages, rich spatiotemporal dynamics, such as Turing and checker-board patterns, appear via spatially non-homogeneous steady-state and Hopf bifurcations.

Keller-Segel Chemotaxis Models: A Review

We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain.

Linearized Implicit Methods Based on a Single-Layer Neural Network: Application to Keller–Segel Models

This paper is concerned with numerical approximation of some two-dimensional Keller–Segel chemotaxis models, especially those generating pattern formations. The numerical resolution of such nonlinear parabolic–parabolic or parabolic–elliptic systems of partial differential equations consumes a significant computational time when solved with fully implicit schemes. Standard linearized semi-implicit schemes, however, require reasonable computational time, but suffer from lack of accuracy. In this work, two methods based on a single-layer neural network are developed to build linearized implicit schemes: a basic one called the each step training linearized implicit method and a more efficient one, the selected steps training linearized implicit method. The proposed schemes, which make use also of a spatial finite volume method with a hybrid difference scheme approximation for convection–diffusion fluxes, are first derived for a chemotaxis system arising in embryology. The convergence of the numerical solutions to a corresponding weak solution of the studied system is established. Then the proposed methods are applied to a number of chemotaxis models, and several numerical tests are performed to illustrate their accuracy, efficiency and robustness. Generalization of the developed methods to other nonlinear partial differential equations is straightforward.

A Positivity Preserving Moving Mesh Finite Element Method for the Keller–Segel Chemotaxis Model

In this paper, we present an efficient adaptive moving mesh finite element method for the numerical solution of the Keller–Segel chemotaxis model. The mesh points are continuously redistributed by a coordinate transformation defined from the computational domain to the physical domain to concentrate the grid nodes in regions of large solution variations in the physical domain. The Keller–Segel equations are discretized using an implicit-explicit finite element method using piecewise polynomials defined on triangular meshes. The spatial discretization scheme is designed with a positivity preserving property, if the initial solutions of the physical model are positive then the computed solutions stay positive at all time levels. Several numerical experiments are presented to demonstrate the performance of the proposed method for solving Keller–Segel model. The numerical results show that the proposed method can reduce the computational cost while improving the overall accuracy of the computed solutions of the Keller–Segel model.

Interactions of solitary pulses of E. coli in a one-dimensional nutrient gradient

Abstract We study the interaction of two bacterial pulses in a one-dimensional nutrient gradient. Simulations of the Keller–Segel chemotaxis model reveal two qualitatively distinct behaviors. As the two pulses approach one another, they either combine and move as a single pulse or, surprisingly, change direction and begin moving away from each other in the direction from which they originated. To study this phenomenon, we introduce a heuristic approximation to the spatial profiles of the pulses in the Keller–Segel model and derive a system of ordinary differential equations approximating the dynamics of the pulse centers of mass and widths. This approximation simplifies analysis of the global dynamics of the bacterial system and allows us to efficiently explore qualitative behavior changes under a range of parameter variations. We end by presenting experimental data showing that populations of E. coli display behavior that qualitatively agrees with our theoretical results.

Energy Dissipative Local Discontinuous Galerkin Methods for Keller–Segel Chemotaxis Model

In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller–Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.

Analytical solutions of the Keller-Segel chemotaxis model involving fractional operators without singular kernel

This paper discusses the application of analytical techniques, namely the Laplace homotopy perturbation method and the modified homotopy analysis transform method, for solving a coupled one-dimensional time-fractional Keller-Segel chemotaxis model. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on the homotopy polynomial. Fractional derivatives with exponential and Mittag-Leffler laws in Liouville-Caputo sense are considered. The effectiveness of both methods is demonstrated by finding the exact solutions of the Keller-Segel chemotaxis model. Some examples have been presented in order to compare the results obtained with both fractional-order derivatives.

