Age-structured population equation ($n_t+n_a=-\mu (a)n$)

The density of individuals at age $a$ evolves by transport with unit ageing speed and loss rate $\mu (a)$. New individuals enter at age zero through the integral birth boundary. Exponential separated profiles lead to the Euler-Lotka equation.

The mean lifetime number of offspring is the birth rate integrated against the probability of surviving to each age. For a continuous age model it is ${\int }_0^{\infty }b(a)\exp [-{\int }_0^a\mu (s)ds]da$. The Euler-Lotka equation compares this with one to locate positive population growth.