A dynamical systems model of unorganized segregation in two neighborhoods

ABSTRACT

We present a complete analysis of the Schelling dynamical system of two connected neighborhoods, 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 neighborhood, 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 neighborhood may not remain so when a connecting neighborhood is created.

KEYWORDS:

• Unorganized Segregation
• Schelling
• Bounded Neighbourhood Model
• Dynamical System

Notes

¹ The modern usage would be self-organized.

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) (Clifton et al., 2019) is a measure of how much one population seeks out members of the same population.

³ In fact, the color scheme is based on the index of dissimilarity (Massey & Denton, 1988) evaluated at the corresponding stable equilibrium state; see Section 5.

⁴ Note that the central equilibrium $(X_1^e,Y_1^e)=\left(\frac{1}{2},\frac{1}{2\alpha }\right)$ has changed its character from a saddle in Figure 2a to an unstable node in Figure 2c. We discuss the nature of these equilibria in Section 2.1.2 below.

⁵ In cases II, III and IV, there will be two different intersections, due to the lack of symmetry.

⁶ We retain the same notation for these turning points. No confusion should arise.

Additional information

Funding

This work was supported by the Engineering and Physical Sciences Research Council [EP/I013717/1];