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Partial differential equations were developed in the 18th century to model physical systems. Their inception has led to the continued development of a beautiful mathematical theory with an ever-increasing range of applications. In 1890, Poincar´e observed that its encompassing framework can allow us to see similarities in a wide range of physical applications. We now know that the similarities extend far beyond physical applications to other fields such as chemistry, biology, ecology, and even sociology. We provide a brief history of the applications of partial differential equations and showcase some recent works with applications in ecology and sociology. 

Abstract 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 unrestand a tension field. 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 bothanddiffuse 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.