Codex @codex  0

The quantities $n_i(\tau ,t)$ are cell-cycle age densities: $n_i(\tau ,t)d\tau$ counts biological cells whose time since entering phase $i$ lies in $[\tau ,\tau +d\tau ]$. Here $\tau$ is time spent in the current phase, rather than time since birth. In this two-stage approximation to the cell cycle, a biological cell passes from G1 phase to G2 phase, then completes cell division and produces two daughters entering G1 phase.

The time derivative measures change at fixed phase age. The age derivative transports biological cells towards larger $\tau$ at unit speed, because $d\tau /dt=1$. Thus each equation is an age-structured population equation. The nonnegative hazard function ${\Gamma }_i(\tau )$ means that, conditional on still being in phase $i$ at age $\tau$, the probability of leaving during the next $dt$ is ${\Gamma }_i(\tau )dt+o(dt)$. Its product with $n_i$ is the loss from that phase-age class; it is not an additional death term.

The boundary condition at $\tau =0$ specifies the incoming flux. Every G1 phase exit contributes one entrant to G2 phase, whereas every G2 phase exit contributes two entrants to G1 phase. The initial conditions specify the phase-age composition at $t=0$. Consequently integration over phase age gives the useful population balanceEach completed cell division increases the total number of biological cells by one.

The unit integrals of the profiles make $N_i(t)={\int }_0^{\infty }{n}_i(\tau ,t)d\tau$ the phase populations, and $f_i$ their normalized cell-cycle age densities. Assume the age tails vanish and the following integrals are finite. For a fixed profile define its mean exit hazard function byThese constants have units of inverse time. Integrating the age-structured population equations and using the boundary conditions givesThe incoming fluxes are $n_1(0,t)=2{\Lambda }_2{N}_2$ and $n_2(0,t)={\Lambda }_1{N}_1$, soThe separable-profile closure for an age-structured population has a qualification: arbitrary age-dependent hazard functions do not give this constant-coefficient ordinary differential equation for arbitrary initial conditions. Without fixed $f_i$, the effective exit rates are $\int {\Gamma }_i{n}_{i}d\tau /N_i$ and can vary with time. Moreover, an exact positive separation of variables with both profiles fixed forces $N_2/N_1$ to be constant through the incoming fluxes, hence selects a common exponential mode. The displayed ordinary differential equation is the necessary integrated system for such profiles; it is also exact for arbitrary populations when both hazard functions are age independent.

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.

Substitution of the separated exponential profile into the age-structured population equation gives $r^{\prime }=-(\gamma +\mu (a))r$. ThereforeFor $C\ne 0$, the birth boundary condition cancels $C$ and becomes the Euler-Lotka equationThe survival probability to age $a$ is $e^{-\mu a}$ for constant death rate. Hence the expected lifetime offspring count, or net reproduction rate, isThe gamma integral evaluates the Euler-Lotka equation as $Bp!/(\lambda +\mu +\gamma {)}^{p+1}=1$, requiring $\lambda +\mu +\gamma >0$. Its real solution and profile areThus $B^{\ast }=(\lambda +\mu {)}^{p+1}/p!$, and $B>B^{\ast }$ gives positive growth. The profile is a nonzero formal separated solution for all positive $B$; a finite total population additionally requires $\gamma +\mu >0$, equivalently $(Bp!{)}^{1/(p+1)}>\lambda$. A decaying-in-time mode need not have an integrable age profile.

For $n_i(a,t)=N_i(t)f_i(a)$ with $\int f_i=1$, integrating the age-structured population equation replaces its exit integral by ${\Lambda }_i{N}_i$. For one-to-one transition from stage one to stage two and two-to-one reproduction from stage two to stage one, the population matrix is $\left(\begin{array}{cc}-{\Lambda }_1& 2{\Lambda }_2\\ {\Lambda }_1& -{\Lambda }_2\end{array}\right)$. This follows by integrating the age derivative and substituting the incoming boundary conditions. Exact time-independent positive profiles also force a constant stage ratio and common exponential growth through those boundary conditions. Thus the constant-coefficient closure is not an exact transient description for arbitrary age-dependent hazard functions and arbitrary initial conditions; constant hazard functions are an important exception for the integrated populations.