Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain

We derive the Virial theorem appropriate to the generalized Smoluchowski-Poisson system describing self-gravitating Brownian particles and bacterial populations (chemotaxis). We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the Virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature T_c in dimension d=2 and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity or chemotactic attraction. We also study the collapse at T=T_c and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for d>2, we show that the evaporation of the system is essentially a pure diffusion slightly slowed-down by self-gravity. We also study the linear dynamical stability of stationary solutions of the generalized Smoluchowski-Poisson system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the dynamical stability of the system. Finally, we propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles and biological populations and recover known models in some particular limits

Hyperbolic Techniques in Modelling, Analysis and Numerics

Several research areas are flourishing on the roots of the breakthroughs in conservation laws that took place in the last two decades. The meeting played a key role in providing contacts among the different branches that are currently developing. All the invitees shared the same common background that consists of the analytical and numerical techniques for nonlinear hyperbolic balance laws. However, their fields of applications and their levels of abstraction are very diverse. The workshop was the unique opportunity to share ideas about analytical issues like the fine-structure of singular solutions or the validity of entropy solution concepts. It turned out that generalized hyperbolic techniques are able to handle the challenges posed by new applications. The design of efficient structure preserving methods turned out to be the major line of development in numerical analysis