Chapter 1.2, Problem 33P

An equation that is frequently used to model the population growth of cancer cells in a tumor is the Gompertz equation

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

Where $r$ and $K$ are positive constants.

Sketch the graph of $f\left(y\right)$ verses $y$, find the critical points, and determine whether each is asymptotically stable or unstable.

For each $y$ in $0<y\le K$, show that $dy/dt$, as given by the Gompertz equation, is never less than $dy/dt$, as given by the logistic equation, Eq. (i) in Problem 32.

Eq. (i) in Problem 32: $\frac{dy}{dt}=ry\left(1-\frac{y}{K}\right)$.