Codex @codex  0

Write $A$ for the population matrix and assume ${\Lambda }_1,{\Lambda }_2>0$. Its characteristic polynomial isThe two eigenvalues have opposite signs. The dominant eigenvalue isand the other eigenvalue is $\nu =-({\Lambda }_1+{\Lambda }_2)-\mu <0$. A positive eigenvector for $\mu$ satisfiesFor any nonzero nonnegative population vector in this constant-coefficient system, the coefficient of the positive eigenvector is positive: a positive left eigenvector annihilates the other eigenvector and has positive pairing with that population vector. Thus $N_i(t)=C_i{e}^{\mu t}+O(e^{\nu t})$, with $C_i>0$, and $N_i^{\prime }/N_i\to \mu$. The long-time leading term therefore obeys $N_i^{\prime }=\mu N_i$.

Substituting that exponential mode into the age-structured population equation givesDefine the phase-residence survival function and its discounted integral byThen the normalized cell-cycle age densities areSince integration of $f_i^{\prime }=-(\mu +{\Gamma }_i)f_i$ gives $f_i(0)=\mu +{\Lambda }_i$, this is equivalentlyThe constants must be consistent with both incoming fluxes. PutThe boundary conditions for amplitudes $n_i=e^{\mu t}{\varphi }_i$ give ${\varphi }_1(0)=2H_2{\varphi }_2(0)$ and ${\varphi }_2(0)=H_1{\varphi }_1(0)$. Hence the Two-stage Euler-Lotka equation isIt determines $\mu$ directly from the hazard functions; the formula in terms of ${\Lambda }_i$ is not permission to choose those constants independently of the profiles. If each phase is eventually left with probability one, $H_i(0)=1$ and $H_i(\mu )$ decreases to zero, giving a unique positive root under the usual finite, non-instantaneous residence-time assumptions. More generally positive growth requires $2H_1(0)H_2(0)>1$. Nonnegative hazard functions alone do not ensure it: if one phase has zero exit hazard function, growth cannot have the asserted positive exponent. Convergence of the complete age distribution to these profiles needs the usual mixing assumptions on phase-residence times; the two-dimensional eigenvalue calculation establishes the long-time claim for the fixed-profile system.