This book presents models written as partial differential equations and originating from various questions in population biology, such as physiologically structured equations, adaptive dynamics, and bacterial movement. Its purpose is to derive appropriate mathematical tools and qualitative properties of the solutions (long time behavior, concentration phenomena, asymptotic behavior, regularizing effects, blow-up or dispersion). Original mathematical methods described are, among others, the generalized relative entropy method - a unique method to tackle most of the problems in population biology, the description of Dirac concentration effects using a new type of Hamilton-Jacobi equations, and a general point of view on chemotaxis including various scales of description leading to kinetic, parabolic or hyperbolic equations.
These lecture notes are based on several courses and lectures given at different places (University Pierre et Marie Curie, University of Bordeaux, CNRS research groups GRIP and CHANT, University of Roma I) for an audience of mathematicians. The main motivation is indeed the mathematical study of Partial Differential Equations that arise from biological studies. Among them, parabolic equations are the most popular and also the most numerous (one of the reasons is that the small size, at the cell level, is favorable to large viscosities). Many papers and books treat this subject, from modeling or analysis points of view. This oriented the choice of subjects for these notes towards less classical models based on integral equations (where PDEs arise in the asymptotic analysis), transport PDEs (therefore of hyperbolic type), kinetic equations and their parabolic limits.