A Mesoscopic approach to Cross-diffusion Modelling in population dynamics

ended

Cross-diffusion aims at modelling the influence of one species on the diffusion of another one and can be seen as an extension of the standard diffusion described by the Laplacian. It finds application in several fields like physics, chemistry and biology. Of particular interest in the literature is the macroscopic cross-diffusion Shigesada–Kawasaki–Teramoto (SKT) system which is considered the prototypical model to describe the dispersive movement of cross-interacting species in population dynamics, leading to the formation of spatial patterns (front invasions, segregation effects). Despite their importance and large use in the scientific community, their derivation from a mesoscopic formulation is still missing in the literature and the admissible ranges for the relevant macroscopic coefficients that would be needed for application purposes have not been yet identified. The novel challenge of MesoCroMo is to introduce new biologically meaningful multi-species Boltzmann-type models for competitive dynamics, obtain the first derivation of the SKT system from these kinetic equations in some suitable hydrodynamic limit, from a theoretical and a numerical point of view, and provide explicit relations for the cross-diffusion and reaction coefficients in terms of the mesoscopic parameters. This will be done by redesigning well-known tools and methods from the kinetic theory of reactive gases and creating an innovative bridge between the concept of relaxation to thermodynamic equilibrium for distribution functions in kinetic theory and that of quasi steady state approximation for fast-reaction limits in dynamical systems. The action will be able to gather mathematicians from different communities, give an impulse to existing lines of research and identify new long-term directions. Combined with the experience of the supervisor, the excellent training and the rich environment provided by the host, it will enhance the researcher towards an outstanding career in academia.