ANALYSIS OF SOME NONLOCAL MODELS IN POPULATION DYNAMICS

This thesis is mainly devoted to the mathematical analysis of some nonlocal models arising in population dynamics. In general, the study of these models meets with numerous difficulties owing to the lack of compactness and of regularizing effects. In this respect, their analysis requires new tools, both theoretical and qualitative. We present several results in this direction. In the first part, we develop a functional analytic toolbox which allows one to handle some quantities arising in the study of these models. In the first place, we extend the characterization of Sobolev spaces due to Bourgain, Brezis and Mironescu to low regularity function spaces of Besov type. This results in a new theoretical framework that is more adapted to the study of some nonlocal equations of Fisher-KPP type. In the second place, we study the regularity of the restrictions of these functions to hyperplanes. We prove that, for a large class of Besov spaces, a surprising loss of regularity occurs. Moreover, we obtain an optimal characterization of the regularity of these restrictions in terms of spaces of so-called “generalized smoothness”. In the second part, we study qualitative properties of solutions to some nonlocal reaction-diffusion equations set in (possibly) heterogeneous domains. In collaboration with J. Coville, F. Hamel and E. Valdinoci, we consider the case of a perforated domain which consists of the Euclidean space to which a compact set, called an “obstacle”, is removed. When the latter is convex (or close to being convex), we prove that the solutions are necessarily constant. In a joint work with J. Coville, we study in greater detail the influence of the geometry of the obstacle on the classification of the solutions. Using tools of the type of those developed in the first part of this thesis, we construct a family of counterexamples when the obstacle is no longer convex. Lastly, in a work in collaboration with S. Dipierro, we study qualitative properties of solutions to nonlinear elliptic systems in variational form. We establish various monotonicity results in a fairly general setting that covers both local and fractional operators.