On a cross-diffusion segregation problem arising from a model of interacting particles

Abstract

We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.

Rigorous mean-field limit and cross-diffusion

“…where σ i > 0 and a ij are real numbers, starting from a stochastic many-particle system for multiple species. System (1) describes the diffusive dynamics of populations subject to segregation effects modeled by the term on the right-hand side [10].…”

An economic cross-diffusion mutualistic model for cities emergence

“…Starting from a general production function, we show that the gradient flow of the corresponding profits function may be approximated by a Lotka-Volterra system. We show that standard economic assumptions imply that this system is of the mutualistic type, fact which is illustrated with the consideration of the specific CES production function, see (9).…”

“…The chemotaxis model is a particular example of reaction cross-diffusion models [17,2,8,5,6,9,12]. Apart from the evolution terms, the equations have two other qualitatively different terms.…”

“…The values of and β are more difficult to estimate. They can be obtained from the definition of the CES production function (9), from setting the scaling variable A to a known value, and from data indicating that the ratio K/Y ≈ 3.5 holds in a variety of situations. Of course, the choice of these parameters is not unique.…”

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

“…Analysis of a simplified model given in [20] particularized to a tumor model is studied in [22][23][24]. Moreover, recent and more general results on the existence of solutions to the Busenberg-Gurtin model are given by Galiano et al in [25]. This paper is devoted to the mathematical and numerical analysis of indirectly transmitted diseases with nonlocal and cross diffusion.…”