Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 2 / 1C / Solution

This is a logistic equation, with positive equilibrium point $N=\alpha$ and initial population above that value. On the positive branch, put $v=1/N$. It satisfies the linear ordinary differential equation $v^{\prime }+\alpha v=1$, so $v={\alpha }^{-1}+C{e}^{-\alpha t}$. The initial condition gives $C=-1/(2\alpha )$, henceThe population decreases monotonically toward the stable equilibrium point; it never reaches it at a finite time.