The Population Analysis System is built around elementary theoretical models that are well-known to ecologists. We begin with a the exponential growth equation, sometimes called the Malthusian Law or the first principle of population dynamics,
where dN / dt is the instantaneous growth rate of the population, N is population density, and R is its per-capita rate of change, usually called the instantaneous or intrinsic rate of increase (note that small r is often used in the literature for the intrinsic rate of increase, but we reserve the lower case for the correlation coefficient). The equation above is easily integrated to yield
where N(t) is the density of the population at time t, N(0) is its initial density, and exp means that the base of the natural logarithm (2.718.....) is raised to the power of the argument (R * t). If we set the time increment to t =1, then we have
or, in more general terms,
We call this a step-ahead forecasting equation because population density at time t can be forecasted from its density one time step previously if R is known. The equation can be rearranged to give
or
This equation provides us with a way to estimate the per-capita rate of increase from population density counts. Thus, a population census over Y years will provide Y-1 estimates of R .