MesoCroMo · A Mesoscopic approach to Cross-diffusion Modelling in population dynamics

A Mesoscopic approach to Cross-diffusion Modelling in population dynamics

The description of natural phenomena with mathematical models is a fundamental step to understand the key physical mechanisms that dictate their evolution. By varying the scale of observation, these mechanisms can be formulated using microscopic first principles and conservation laws, and their effect is then made visible in the macroscopic equations that govern the evolution of the relevant observable quantities characterizing the phenomenon under study. A typical example is provided by gas dynamics, where the collisions between molecules determine the behavior of the physical quantities (density, temperature, pressure, etc.) describing the whole gaseous mixture. It is then crucial to determine the rigorous mathematical connections between these alternative formulations, to ensure that they are in fact coherent. The challenging task of the MesoCroMo project has been to construct a new modeling approach able to characterize, across multiple scales of observation, the evolution of multi-species systems in the context of population dynamics. In particular, the proposed approach had to be general and flexible enough to be adapted in a wide variety of settings, most notably for ecological and epidemiological applications. The ultimate goal of the project was the justification of macroscopic cross-diffusion models, as suitable asymptotic limits of Boltzmann-like equations. These models find numerous real-world applications due to their ability to capture relevant collective behaviors, emerging from the complex spatial movement of a species when its diffusion is influenced by the presence of another. As such, the project will also stand as a powerful mean to bridge the mathematical interest with other disciplines like physics, chemistry and biology. The specific objectives of the MesoCroMo project have been to 1) use the classical kinetic theory of reactive gaseous mixtures as a benchmark to set up an appropriate microscopic framework for inelastic interactions in multi-species populations, and to introduce new biologically meaningful Boltzmann-like operators modeling such interactions at a mesoscopic level; 2) determine simplified Fokker-Planck-type equations with kinetic nonconservative reaction operators describing birth/death processes and cooperative/competitive dynamics, and derive the corresponding macroscopic models; 3) extend the Fokker-Planck equations to include spatially inhomogeneous dynamics, and investigate the coupled diffusion/fast-reaction regimes of its parameters to derive macroscopic cross-diffusion-like systems; 4) characterize the equilibria of these simplified kinetic equations, study the stability of the equilibria, and analyze the trends to equilibrium of the kinetic solutions; 5) develop and implement an asymptotic-preserving numerical scheme able to simulate the evolution of the newly introduced Fokker-Planck equations and to capture their limit macroscopic models.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз