Search

Riots originating during, or in the aftermath of, sports events can incur significant costs in damages, as well as large-scale panic and injuries. A mathematical description of sports riots is therefore sought to better understand their propagation and limit these physical and financial damages. In this work, we present an agent-based modelling framework that describes the qualitative features of populations engaging in riotous behaviour. Agents, pertaining to either a 'rioter' or a 'bystander' sub-population, move on an underlying lattice and can either be recruited or defect from their respective subpopulation. In particular, we allow these individual-level recruitment and defection processes to vary with local population density. This agent-based modelling framework provides the unifying link between multi-population stochastic models and density-dependent reaction processes. Furthermore, the continuum description of this ABM framework is shown to be a system of nonlinear reaction-diffusion equations and faithfully agrees with the average ABM behaviour from individual simulations. Finally, we determine the unique correspondence between the underlying individual-level recruitment and defection mechanisms with their population-level counterparts, providing a link between local-scale effects and macroscale rioting phenomena.

We examine travelling wave solutions of the reaction-diffusion equation, $\partial_t u= R(u) + \partial_x \left[D(u) \partial_x u\right]$, with a Stefan-like condition at the edge of the moving front. With only a few assumptions on $R(u)$ and $D(u)$, a variety of new compactly supported travelling waves arise in this Reaction-Diffusion Stefan model. While other reaction-diffusion models admit compactly supported travelling waves for a unique wavespeed, we show that compactly supported travelling waves in the Reaction-Diffusion Stefan model exist over a range of wavespeeds. Furthermore, we determine the necessary conditions on $R(u)$ and $D(u)$ for which compactly supported travelling waves exist for all wavespeeds. Using asymptotic analysis in various distinguished limits of the wavespeed, we obtain approximate solutions of these travelling waves, agreeing with numerical simulations with high accuracy.

We examine travelling wave solutions of the Porous-Fisher model, $\partial_t u(x,t)= u(x,t)\left[1-u(x,t)\right] + \partial_x \left[u(x,t) \partial_x u(x,t)\right]$, with a Stefan-like condition at the moving front, $x=L(t)$. Travelling wave solutions of this model have several novel characteristics. These travelling wave solutions: (i) move with a speed that is slower than the more standard Porous-Fisher model, $c<1/\sqrt{2}$; (ii) never lead to population extinction; (iii) have compact support and a well-defined moving front, and (iv) the travelling wave profiles have an infinite slope at the moving front. Using asymptotic analysis in two distinct parameter regimes, $c \to 0^+$ and $c \to 1/\sqrt{2}\,^-$, we obtain closed-form mathematical expressions for the travelling wave shape and speed. These approximations compare well with numerical solutions of the full problem.