= Solution
Write $A$ for the population <matrix> and assume $\Lambda_1,\Lambda_2>0$. Its <characteristic polynomial> is
$$
\det(sI-A)=(s+\Lambda_1)(s+\Lambda_2)-2\Lambda_1\Lambda_2
=s^2+(\Lambda_1+\Lambda_2)s-\Lambda_1\Lambda_2.
$$
The two <eigenvalues> have opposite signs. The <dominant eigenvalue> is
$$
\boxed{\mu=\frac{\sqrt{\Lambda_1^2+6\Lambda_1\Lambda_2+\Lambda_2^2}-\Lambda_1-\Lambda_2}{2}>0,}
$$
and the other <eigenvalue> is $\nu=-(\Lambda_1+\Lambda_2)-\mu<0$. A positive <eigenvector> for $\mu$ satisfies
$$
\boxed{\frac{N_2}{N_1}\longrightarrow\frac{\Lambda_1}{\mu+\Lambda_2}
=\frac{\mu+\Lambda_1}{2\Lambda_2}.}
$$
For 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 $N_i'/N_i\to\mu$. The long-time leading term therefore obeys $N_i'=\mu N_i$.
Substituting that exponential mode into the <age-structured population equation> gives
$$
f_i'(\tau)=-[\mu+\Gamma_i(\tau)]f_i(\tau).
$$
Define the phase-residence <survival function> and its discounted integral by
$$
S_i(\tau)=\exp\left[-\int_0^\tau\Gamma_i(a)\,da\right],\qquad
Z_i(\mu)=\int_0^\infty e^{-\mu a}S_i(a)\,da.
$$
Then the normalized <cell-cycle age densities> are
$$
\boxed{f_i(\tau)=\frac{e^{-\mu\tau}S_i(\tau)}{Z_i(\mu)},\qquad i=1,2.}
$$
Since <integration> of $f_i'=-(\mu+\Gamma_i)f_i$ gives $f_i(0)=\mu+\Lambda_i$, this is equivalently
$$
\boxed{f_i(\tau)=(\mu+\Lambda_i)\exp\left[-\mu\tau-\int_0^\tau\Gamma_i(a)\,da\right].}
$$
The constants must be consistent with both incoming <fluxes>. Put
$$
H_i(\mu)=\int_0^\infty\Gamma_i(a)e^{-\mu a}S_i(a)\,da
=1-\mu Z_i(\mu).
$$
The <boundary conditions> for amplitudes $n_i=e^{\mu t}\phi_i$ give $\phi_1(0)=2H_2\phi_2(0)$ and $\phi_2(0)=H_1\phi_1(0)$. Hence the <Two-stage Euler-Lotka equation> is
$$
\boxed{2H_1(\mu)H_2(\mu)=1.}
$$
It 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.
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