Dynamical systems of self-organized segregation

ABSTRACT

We re-consider Schelling’s (1971) bounded neighborhood model as put into the form of a dynamical system by Haw and Hogan (2018). The aim is to determine how tolerance can prevent (or lead to) segregation. In the case of a single neighborhood, we explain the occurring bifurcation set, thereby correcting a scaling error. In the case of two neighborhoods, we correct a major error and derive a dynamical system that does satisfy the modeling assumptions made by Haw and Hogan (2020), staying as close as possible to their construction. We find that stable integration is then only possible if the populations in the two neighborhoods have the option to be in neither neighborhood. In the absence of direct movement between the neighborhoods, the problem is furthermore equivalent to independent single neighborhood problems.

KEYWORDS:

• Bounded neighborhood model
• structural stability
• unorganized segregation

Acknowledgments

We would like to thank Dr Özge Bilgili for guiding us toward a more-nuanced understanding of the complexity of the effects of residential segregation. We also thank the anonymous referee for improving the presentation of our results.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Also known as saddle-node bifurcations.

² So small that making $\mathit{\delta }$ even smaller or $10\mathrm{\%}$ larger does not change the shape of ${\mathcal{W}}_{\delta }^s(x_e)$.

³ Next to $\mathit{\gamma }$ as the scaling replaces the tolerance $a_2$ in (18b) by $\mathit{\alpha \gamma }=a_2\mathit{k}$.

⁴ In particular, we do not exclude the possibility that there is a district in the city with only members of the $\mathit{X}$–population or a district with only members of the $\mathit{Y}$–population.