Dynamical systems of self-organized segregation (Q6577558)

The primary aim of this paper is to add mathematical rigor to a previous work by \textit{D. J. Haw} and \textit{J. Hogan} [J. Math. Sociol. 42, No. 3, 113--127 (2018; Zbl 1485.91174)], which puts the classical bounded neighborhood (BN) model of \textit{T. C. Schelling} [J. Math. Sociol. 1, 143--186 (1971; Zbl 1355.91061)] into the form of dynamical systems.\N\NThe authors dedicate a significant portion of the introduction to explaining how the intended application to sociology can be modeled mathematically using dynamical systems. They also emphasize the importance of the concept of structural stability in such a model, pointing out how instability renders a mathematical statement unsuitable for interpretation.\N\NThe main mathematical content of this article is divided into two parts:\N\N(i) The first part is a reworking of Haw and Hogan's dynamical system for a single bounded neighborhood (BN), with the aim of providing a precise sufficient condition for structural stability. Notably, a single BN contains a single majority population X and a single minority population Y, both of which can move in and out of a population ``reservoir''. They offered a different way to understand a nondimensionalization step, but the conclusions are ultimately similarly to Haw and Hogan's.\N\N(ii) The second part begins with a criticism of Haw and Hogan's model for two interacting bounded neighborhoods, in that population conservation was not respected even though they considered the system closed, i.e., with no population ``reservoir''. The authors pointed out that the key mathematical issue is the change of sign of a cubic function of X and Y, and offered a fix by redefining the nonlinear dynamics using a rectifier \(f^+ = (f+|f|)/2\) (they actually used piecewise function with verbose notations for the subsets where the sign does not change, which we omit in this summary). The remaining work is a detailed analysis of the modified non-differentiable, but Lipschitz, system that they propose.\N\NThe methods of proof are quite standard for a mathematical analysis of dynamical systems, and they provided additional motivation and details for potential readers who may not be well-versed in terms like ``stable manifold''. However, the import of the mathematical analyses remains of interest mainly to applied mathematicians who are interested in the novelty of the bifurcation phenomena, as most conclusions are negative. While the authors pointed out flaws in Haw and Hogan's analysis, their version of the corrected two-BN model contains non-isolated equilibria in a large region of the parameter space, rendering the model less suitable for application. While the authors pointed out avenues to improve the model with respect improve its interpretative power for the intended sociological applications, they are not pursued in details, but reside in a short summary in the last section.