Chemotaxis Equations Research Articles

وجود حل محلي لنظام الجذب الكيميائي والتنافر مع الانتشار غير الخطي والمصدر اللوجستي

In this paper, we are concerned with a model arising from biology, which is a coupled system of chemotaxis equations and viscous compressible fluid equations through transport and external forcing. The local existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for an attraction-repulsion chemotaxis model system over three space dimensions, we obtain local existence and uniqueness of convergence on classical solutions near constant states. We prove local existence of unique solutions in three dimensions by using energy estimates.

Fully discrete stabilized mixed finite element method for chemotaxis equations on surfaces

In this paper, we study fully discrete stabilized mixed finite element methods for solving chemotaxis equations on surfaces. By defining two global fluxes as new vector-valued variables, the original equations are firstly transformed into an equivalent first order mixed form. Then, employing the stabilized mixed finite element method for the spatial discretization, and the backward Euler and backward differential formulation of second order (BDF2) schemes for the temporal discretization, respectively, we propose the first and second order fully discrete stabilized finite element methods for the models. Furthermore, by utilizing a modified scalar auxiliary variable (SAV) method to balance the errors in the stabilizing and decoupling processes, we prove the stabilities and the inf–sup conditions of the stabilized methods. Finally, several numerical examples are presented to verify the efficiencies of the proposed methods.

A spatial model to understand tuberculosis granuloma formation and its impact on disease progression

Abstract Tuberculosis (TB) is caused by a bacterium called Mycobacterium tuberculosis (Mtb). When Mtb enters inside the pulmonary alveolus, it is phagocytosed by the alveolar macrophages, followed by a cascade of immune responses. This leads to the recruitment and accumulation of additional macrophages and T cells in the pulmonary tissues. A key outcome of this is the formation of granuloma, the hallmark of TB infection. In this paper, we develop a mathematical model of the evolution of granuloma by a system of partial differential equations that is based on the classical Keller–Segel chemotaxis equation. We investigate the effect of different parameters on the formation of granuloma. We present numerical simulation results that illustrate the impact of different parameters. The implication of our result on the disease progression is also discussed.

Boundedness of solutions for parabolic-elliptic predator-prey chemotaxis-fluid system with logistic source term

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics{(n1)t+u⋅∇n1=Δn1−χ∇⋅(n1∇w)+n1(λ1−μ1n1+an2),x∈Ω,t>0,(n2)t+u⋅∇n2=Δn2+ξ∇⋅(n2∇z)+n2(λ2−μ2n2−bn1),x∈Ω,t>0,wt+u⋅∇w=Δw−w+n2,x∈Ω,t>0,u⋅∇z=Δz−z+n1,x∈Ω,t>0,ut+∇P=Δu+(n1+n2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under no-flux boundary conditions for n1, n2, w and z in three-dimensional bounded domains and no-slip boundary conditions for u, this is∂n1∂ν=∂n2∂ν=∂w∂ν=∂z∂ν=0,u=0,x∈∂Ω,t>0, where χ>0, ξ>0, μ1≥0, μ2≥0, λ1≥0, λ2≥0, a≥0, b≥0 and ϕ∈W2,∞(Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution.

Solutions to the 1D coupled chemotaxis equations using the generalised exponential rational function method

Solutions to the 1D coupled chemotaxis equations using the generalised exponential rational function method

Nonlocal Symmetries of the System of Chemotaxis Equations with Derivative Nonlinearity

Nonlocal Symmetries of the System of Chemotaxis Equations with Derivative Nonlinearity

Analyzing the Solution of Chemotaxis Equations with Logistic Source Term

When the parameters have the characteristics of sensitivity and specificity, while the traditional analysis method is the analysis of equations, under the influence of the characteristics of parameters, because the ability to consider the overall existence and limitations of the equation is weak, this leads to the deviation of the analysis results beyond the allowable range. With the chemotactic equations with logistic source terms, a new analysis method is proposed, on the basis of setting the optimal adjustment parameters of logistic source term, the method evaluates the well-posed behavior of the solution of the chemotactic equation system with logistic source term by quantitative analysis of the global existence and limitations of the solution. The experimental results show that the proposed analysis method, after the analysis of the overall existence and limitations of the equation, has smaller deviation in the results of the appropriate qualitative evaluation and meets the analysis requirements of the solution problem of the chemotaxis equations. It can be seen that the analysis method is more practical.

Optimal control of a chemotaxis equation arising in angiogenesis

<abstract><p>In this paper we consider an optimal control for an equation that models a crucial step in the tumor development, the angiogenesis. We show the existence of an optimal control, we characterize the optimal control as a solution of the optimality system and we show the uniqueness of the optimal control for short times.</p></abstract>

Chemotactic traveling waves with compact support

A logarithmic model type chemotaxis equation is introduced with porous medium diffusion and a population dependent consumption rate. The classical assumption that individual bacterium can sense the chemical gradient is not taken. Instead, the chemotactic term appears by assuming that the migration distance is inversely proportional to the amount of food if food is the reason for migration. The existence and uniqueness of a traveling wave solution of the model are obtained. In particular, solutions have interfaces that divide into constant and non-constant regions. In particular, the profile of the population distribution has compact support. Numerical simulations are provided and compared with analytic results.

