A dynamical systems model of unorganized segregation in two neighborhoods

Abstract

We present a complete analysis of the Schelling dynamical system [5] of two connected neighbourhoods, with or without population reservoirs, for different types of linear and nonlinear tolerance schedules.We show that stable integration is only possible when the minority is small and combined tolerance is large. Unlike the case of the single neighbourhood, limiting one population does not necessarily produce stable integration and may destroy it.We conclude that a growing minority can only remain integrated if the majority increases its own tolerance. Our results show that an integrated single neighbourhood may not remain so when a connecting neighbourhood is created. 1 The modern usage would be self-organised. 2 Tolerance is a measure of how members of one population remain in an area where there is another population present. In contrast, homophily (or self-segregation) [1] is a measure of how much one population seeks out members of the same population.

Dynamical systems of self-organized segregation

“…2. Correct the differential equations for the two neighbourhood model as put forward in (Haw and Hogan, 2020). We conclude that Haw and Hogan's two neighbourhood model is not a good model as it leads to dynamical systems that are not structurally stable.…”

“…In this section we also correct the scaling in Haw and Hogan's (2018) paper and provide an in depth qualitative analysis. Section 3 contains our evaluation of Haw and Hogan's (2020) two neighbourhood model. We argue that the given mathematical formulation inaccurately depicts their described model.…”

“…It turns out that Haw and Hogan's two neighbourhood model describes dynamical systems that are not structurally stable, which demonstrates that their model is not a good mathematical model of a real life system. Section 4 gives an interpretation of the results and also touches upon Haw and Hogan's (2020) two neighbourhood model with a reservoir population. Section 5 suggests possible areas of improvement for the single and the two neighbourhood model and future directions of research.…”

“…The BN is preferred over the reservoir by every individual and individuals do not take into consideration the ratio of the two populations in the reservoir. Note that we do not follow Haw and Hogan (2020) who use one reservoir for each population and consider both reservoirs to be segregated. Rather we think of the reservoir as being the rest of the city.…”

“…In § 3.3 we perform a qualitative analysis of the corrected system. We concentrate on Case I of (Haw and Hogan, 2020) where both BN have the same X-tolerance schedule R X and the same Y -tolerance schedule R Y . The problems already become apparent in this case, which allows us to simplify the notation.…”