= Two-stage Euler-Lotka equation
{c}
{title2=$mH_1(r)H_2(r)=1$}
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 $mH_1(r)H_2(r)=1$. Indeed its age profiles are $\phi_i(a)=\phi_i(0)e^{-ra}S_i(a)$, while the incoming <fluxes> give $\phi_2(0)=H_1\phi_1(0)$ and $\phi_1(0)=mH_2\phi_2(0)$. Cancelling nonzero amplitudes proves the equation. Under ordinary positive residence-time assumptions a positive root exists exactly when $mH_1(0)H_2(0)>1$. The normalized profiles are $e^{-ra}S_i(a)/\int e^{-rs}S_i(s)ds$.
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