A dynamical systems model of unorganised segregation

TL;DR

This paper models unorganised segregation using a dynamical systems approach to Schelling's model, providing quantitative analysis, exact criteria for population mixing, and insights into neighborhood tipping.

Contribution

It introduces a dynamical systems framework for Schelling's model, generalizes qualitative results, and derives exact conditions for stable population mixes and tipping points.

Findings

Exact formulas for stable mixed populations in unlimited movement scenarios

Identification of basin of attraction explanations for neighborhood tipping

Numerical simulations illustrating nonlinear tolerance schedule effects

Abstract

We consider Schelling's bounded neighbourhood model (BNM) of unorganised segregation of two populations from the perspective of modern dynamical systems theory. We derive a Schelling dynamical system and carry out a complete quantitative analysis of the system for the case of a linear tolerance schedule in both populations. In doing so, we recover and generalise Schelling's qualitative results. For the case of unlimited population movement, we derive exact formulae for regions in parameter space where stable integrated population mixes can occur. We show how neighbourhood tipping can be adequately explained in terms of basins of attraction. For the case of limiting population movement, we derive exact criteria for the occurrence of new population mixes and identify the stable cases. We show how to apply our methodology to nonlinear tolerance schedules, illustrating our approach with numerical simulations. We associate each term in our Schelling dynamical system with a social meaning. In particular we show that the dynamics of one population in the presence of another can be summarised as follows {rate of population change} = {intrinsic popularity of neighbourhood} - {finite size of neighbourhood} - {presence of other population} By approaching the dynamics from this perspective, we have a complementary approach to that of the tolerance schedule.