A basic goal of theoretical ecology is to understand how the interactions of individual organizations with each other under a certain inhabiting environment determine the spatiotemporal structure of distribution of populations. The diffusive logistic equation, which describes the spatial and temporal distribution of the population density of a single species, is one of the fundamental reaction-diffusion equation models in population biology. In this thesis, we are concerned with the periodic logistic equation with homogeneous Neumann boundary conditions. The theory for this basic case forms the foundation for further investigation of multispecies problems. The thesis consists of four chapters.