Write

A

for the population matrix and assume

Λ_{1}, Λ_{2} > 0

. Its characteristic polynomial is

det (s I - A) = (s + Λ_{1}) (s + Λ_{2}) - 2 Λ_{1} Λ_{2} = s^{2} + (Λ_{1} + Λ_{2}) s - Λ_{1} Λ_{2} .

(1)

The two eigenvalues have opposite signs. The dominant eigenvalue is

μ = \frac{Λ _{1}^{2} + 6 Λ _{1} Λ _{2} + Λ _{2}^{2} - Λ _{1} - Λ _{2}}{2} > 0,

(2)

and the other eigenvalue is

ν = - (Λ_{1} + Λ_{2}) - μ < 0

. A positive eigenvector for

μ

satisfies

\frac{N _{2}}{N _{1}} ⟶ \frac{Λ _{1}}{μ + Λ _{2}} = \frac{μ + Λ _{1}}{2 Λ _{2}} .

(3)

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^{μ t} + O (e^{ν t})

, with

C_{i} > 0

, and

N_{i}^{'} / N_{i} \to μ

. The long-time leading term therefore obeys

N_{i}^{'} = μ N_{i}

Substituting that exponential mode into the age-structured population equation gives

f_{i}^{'} (τ) = - [μ + Γ_{i} (τ)] f_{i} (τ) .

(4)

Define the phase-residence survival function and its discounted integral by

S_{i} (τ) = exp [- \int_{0}^{τ} Γ_{i} (a) d a], Z_{i} (μ) = \int_{0}^{\infty} e^{- μ a} S_{i} (a) d a .

(5)

Then the normalized cell-cycle age densities are

f_{i} (τ) = \frac{e ^{- μτ} S _{i} ( τ )}{Z _{i} ( μ )}, i = 1, 2.

(6)

Since integration of

f_{i}^{'} = - (μ + Γ_{i}) f_{i}

gives

f_{i} (0) = μ + Λ_{i}

, this is equivalently

f_{i} (τ) = (μ + Λ_{i}) exp [- μτ - \int_{0}^{τ} Γ_{i} (a) d a] .

(7)

The constants must be consistent with both incoming fluxes. Put

H_{i} (μ) = \int_{0}^{\infty} Γ_{i} (a) e^{- μ a} S_{i} (a) d a = 1 - μ Z_{i} (μ) .

(8)

The boundary conditions for amplitudes

n_{i} = e^{μ t} ϕ_{i}

give

ϕ_{1} (0) = 2 H_{2} ϕ_{2} (0)

and

ϕ_{2} (0) = H_{1} ϕ_{1} (0)

. Hence the Two-stage Euler-Lotka equation is

2 H_{1} (μ) H_{2} (μ) = 1.

(9)

It determines

μ

directly from the hazard functions; the formula in terms of

Λ_{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} (μ)

decreases to zero, giving a unique positive root under the usual finite, non-instantaneous residence-time assumptions. More generally positive growth requires

2 H_{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.