PAS programs are constructed from simple theoretical models of population dynamics. Single species programs are developed around the principles of logistic population growth. Two species programs extend the basic idea of logistic growth to two interacting species.
The most elementary form of the single-species logistic model in discrete time is
where N(t) is the density of the population at time t, R is the actual per-capita rate of change of the population over the time interval t-1 to t, A is the maximum per-capita rate of change of the population in a given environment, and C is the marginal impact of an individual organism on the rate of change of its cohorts. The first equation describes the recursive process of positive feedback during population growth (the first principle or Malthusian Law), and the second the feedback due to density-dependent forces acting on the per-capita rate of change (the third principle or population regulation). We refer to the second equation as the R-function. Because the Malthusian Law is the same for all populations while the R-function may be different for different species or even for the same species living in different places, the major task of PAS programs is to identify the appropriate R-function for a specific population of organisms. Details on the logistic equation and its generalization to include time-lags, non-linear feedback, underpopulation effects, and more than one equilibrium point, can be found in the Single-Species PAS programs P1a and P1b.