The first reaction-diffusion equation developed and studied is the Fisher-KPP equation. Introduced in 1937, it accounts for the spatial spreading and growth of a species. Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion. Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al. I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions. Afterwards, I will describe the model for the cane toads equations and give new results regarding this model. In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads. The talk is based on a joint work with Bouin and Ryzhik.
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