PAPP103 - S09: Models of populations with variable growth rates

Generalized stable population models

Stable population theory states that in a closed population that is growing at rate r for every age group:

$$P_x(0)=B_0{{\displaystyle e}}^{-r(x+0.5)}\frac{L_x}{l_0}$$

This equation expresses P_x(0), the current size of the population aged x in completed years, in terms of the number of births occurring x completed years ago and L_x, the probability of surviving from birth into age group x. The births that occurred x years ago, in turn, are expressed as current births, B₀, adjusted for x+0.5 years growth in births at rate r. If r = 0, one obtains the stationary population in which P_x(0) = L_x if B₀ is taken as the radix of the life table. Note that the fact that the growth rate is the same at every age in a stable population implies that its age distribution is unchanging and vice versa.

But what if the population is neither stationary nor stable? What if the growth rate of the population varies by age? Somewhat surprisingly, it turns out that the size of the population by age is still a function of the current births, current life table and current age-specific growth rates of the population. Thus, although the population’s changing age structure is the product of a history of changing vital rates and age structural change, one does not need to know anything about its history to relate the current age structure to the current vital rates. Instead, the impact of that history on the current age structure is captured in the current age-specific growth rates.