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 pro cesses. The spatial aspect is incorporated using compartmental analysi s and is developed for two spatial areas (or compartments). The exact bivariate distributions for such processes are intractable; hence appr oximating distributions are constructed by matching cumulants. A basic Markovian model with exponential waiting times between births is inve stigated first. The individual effects of swarming, multiple births, a nd Erlang distributed waiting times, all of which enhance the biologic al realism, are investigated. A full model which includes all of these effects is then studied. The models are illustrated with observed dat a 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 epi sode, and with waiting times following an Erlang distribution with sha pe parameter 5 is found to provide the best description of the observe d data. The methodology is very general and should have broad applicat ion for other biological population models involving dispersal and gro wth.