We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor mode...

The paper is concerned with development of a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. The first step in the deriva...

We study the regularity and large-time behavior of a crowd species driven by chemo-tactic interactions. What distinguishes different is way they interact with rest crowd: collective motion chemical reactions which end up in coupled system parabolic Patlak-Keller-Segel equations. show that densities diffuse to zero provided interactions between satisfy certain sub-critical condition; latter inti...

In this paper a one-dimensional Keller-Segel model with a logarithmic chemotactic-sensitivity and a non-diffusing chemical is classified with respect to its long time behavior. The strength of production of the non-diffusive chemical has a strong influence on the qualitative behavior of the system concerning existence of global solutions or Dirac-mass formation. Further, the initial data play a...

This note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-LittlewoodSobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate o...

A brief historical survey of the development of asymptotic and analytical methodologies for the analysis of spatio-temporal patterns in reaction-diffusion (RD) and related systems is given. Although far from complete, the bibliography is hopefully representative of some of the advances in this area over the past 40 years. Within the scope of this survey, some of the key research contributions o...

The downstream development in both space and time of baroclinic instability is studied in a nonlinear channel model on the f -plane. The model allows the development of the instability to be expressed on space and time scales that are long compared to the growth rates and wavelengths of the most unstable wave. The unstable system is forced by time-varying boundary conditions at the origin of th...

We analyse an operator arising in the description of singular solutions to two-dimensional Keller-Segel problem. It corresponds linearised parabolic self-similar variables, close a concentrated stationary state. This is two-scale problem, with vanishing thin transition zone near origin. Via rigorous matched asymptotic expansions, we describe eigenvalues and eigenfunctions precisely. also show s...

Abstract The interplay between cellular growth and cell-cell signaling is essential for the aggregation proliferation of bacterial colonies, as well self-organization cell tissues. To investigate this interplay, we focus here on collective properties dividing chemotactic colonies by studying their long-time large-scale dynamics through a renormalization group (RG) approach. RG analysis reveals ...

The interaction among leukocytes is at the basis of the innate and adaptive immune-response and it is largely ascribed to direct cell-cell contacts. However, the exchange of a number of chemical stimuli (chemokines) allows also non-contact interaction during the immunological response. We want here to evaluate the extent of the effect of the non-contact interactions on the observed leukocyte-le...