Propagation in reaction-diffusion equations with fractional diffusion

TL;DRAbstract

This thesis focuses on the long time behaviour of solutions to Fisher-KPP reaction-diffusion equations involving fractional diffusion. This type of equation arises, for example, in spatial propagation or spreading of biological species (rats, insects,...). In population dynamics, the quantity under study stands for the density of the population. It is well-known that, under some specific assumptions, the solution tends to a stable state of the evolution problem, as time goes to infinity. In other words, the population invades the medium, which corresponds to the survival of the species, and we want to understand at which speed this invasion takes place. To answer this question, we set up a new method to study the speed of propagation when fractional diffusion is at stake and apply it on three different problems. Part I of the thesis is devoted to an analysis of the asymptotic location of the level sets of the solution to two different problems : Fisher-KPP models in periodic media and

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