 Two-stage Euler-Lotka equation

For serial stages with survival functions $S_i(a)=\exp [-{\int }_0^a{\Gamma }_i(s)ds]$, set $H_i(r)={\int }_0^{\infty }{\Gamma }_i(a)S_i(a)e^{-ra}da$. If completion of stage two produces $m$ new stage-one individuals, a common exponential mode requires $m{H}_1(r)H_2(r)=1$. Indeed its age profiles are ${\varphi }_i(a)={\varphi }_i(0)e^{-ra}{S}_i(a)$, while the incoming fluxes give ${\varphi }_2(0)=H_1{\varphi }_1(0)$ and ${\varphi }_1(0)=m{H}_2{\varphi }_2(0)$. Cancelling nonzero amplitudes proves the equation. Under ordinary positive residence-time assumptions a positive root exists exactly when $m{H}_1(0)H_2(0)>1$. The normalized profiles are $e^{-ra}{S}_i(a)/\int e^{-rs}{S}_i(s)ds$.