In this work, we study two different models that appear in population dynamics. The first problem is concerned with an integral equation that models discrete time dynamics of a population in a patchy landscape. The patches in the domain are reflected through the discontinuity of the kernel of the integral operator at a finite number of points in the whole domain. We prove the existence and uniqueness of a stationary state under certain assumptions on the principal eigenvalue of the linearized integral operator and the growth term as well. We also derive criteria under which the population undergoes extinction (in which case the stationary solution is 0 everywhere). In the second problem we consider a reaction-diffusion model with a drift term in a bounded domain. Given a time T, we prove the existence and uniqueness of an initial datum that maximizes the total mass [equation] in the presence of an advection term. In a population dynamics context, this optimal initial datum can be understood as the best distribution of the initial population that leads to a maximal the total population at a prefixed time T. We also compare the total masses at a time T in two cases: depending on whether an advection term is present in the medium or not. We prove that the presence of a large enough advection enhances the total mass.
