Traveling wave solutions in a model for social outbursts in a tension-inhibitive regime

TL;DR

This paper investigates non-monotone traveling wave solutions in a reaction-diffusion social unrest model, focusing on the tension-inhibitive regime relevant to real-world riots, using geometric singular perturbation theory.

Contribution

It analyzes the existence of complex wave solutions in a social unrest model under tension-inhibitive conditions, applying geometric singular perturbation theory to different diffusion regimes.

Findings

Existence of non-monotone traveling wave solutions in the model.

In the slow diffusion of tension case, dynamics reduce to Fisher-KPP equation.

The bandwagon effect's role is crucial in wave formation.

Abstract

In this work we investigate the existence of non-monotone traveling wave solutions to a reaction-diffusion system modeling social outbursts, such as rioting activity, originally proposed in arXiv:1502.04725v3. The model consists of two scalar values, the level of unrest $u$ and a tension field $v$. A key component of the model is a bandwagon effect in the unrest, provided the tension is sufficiently high. We focus on the so-called tension inhibitive regime, characterized by the fact that the level of unrest has a negative feedback on the tension. This regime has been shown to be physically relevant for the spatiotemporal spread of the 2005 French riots. We use Geometric Singular Perturbation Theory to study the existence of such solutions in two situations. The first is when both $u$ and $v$ diffuse at a very small rate. Here, the time scale over which the bandwagon effect is observed plays a key role. The second case we consider is when the tension diffuses at a much slower rate than the level of unrest. In this case, we are able to deduce that the driving dynamics are modeled by the well-known Fisher-KPP equation.