Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision

Abstract

This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.

Stability of a delayed SARS-CoV-2 reactivation model with logistic growth and adaptive immune response

“…Model (5) – (10) assumes that viruses and cells are well mixed and neglected the mobility of them. Viruses and cells can move and go from high concentration regions to low concentration regions [56] , [57] , [58] , [59] . Taking into account the mobility of viruses and cells, model (5) – (9) can be extended as: where , is the Laplacian operator and is the diffusion coefficient corresponding to compartment of the model.…”

Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data

“…Depending on the level of information and spatio-temporal scale of analysis, a vast number of mathematical models have been proposed ranging from the molecular/individual [11,12,13,14,15] to the continuum/macroscopic scale [16,17,18,19,13,20,21] (for an extensive review of both modelling approaches see [22,23,13,21]. The celebrated Keller-Segel [17] PDE derived for the macroscopic description of the population density evolution, coupled with the concommitant chemoattractant field, constitutes the cornerstone in the field.…”

Pattern formation of a pathway-based diffusion model: linear stability analysis and an asymptotic preserving method

“…The anisotropic diffusion term is of particular interest since it appears in other models and applications in the literature. For example, the cross diffusionreaction limit model of Shigesada-Kawasaki-Teramoto system [7,14], the stationary Fokker-Planck-Kolmogorov system [4], the Kermack-McKendrick model with nonlocal source terms [3], the predator-prey model with nonlinear cross-diffusion [6] and so on.…”