Connecting logistic probability models with basic dynamic processes

The logistic distribution is a popular probability model yet it is usually not motivated with reference to the processes under study. While its popularity can be attributed to its simplicity, it can also be derived from basic contextual considerations. Although it has been shown that logistic growth is the limiting form of a class of Markov processes on a lattice, the connections of these microscopic models with statistics are less widely appreciated. We review some history of logistic distributions in classical models of infection, and describe how the apparent density dependence emerges as a consequence of a particular lattice embedding. We then review how logistic growth arises from microscopic random behavior. We also derive a "square-logistic" model from basic considerations. Finally, we describe how the underlying discrete model relates to ordinary logistic regression modeling of time dependence.