 Finite-time extinction in a sublinear Bernoulli equation

For $y^{\prime }=y-A{y}^p$, $A>0$ and $0<p<1$, put $u=y^{1-p}$ while $y>0$. Then $u^{\prime }=(1-p)(u-A)$. If $0<y_0<A^{1/(1-p)}$, $u$ decreases to zero at the displayed finite time. The positive branch joins the zero solution with zero first derivative, giving a nonnegative forward solution. The equilibrium at zero is one-sided stable even though the vector field is not locally Lipschitz continuous there; the positive equilibrium is unstable.