Euler–Lotka equation

In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an estimation of how a population is growing.

The field of mathematical demography was largely developed by Alfred J. Lotka in the early 20th century, building on the earlier work of Leonhard Euler. The Euler–Lotka equation, derived and discussed below, is often attributed to either of its origins: Euler, who derived a special form in 1760, or Lotka, who derived a more general continuous version. The equation in discrete time is given by

${\displaystyle 1=\sum _{a=1}^{\omega }\lambda ^{-a}\ell (a)b(a)}$

where ${\displaystyle \lambda }$ is the discrete growth rate, ℓ(a) is the fraction of individuals surviving to age a and b(a) is the number of offspring born to an individual of age a during the time step. The sum is taken over the entire life span of the organism.