LotkaEuler

The Euler-Lotka equation, also known as the Lotka equation, is a fundamental relation in demography and population dynamics that connects survival, fertility, and intrinsic population growth. It is named after Leonhard Euler and Alfred J. Lotka, who developed related formulations in the early 20th century.

Discrete form: Given a life table with survivorship l_x (probability of surviving to age x) and age-specific fertility m_x (average number of daughters produced by a female of age x), the intrinsic growth rate r satisfies the equation

sum_{x=0}^{ω} e^{-r x} l_x m_x = 1,

where ω is the maximum age. Solving for r yields the intrinsic rate of increase per time unit, and the associated stable age distribution describes how the population is composed when growth is exponential at rate r.

Continuous form: If l(x) is survivorship and m(x) is fertility at age x, the equation becomes

∫_0^{ω} e^{-r x} l(x) m(x) dx = 1.

Usage and interpretation: The equation is used to estimate the intrinsic growth rate r from observed life-tables and fertility schedules, or conversely to infer the required fertility pattern to achieve a desired growth rate. The net reproductive rate R0 = ∑ l_x m_x (or ∫ l(x) m(x) dx in the continuous case) indicates whether the population is expected to grow (R0 > 1), decline (R0 < 1), or remain stable (R0 = 1) in the absence of migration.

Limitations: The Euler-Lotka framework assumes a stable age distribution, constant survival and fertility over time, and closed populations. Real-world data and changing environments can violate these assumptions, affecting applicability.