Global existence and asymptotic stability of the fractional chemotaxis–fluid system in [formula omitted

In this paper, we consider a model arising from biology, consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in R3. We establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument. Moreover, the decay rates of the solutions and their higher-order spatial derivatives are established towards the equilibrium by introducing the negative Sobolev space Ḣ−s(0⩽s<32).

Decay estimate of solutions to the coupled chemotaxis–fluid equations in [formula omitted

We are concerned with a model arising from biology, which is coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. We study the large time behavior of solutions near a constant states to the chemotaxis-Navier–Stokes system in R3. Appealing to a pure energy method, we first obtain a global existence theorem by assuming that the H3 norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms Ḣ−s(0≤s<32) or homogeneous Besov norms Ḃ2,∞−s(0<s≤32), we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual Lp−L2(1≤p≤2) type of the decay rates without requiring that the Lp norm of initial data is small.

Numerical analysis of a chemotaxis–swimming bacteria model on a general triangular mesh

This paper is devoted to the numerical study of a model arising from biology, consisting of chemotaxis equations coupled to Navier–Stokes flow through transport and external forcing. A detailed convergence analysis of this chemotaxis–fluid model by means of a suitable combination of the finite volume method and the nonconforming finite element method is investigated. In the case of nonpositive transmissibilities, a correction of the diffusive fluxes is necessary to maintain the monotonicity of the numerical scheme. Finally, many numerical tests are given to illustrate the behavior of the anisotropic Keller–Segel–Stokes system.

A regularity condition and temporal asymptotics for chemotaxis-fluid equations

We consider two dimensional chemotaxis equations coupled to the Navier–Stokes equations. We present a new localized regularity criterion that is localized in a neighborhood at each point. Secondly, we establish temporal decays of the regular solutions under the assumption that the initial mass of biological cell density is sufficiently small. Both results are improvements of previously known results given in Chae et al (2013 Discrete Continuous Dyn. Syst. A 33 2271–97) and Chae et al (2014 Commun. PDE 39 1205–35)

INVESTIGATION OF A STRUCTURED FISHER'S EQUATION WITH APPLICATIONS IN BIOCHEMISTRY.

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.

Critical spaces for quasilinear parabolic evolution equations and applications

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal Lp-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier–Stokes problem, convection–diffusion equations, the Nernst–Planck–Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.

Nonlinear flux-limited models for chemotaxis on networks

In this paper we consider macroscopic nonlinear moment models for the approximation of kinetic chemotaxis equations on a network. Coupling conditions at the nodes of the network for these models are derived from the coupling conditions of kinetic equations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations an asymptotic well-balanced schemes is extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

Monte Carlo simulation for kinetic chemotaxis model: An application to the traveling population wave

A Monte Carlo simulation of chemotactic bacteria is developed on the basis of the kinetic model and is applied to a one-dimensional traveling population wave in a microchannel. In this simulation, the Monte Carlo method, which calculates the run-and-tumble motions of bacteria, is coupled with a finite volume method to calculate the macroscopic transport of the chemical cues in the environment. The simulation method can successfully reproduce the traveling population wave of bacteria that was observed experimentally and reveal the microscopic dynamics of bacterium coupled with the macroscopic transports of the chemical cues and bacteria population density. The results obtained by the Monte Carlo method are also compared with the asymptotic solution derived from the kinetic chemotaxis equation in the continuum limit, where the Knudsen number, which is defined by the ratio of the mean free path of bacterium to the characteristic length of the system, vanishes. The validity of the Monte Carlo method in the asymptotic behaviors for small Knudsen numbers is numerically verified.

A Hydrodynamic Limit for Chemotaxis in a Given Heterogeneous Environment

In this paper the first equation within a class of well known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically the limiting procedure and its proofs are based on results by Koukkus [18] and Kipnis/Landim [17]. Numerical simulations extend and illustrate the theoretical findings.

Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions

The aim of this work is to investigate the application of partial moment approximations to kinetic chemotaxis equations in one and two spatial dimensions. Starting with a kinetic equation for the cell densities we apply a half-/quarter-moments method with different closure relations to derive macroscopic equations. Appropriate numerical schemes are presented as well as numerical results for several test cases. The resulting solutions are compared to kinetic reference solutions and solutions computed using a full moment method with a linear superposition strategy.

Memory Effect in Chemotaxis Equation

Diffusion-Reaction (DR) equation has been used to model a large number of phenomena in nature. It may be mentioned that a linear diffusion equation does not exhibit any traveling wave solution. But there are a vast number of phenomena in different branches not only of science but also of social sciences where diffusion plays an important role and the underlying dynamical system exhibits traveling wave features. In contrast to the simple diffusion when the reaction kinetics is combined with diffusion, traveling waves of chemical concentration are found to exist. This can affect a biochemical change, very much faster than straight diffusional processes. This kind of coupling results into a nonlinear (NL) DR equation. In recent years, memory effect in DR equation has been found to play an important role in many branches of science. The effect of memory enters into the dynamics of NL DR equation through its influence on the speed of the travelling wavefront. In the present work, chemotaxis equation with source term is studied in the presence of finite memory and its solution is compared with the corresponding chemotaxis equation without finite memory. Also, a comparison is made between Fisher-Burger equation and chemotaxis equation in the presence of finite memory. We have shown that nonlinear diffusion-reaction-convection equation is equivalent to chemotaxis equation.