Check that if $f$ is such a differentiable and strictly increasing function that $f(K)=0$, then the solution of equality $N'(t)=f(N(t)) N(t)$ have the following property: $\lim_{t\to\infty} N(t)=K$.
It's the fact that $N'(t)>0$, when $N(t)<K$,
and $N'(t)<0$, when $N(t)>K$.
The problem is kind of relates to Verhulst Model.
I need help with this exercise, any will be appreciated.