Exponential density-dependent birth model
= Exponential density-dependent birth model
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].