Introduction to diffusive logistic equations in population dynamics

The purpose of this paper is to provide a careful and accessible exposition of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. Moreover we prove that a population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to Thomas Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by P. F. Verhulst.