In this paper we use the di usion-limit expansion of transport equations developed earlier [23] to study the limiting equation under a variety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modi cation of the turning rate, the movement speed or the preferred direction of movement. Depending on the strength of the bias, it leads to anis...

We study a model equation that mimics convection under rotation in a fluid with temperaturedependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Kiippers-Lortz instability [Kuppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a wea...

Abstract. In this paper, we propose a general framework for designing numerical schemes that have both well-balanced (WB) and asymptotic preserving (AP) properties, for various kinds of kinetic models. We are interested in two different parameter regimes, 1) When the ratio between the mean free path and the characteristic macroscopic length ε tends to zero, the density can be described by (adve...

The \closure problem" is a well known and widely discussed problem in transport theory. From the full transport equation one can derive a (innnite) sequence of hyperbolic subsystems for the moments. The question arises how to close the system for the rst n moments. In case of Boltzmann equations it can be answered in the theory of extended thermodynamics. Here we consider transport equations wh...

A hybrid stochastic individual-based model of proliferating cells with chemotaxis is presented. The expressed by a branching diffusion process coupled to partial differential equation describing concentration chemotactic factor. It shown that in the hydrodynamic limit when number goes infinity converges solution nonconservative Patlak-Keller-Segel-type system. nonlinear mean-field defined and i...

For a specific choice of the diffusion, the parabolic-elliptic PatlakKeller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass Mc > 0 such that all the solutions with initial data of mass smaller or equal to Mc exist globally while the solution blows up in finite time ...

In two space dimensions, the parabolic-parabolic Keller-Segel system shares many properties with the parabolic-elliptic Keller-Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M(c). However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mas...