= Solution
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 balance
$$
\boxed{\frac{d}{dt}(N_1+N_2)=\int_0^\infty\Gamma_2(\tau)n_2(\tau,t)\,d\tau.}
$$
Each completed <cell division> increases the total number of <biological cells> by one.
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