calculus – Deriving logistic growth equation from the exponential

I’m following along with this: https://jdyeakel.github.io/teaching/ecology/section9/ and I just want to make sure my derivation is correct since this website seems to use prime notation not for derivatives, but for different values. (And that really took me an embarrassingly long time to figure out)

So, it’s as if we start off with exponential growth $frac{dN}{dt}=kN$ and then, for small population $N$, $k=b_0 – d_0$ (where those $0$‘s are the initial values, or y-intercepts).

So the equation becomes $frac{dN}{dt}=(b_0 – d_0)N$ but then, as population increases, we don’t want constant values, but linear equations $b$ and $d$. And these linear equations are are $b_0 -aN$ and $d_0 + cN$

Now we’d get $frac{dN}{dt}=((b_0 -aN) – (d_0 + cN))N$

So I guess my question is “is this how you make the connection between exponential growth and logistic growth”? Do we start with one and then build the other one on top of it, or should it be derived in a completely different way. In other words:

  • Do we start with exponential and then convert it into logistic

  • Do we start with constants for birth and death rate and then make them into lines

  • Do we start with the assumption things aren’t density-dependent, and then make it so that it is