Past exam of the further mathematical biology course of the University of Oxford / 2023 / 1 / d / Solution

Because the hazard functions are constant, normalization gives ${\Lambda }_1=2\alpha$ and ${\Lambda }_2=3\alpha$. The population characteristic polynomial factors asThusThese exponential distributions describe ages among currently present biological cells. Their rates differ from the phase-exit rates because exponential growth overrepresents younger entrants. The stable phase ratio is $N_2/N_1=1/2$, so the G1 phase and G2 phase fractions approach $2/3$ and $1/3$.

Increasing $\alpha$ rescales all times by $1/\alpha$: the population undergoes faster exponential growth, its doubling time is $\boxed{(\log 2)/\alpha }$, and the profile means $1/(3\alpha )$ and $1/(4\alpha )$ decrease. The phase fractions do not depend on $\alpha$. The population matrix has exact solutionso population-ratio transients decay relative to the growing term at rate $7\alpha$. This exact population formula does not require initially stationary cell-cycle age densities.