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

The per-capita net growth rate is the birth rate minus the death rate,Multiplying by the population size gives the ordinary differential equationThis is the exponential density-dependent birth model.

At a positive equilibrium, division by $N$ givesSince $b>d$, this has the positive solution

Write the vector field as $F(N)=N(b{e}^{-aN}-d)$. Its derivative isAt $N=N^{\ast }$, the first two terms cancel and $b{e}^{-a{N}^{\ast }}=d$, soThe one-dimensional linear stability analysis therefore shows that $N^{\ast }$ is locally asymptotically stable.