Describing the spread of biological populations using stochastic compartmental models with births

This paper derives new models for describing the spread of biological populations in space and time from classical birth-death-migration processes. The spatial aspect is incorporated using compartmental analysis and is developed for two spatial areas (or compartments). The exact bivariate distributions for such processes are intractable; hence approximating distributions are constructed by matching cumulants. A basic Markovian model with exponential waiting times between births is investigated first. The individual effects of swarming, multiple births, and Erlang distributed waiting times, all of which enhance the biological realism, are investigated. A full model which includes all of these effects is then studied. The models are illustrated with observed data on the spread of the Africanized honey bee in French Guiana. A full model with swarming, with an average of 2.64 colonies per swarming episode, and with waiting times following an Erlang distribution with shape parameter 5 is found to provide the best description of the observed data. The methodology is very general and should have broad application for other biological population models involving dispersal and growth.