Thesis

My manuscript : Unconventional diffusion models in ecology and evolutionary biology involving fragmented environments - A.L.

In this thesis, we are interested in a qualitative mathematical study of problems from ecology and evolutionary biology. We study the influence of a non-local dispersion for a biological species living in a patchy environment. More precisely, we first establish a criterion whose ensures the survival of a biological species which dynamics are driven by a fractional Fisher-KPP equation in a fragmented domain with Dirichlet exterior conditions. This criterion relies on the sign of the principal eigenvalue of subsets included in the fragmented domain. Moreover, we demonstrate an existence and uniqueness result of the stationary state of a Fisher-KPP equation in general patchy domains belonging to the class of non-negative, bounded and non-trivial solutions. In the particular case of a periodic and patchy domain, we establish the existence of invasion phenomena with exponential speed. Finally, we consider a model dealing with a phenotypically structured biological species living in a patchy environment. This species is subject to small mutations of the phenotype and to local and non-local spatial dispersion. We demonstrate the emergence of phenotypical dominant traits as the mutations become small.