Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 6A (Mathematical Biology)

The delay $\tau$ is the time between an individual's presence in the population and its contribution to the current growth rate, for example a maturation or gestation time. Thus reproduction at time $t$ is determined by the reproductive population at time $t-\tau$.

For $0\le t\le \tau$, the delayed argument lies in the prescribed history, so $n^{\prime }(t)=r$ and $n(t)=1+rt$. For $\tau \le t\le 2\tau$,Integrating from $\tau$ and using $n(\tau )=1+r\tau$ gives

Set $\omega =\pi /(2\tau )$. The condition $r\tau =-\pi /2$ says $r=-\omega$. ThensatisfiesIt is periodic with period $2\pi /\omega =4\tau$. Every nonzero such solution is negative during half of each period, so it cannot represent a population size.