Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling Species

Abstract

Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend the analysis to a system for two species interacting with each other according to different inner-and intra-species attractions. Under suitable conditions on this self-and crosswise attraction an interesting effect can be observed, namely phase separation into neighbouring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is a connected interval.Preliminary results indicate the existence of phase separation, i.e. spatial sorting of the different species. A detailed analysis is given in one spatial dimension. The existence and shape of segregated stationary solutions is shown via the Krein-Rutman theorem. Moreover, for small repulsion/nonlinear diffusion, also uniqueness of these stationary states is proved.

A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation

“…The rigorous derivation for one single cell type can be done following the blueprint in [30], see [6,5] and the references therein too. This basic model for two populations shows very rich dynamical properties and complex set of stationary states and stability [8,10,13,7]. However, it does not establish any upper bound on the maximal density of cells, density saturation, that is sensible for cell population models and it does not include reaction terms to take into account cell apoptosis and cell division/growth.…”

Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion

“…Let us point out that there other ways of including volume effects such as the volume filling assumption [40] differing from the volume exclusion considered here [14]. This basic model shows very rich dynamical properties and complex set of stationary states and metastability both for one species and multispecies cases [12,13,16,18,19] dealing with other attractive kernels instead of the classical chemotaxis kernels. As usual in chemotaxis modeling, the previous equation is coupled with a reaction-diffusion equation of the chemoattractant v(x, t) typically created and degraded linearly as…”

Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles

“…First-order models with nonlocal attractive and repulsive terms to include cell adhesion and volume effects have been already proposed in the literature of cell interactions for cancer invasion models ( Colombi et al, 2015a( Colombi et al, , 2017Domschke et al, 2014;Gerisch and Chaplain, 2008;Painter et al, 2015 ), zebrafish lateral line patterning ( Volkening and Sandstede, 2015 ), and cell sorting in heterogenous cell populations ( Burger et al, 2017;Carrillo et al, 2017;Murakawa and Togashi, 2015 ). Some of this works do not deal with agent-based models but with their continuum macroscopic limit or even with an hybridization of both description levels ( Colombi et al, 2015a( Colombi et al, , 2017.…”

“…Some of this works do not deal with agent-based models but with their continuum macroscopic limit or even with an hybridization of both description levels ( Colombi et al, 2015a( Colombi et al, , 2017. For instance, at the macroscopic level, repulsion is taken into account by (nonlinear) diffusion or drift saturation terms ( Burger et al, 2017;Calvez and Carrillo, 2006;Carrillo et al, 2017;Hillen and Painter, 2001;Painter and Hillen, 2002;Shigesada et al, 1979 ), which can be obtained from particle-based nonlocal repulsive models in the right scaling limit ( Bodnar and Velázquez, 2013;Oelschläger, 1990 ).…”