A deer population grows logistically and is harvested at a rate proportional to its population size. The dynamics of population growth is modeled by
$P' = rP (1-\frac{P}{K}) - hP$
where $r$ (the rate of growth), $K$ (capacity of the environment) and $h$ (harvesting rate) are constants
For $h > 0$, use a bifurcation diagram to explain the effects on the equilibrium deer population when h is slowly increased from a small value to a large value.
Question:
How do I use changing variables to simplify the model (only left with one variable and the constant $h$) so that I am able to find the equilibria?
For example in the following model: $P' = rP (1-\frac{P}{K}) - H$
I am able to change varibles via $x = \frac{P}{K}$ and $ \mu = rt$, and got $\frac{dx}{d\mu } = x(1-x) - h$ where h is a constant $h = \frac{H}{rK}$ in order to find its equilibria.
Or, there might be other approaches for this question.
I prefer to think about this through the relatively systematic procedure of nondimensionalization.
Introduce $P=P_C R,t=t_C s$, for unknown quantities $P_C$ with units of population and $t_C$ with units of time. Insert this into the equation to get
$$\frac{P_C}{t_C} \frac{dR}{ds} = r P_C R \left ( 1 - \frac{P_C R}{K} \right ) - h P_C R.$$
Cancel coefficients:
$$\frac{dR}{ds}=r t_C R \left ( 1 - \frac{P_C R}{K} \right ) - h t_C R$$
Now you want to choose $P_C$ and $t_C$ to simplify this equation. You can do that by making some of the coefficients be $1$; there are two reasonable ways to do this for this equation. For your bifurcation study, one of them is more convenient than the other, and that is what you almost correctly did, namely to have $t_C=\frac{1}{r}$ and $P_C=K$. Then you have
$$\frac{dR}{ds}=R(1-R)-aR$$
where $a=\frac{h}{r}$ is a bifurcation parameter.