Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 6A (Mathematical Biology)

Of the $a_t$ immature cells, a fraction $\lambda$ leaves by maturation and the rest remains immature. Of the $b_t$ mature cells, a fraction $\mu$ divides and is replaced by four immature cells, while a fraction $k$ dies; the remainder stays mature. These contributions give exactly

Once a cell is mature, its eventual competing outcomes are division and death. Conditional on one of them occurring, division has probability $\mu /(\mu +k)$ and creates four immature offspring. Thus the expected lifetime offspring number isassuming maturation eventually occurs.

Let $N_t=a_t+b_t$. Adding the two recurrences givesSince $b_t\ge 0$, total population can increase only if $3\mu >k$. This is exactlyso the population-growth and offspring criteria agree.

For $t\ge T$, set $\lambda =0$. Mature cells then decay geometrically:while their divisions add $4\mu b_{T+j}$ to the permanently immature population. If $\mu +k>0$, mature cells disappear andIf $\mu =k=0$, no further change occurs and the limit is $a_T+b_T$.