Mathematical Modelling of Population Dynamics of Social Protests

Abstract

Mathematical modeling of riots and protests is now becoming a powerful tool in providing a better understanding of the dynamics of social unrest with the eventual goal to ensure the sustainable development of human society. Currently, however, most of the existing studies in the considered research area are based on either non-spatial or spatially implicit models, whereas in a large number of cases dynamics of social protests clearly exhibit spatial heterogeneity. To bridge the existing gap, here we explore spatial-temporal patterns of social protests using a reaction-diffusion modeling framework. Our model variables are the number of protesters and the cumulative amount of damage made as an outcome of the protest. The system has been studied analytically as well as by means of extensive numerical simulation in one-dimensional and two-dimensional space. We show that the proposed model exhibits a variety of dynamical regimes including stationary patterns with round hot spots as well as complex labyrinthine-like structures. The system also predicts the various types propagating waves of protests with regular and irregular fronts as well as a patchy spread, where protests spread in space via irregular motion and interaction of separate patches of high numbers of protestors without the formation of any continuous front, the number of protestors between patches being nearly zero. We reveal the structure of the model's parameter space, identifying the range of key parameters for which particular dynamical regimes are possible. Along with the reaction-diffusion model, considering continuous space, we also consider the model of protests on discrete networks of different natures. We find that Turing instability can lead to pattern formation on networks, which opens up an exciting possibility to explore it as a generation mechanism in a large number of social unrest contexts. The network-based model also shows a variety of non-Turing patterns including both stationary and non-stationary complex dynamics, where different nodes behave differently. Finally, we consider a realistic network to model the yellow vests movement in France.