Abstract. We code the wave-front tracking algorithm for Hughes' model of pedestrian motion with generalized running cost. This model displays a non-classical dynamic at the splitting point between the two directions of motion. The wave-front tracking scheme provides us with reference solutions to test numerically the convergence of classical finite volume schemes, which do not treat explicitly the dynamics at the turning point.

Our understanding of how roots develop in soil may be at the eve of significant transformations. The formidable expansion of imaging technologies enables live observations of the rhizosphere micro-pore architecture at unprecedented resolution. Granular matter physics provides ways to understand the microscopic fluctuations of forces in soils, and the increasing knowledge of plant mechanobiology may shed new lights on how roots perceive soil heterogeneity. This opinion paper exposes how recent scientific achievements may contribute to refresh our views on root growth in heterogeneous environments.

In this article, we present a simplified model to describe the dynamics of two groups of pedestrians moving in opposite directions in a corridor. The model consists of a 2 x 2 system of conservation laws of mixed hyperbolic-elliptic type. We study the basic properties of the system to understand why and how bounded oscillations in numerical simulations arise. We show that Lax-Friedrichs scheme ensures the invariance of the domain and we investigate the existence of measure-valued solutions as limit of a subsequence of approximate solutions.

Although migrations are essential for soil microorganisms to exploit scarce and heterogeneously distributed resources, bacterial mobility in soil remains poorly studied due to experimental limitations. In this study, time-lapse images collected using live microscopy techniques captured collective and coordinated groups of B. subtilis cells exhibiting "crowd movement". Groups of B. subtilis cells moved through transparent soil (nafion polymer with particle size resembling sand) toward plant roots and re-arranged dynamically around root tips in the form of elongating and retracting "flocks" resembling collective behaviour usually associated with higher organisms (e.g., bird flocks or fish schools). Genetic analysis reveals B. subtilis flocks are likely driven by the diffusion of extracellular signalling molecules (e.g., chemotaxis, quorum sensing) and may be impacted by the physical obstacles and hydrodynamics encountered in the soil like environment. Our findings advance understanding of bacterial migration through soil matrices and expand known behaviours for coordinated bacterial movement.