Back to the future: a simplified and intuitive derivation of the Lotka-Euler equation

The Lotka-Euler equation is a mathematical expression used to study population dynamics and growth, particularly in the context of demography and ecology. The growth rate $\lambda$ is the speed at which an individual produce their offspring. It is essentially a birth process, and here it is shown that by reversing the process to a death process, in which individuals die at a rate $\lambda^{-1}$, the derivation of the Lotka-Euler equation becomes more intuitive and direct, both in discrete and continuous time.