Lotka's stable population equations
Theoretical derivation of the stable population growth rate
In the previous section, we calculated the proportional age distribution of a stable population from a stationary population using an assumed value for the growth rate.
However, the growth rate of a stable population is not an arbitrary quantity, it is determined jointly by the age-specific fertility rates and mortality pattern. Fertility generally has a more significant impact on the growth rate than mortality. In this section we present the theoretical equations from which the stable population growth rate can be calculated. Once the growth rate has been determined from these theoretical relationships, we can use the calculated value for the growth rate as an input to the procedure described in the previous section.
Lotka described stable population growth and age structure in three interconnected equations, whose meaning we will explore in turn.
- The first of Lotka’s equations expresses algebraically the relationship between the age structure of a stable population, its growth rate, and the underlying life table – the same relationships used in the previous session where we derived a stable population from a stationary population.
- The second equation relates the crude birth rate in a stable population to its age structure
- Lotka’s third equation, also known as the characteristic equation of a stable population, explains the inter-relationship between the stable population growth rate, age-specific fertility and life table survival.
As well as describing the structure and growth of stable populations based on their fertility and mortality patterns, Lotka proved that stable populations grow exponentially. We will not present this mathematical proof, as the proof requires an understanding of integral calculus beyond the scope of this course. We will assume that the growth in a stable population is exponential, and show how Lotka’s equations can be used to measure the magnitude of the exponential growth rate.