Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / B2 / c / Solution

This is a stage-structured population model with a census once per maturation interval. Only adults present at census $t$ contribute to the juvenile cohort at census $t+1$; each contributes $r$ new juveniles on average. Thus $r$ is an effective fecundity, including any survival of offspring needed to reach the next census. Juveniles present at $t$ do not contribute a separate birth term during this interval.

The term $c{y}_t$ consists of juveniles that survive and mature into adults by the next census, so $c$ is their survival-to-maturity fraction. The term $(1-d)z_t$ consists of surviving adults; $d$ is the adult mortality fraction per interval. All surviving juveniles leave the juvenile compartment in one step, adults remain adults, and there is no immigration or density dependence. The reproduction term is applied to adults counted at the start of the interval, rather than only to those surviving to its end.