Chapter 1.2, Problem 32P

If in the exponential model for population growth, $dy/dt=ry$, the constant growth rate $r$ is replaced by a growth rate $r\left(1-y/K\right)$ that decreases linearly as the size of the population increases, we obtain the logistic model for population growth,

$\frac{dy}{dt}=ry\left(1-\frac{y}{K}\right)$, (i)

in which $K$ is referred to as the carrying capacity of the population. Sketch the graph of $f\left(y\right)$, find the critical points, and determine whether each is asymptotically stable or unstable.