Chapter 2.5, Problem 1P

Suppose that a certain population obeys the logistic equation $dy/dt=ry[1-(y/K)]$.

If $y_0=K/3$, find the time $\tau$ at which the initial population has doubled. Find the value of $\tau$ corresponding to $r=0.025$ per year.

If $y_0/K=\alpha$, find the time $T$ at which $y(T)/K=\beta$, where $0<\alpha ,\beta <1$. Observe that $T\to \infty$ as $\alpha \to 0$ or as $\beta \to 1$. Find the value of $T$ for $r=0.025$ per year, $\alpha =0.1$, and $\beta =0.9$.