= 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> $v'+\alpha v=1$, so $v=\alpha^{-1}+Ce^{-\alpha t}$. The initial condition gives $C=-1/(2\alpha)$, hence
$$
\boxed{N(t)=\frac{\alpha}{1-\tfrac12e^{-\alpha t}}
=\frac{2\alpha}{2-e^{-\alpha t}},\qquad \lim_{t\to\infty}N(t)=\alpha.}
$$
The population decreases monotonically toward the stable <equilibrium point>; it never reaches it at a finite time.
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