A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions

In this paper we consider the quasilinear chemotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+f(u),x∈Ω,t>0,0=Δv−μ(t)+u,x∈Ω,t>0, with homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with n≥2, where χ>0, μ(t):=1|Ω|∫Ωu(x,t)dx and f∈C([0,∞))∩C1((0,∞)) is a logistic source of the form f(s)=as−bsκ with a≥0,b>0, κ>1 and s≥0, and the diffusion D∈C2([0,∞)) is supposed to satisfyD(s)≥D0s−mfor alls>0 with some D0>0 and m∈R. Given any b>0, when the logistic source is strong enough in the sense thatκ>m+3−4n+2andκ>2, it is shown that for any initial data u0∈C0(Ω¯) and n≥2 the problem possesses a unique global bounded classical solution. However, whenD(s)=D0s−mfor alls>0 with 4n−1<m≤0 in the sense that n≥5, and the effect of logistic source is weaker in the sense thatκ∈(1,(3−m)n−22n−2), it is shown that for arbitrary prescribed M0>0 there exists initial data u0∈C∞(Ω¯) satisfying ∫Ωu0=M0 such that the corresponding solution (u,v) of the system blows up in finite time in a ball Ω=B0(R)⊂Rn with some R>0. This result extends the blow-up arguments of the Keller–Segel chemotaxis model with logistic cell kinetics in Winkler [39] to more general quasilinear case. Moreover, since there is a gap in the proof of Zheng et al. [46], it also presents modified results for the mistake.

Existence of global solution to a two-species Keller–Segel chemotaxis model

In this paper, we investigate the global existence of nonnegative solutions of a two-species Keller–Segel model with Lotka–Volterra competitive source terms. By raising the regularity of a solution from [Formula: see text] to [Formula: see text], the existence and uniqueness of the classical global in time solution to this chemotaxis model is proved for any chemotactic coefficients [Formula: see text] when the space dimension is one. Furthermore, it is shown that the model has a unique classical global solution in two and three space dimensions if the chemotactic coefficients [Formula: see text] and [Formula: see text] are small as compared to the diffusion coefficient [Formula: see text] of the chemoattractant.

Numerical Modelling of C. elegans L1 Aggregation

We attempt to explain the aggregation behavior of starved C. elegans L1s (Artyukhin et al., 2015) by a variation of the Keller-Segel chemotaxis model (1970). Each worm releases a diffusible, unstable attractant. The worms chemotax towards the attractant. By using a continuous density function to approximate the spatial distribution of worms, behavior may be described by a nonlinear system of partial differential equations. To determine whether such a model can account for C. elegans L1 behavior, we solve this PDE system numerically. In the original Keller-Segel model, density is unbounded, leading to unrealistic results. We modified the model in such a way as to limit the maximum density. We also developed numerical methods designed to work well near equilibrium. With these changes our model can reproduce some but not all aspects of the behavior.

Persistence property in a two-species chemotaxis system with two signals

This paper deals with a two-species chemotaxis system with two different signals under homogeneous Neumann boundary conditions in a bounded convex domain with the non-negative initial data. This system is a generalization of the classical Keller-Segel chemotaxis models to the case of two species which are attracted by two different chemical signals. Under suitable conditions, it is proved that for any non-negative global classical solutions, the masses of two species do not extinct at any time.

Asymptotic nonlinear stability of a composite wave of two traveling waves to a chemotaxis model

In this paper, we investigate the asymptotic stability of a composite wave consisting of two traveling waves to a Keller–Segel chemotaxis model with logarithmic sensitivity and nonzero chemical diffusion. We show that the composite wave is asymptotically stable under general initial perturbation, which only be needed small in H1-norm. This improves previous results. Copyright © 2017 John Wiley & Sons, Ltd.

Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth

This paper investigates the formation of time--periodic and stable patterns of a two--competing--species Keller--Segel chemotaxis model with a focus on the effect of cellular growth. We carry out rigorous Hopf bifurcation analysis to obtain the bifurcation values, spatial profiles and time period associated with these oscillating patterns. Moreover, the stability of the periodic solutions is investigated and it provides a selection mechanism of stable time--periodic mode which suggests that only large domains support the formation of these periodic patterns. Another main result of this paper reveals that cellular growth is responsible for the emergence and stabilization of the oscillating patterns observed in the $3\times3$ system, while the system admits a Lyapunov functional in the absence of cellular growth. Global existence and boundedness of the system in 2D are proved thanks to this Lyapunov functional. Finally, we provide some numerical simulations to illustrate and support our theoretical findings.