Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 6E / a / i / Solution

Susceptible-infective model with exponential density-dependent birth

If a population $N$ has per-capita birth rate $be^{-aN}$ and per-capita death rate $d$, then
$$
\dot N=N\left(be^{-aN}-d\right).
$$
When $b>d$, its positive <equilibrium of an autonomous differential equation>[equilibrium] is
$$
N^*=\frac1a\log\frac bd,
$$
and the derivative of the scalar vector field there is $-adN^*<0$, so the equilibrium is locally <asymptotic stability>[asymptotically stable].

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 Incoming links: Exponential density-dependent birth model