Numerical approximation of one model of bacterial self-organization

Abstract

This paper presents finite difference approximations of one dimensional in space mathematical model of a bacterial self-organization. The dynamics of such nonlinear systems can lead to formation of complicated solution patterns. In this paper we show that this chemotaxisdriven instability can be connected to the ill-posed problem defined by the backward in time diffusion process. The method of lines is used to construct robust numerical approximations. At the first step we approximate spatial derivatives in the PDE by applying approximations targeted for special physical processes described by differential equations. The obtained system of ODE is split into a system describing separately fast and slow physical processes and different implicit and explicit numerical solvers are constructed for each subproblem. Results of numerical experiments are presented and convergence of finite difference schemes is investigated.

A Preconditioned Iterative Solution Scheme for Nonlinear Parabolic Systems Arising in Air Pollution Modeling

“…In fact, this approach can be used in a similar way for other time-dependent parabolic reaction-convection-diffusion systems arising in the context of chemical or biological interactions. For instance, models describing chemotaxis-growth systems, when nonlinear advection part of the model starts to be a leading mechanism of dynamics, are described in [3,12], and various other reaction-convection-diffusion systems are given in [5].…”

“…Some of the coefficients belong to photochemical reactions (the ones with the term hν), which means that these reactions depend on the light, more precisely on the position of the Sun relative to the horizon: in k 2 , k 5 and k 7 the angle θ denotes the solar zenith angle, which is the angle of the Sun measured from vertical. 6.0 · 10 −12 k 6 1.6 · 10 −14 k 2 7.8 · 10 −05 · exp(−0.87/ cos θ) k 7 1.9 · 10 −04 · exp(−1.9/ cos θ) k 3 8.0 · 10 −12 k 8 2.3 · 10 −10 k 4 8.0 · 10 −12 k 9 1.0 · 10 −11 k 5 1.0 · 10 −02 · exp(−0.39/ cos θ) k 10 2.9 · 10 −13…”

Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

“…If ρ → 1, the nonlocal Riemann-Liouville integral condition reduces to a nonlocal integral condition which plays an important role in computational fluid dynamics, ill-posed problems and mathematical models [62].…”

Computational modeling of the bacterial self-organization in a rounded container: The effect of dimensionality

“…The mathematical model (23)- (25) has been successfully used to study the bacterial self-organization of luminous E. coli along the contact line of the circular container as detected by bioluminescence imaging [2,38,40]. Here this model was derived as a very special case of the common 3D mathematical model (10)-(15).…”