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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. 

Understanding the factors that drive species to move and develop territorial patterns is at the heart of spatial ecology. In many cases, mechanistic models, where the movement of species is based on local information, have been proposed to study such territorial patterns. In this work, we introduce a nonlocal system of reaction-advection-diffusion equations that incorporate the use of nonlocal information to influence the movement of species. One benefit of this model is that groups are able to maintain coherence without having a home-center. As incorporating nonlocal mechanisms comes with analytical and computational costs, we explore the potential of using long-wave approximations of the nonlocal model to determine if they are suitable alternatives that are more computationally efficient. We use the gradient flow-structure of the both local and nonlocal models to compute the equilibrium solutions of the mechanistic models via energy minimizers. Generally, the minimizers of the local models match the minimizers of the nonlocal model reasonably well, but in some cases, the differences in segregation strength between groups is highlighted. In some cases, as we scale the number of groups, we observe an increased savings in computational time when using the local model versus the nonlocal counterpart. 

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29.