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

• Published in: Studies in applied mathematics (Cambridge) 2021-08, Vol.147 (2), p.650-674
• Main Authors: Bakhshi, Marzieh, Ghazaryan, Anna, Manukian, Vahagn, Rodriguez, Nancy
• Format: Article
• Language: English
• Subjects: Fisher equation; geometric singular perturbation theory; KPP equation; Negative feedback; Outbursts; Perturbation theory; riots; Singular perturbation; traveling front; Traveling waves
• ISSN: 0022-2526; 1467-9590
• DOI: 10.1111/sapm.12394

In this work, we investigate the existence of nonmonotone traveling wave solutions to a reaction‐diffusion system modeling social outbursts, such as rioting activity, originally proposed in Berestycki et al (Netw Heterog Media. 2015;10(3):443–475). 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–Kolmogorov‐Petrovsky‐Piskunov (KPP) equation.