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.

For $n_i(a,t)=N_i(t)f_i(a)$ with $\int f_i=1$, integrating the age-structured population equation replaces its exit integral by ${\Lambda }_i{N}_i$. For one-to-one transition from stage one to stage two and two-to-one reproduction from stage two to stage one, the population matrix is $\left(\begin{array}{cc}-{\Lambda }_1& 2{\Lambda }_2\\ {\Lambda }_1& -{\Lambda }_2\end{array}\right)$. This follows by integrating the age derivative and substituting the incoming boundary conditions. Exact time-independent positive profiles also force a constant stage ratio and common exponential growth through those boundary conditions. Thus the constant-coefficient closure is not an exact transient description for arbitrary age-dependent hazard functions and arbitrary initial conditions; constant hazard functions are an important exception for the integrated populations.

A cell-cycle age density counts biological cells per unit time spent in one specified phase of the cell cycle. Its integral over phase age is the population in that phase. Phase age resets on entry into the next phase, so it differs from chronological age since birth. A phase-exit hazard function supplies loss from the phase and incoming flux to the next phase.

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.