The rate of change in population density over a unit of time, say a year, is

where N_t is the density of the population in the current year, N_t-1 is its density at the same time in the previous year, B is the number of births per individual during the year and D is the number of deaths per capita during the same year*.

The per-capita or specific rate of change is

The finite per-capita rate of change is

The instantaneous (logarithmic or intrinsic) per-capita rate of change is

The step-ahead forecasting equation is

* Actually it is a little bit more complicated than this because per-capita death rates are calculated over a finite interval of time, t, and so must also include the death of those organisms alive at the beginning of the time period (the parents, N_t-1) and those born during the time period (the offspring, B). Hence, the total death rate per-capita is actually D_p + D_oB, where D_P is the per-capita death rate of parents and D_o is the per-capita death rate of offspring (notice that both D_p and D_o are probabilities of dying over the finite time period and so their values are always between 0 and 1). Thus, the total change in population density over a time interval is given by

N_t = N_t-1[1 + B - (D_p + D_oB)] = N_t-1 - D_pN_t-1 + BN_t-1- D_oBN_t-1

or, in plain words,

total change = original parents - dead parents + total offspring - dead offspring.

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