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 (conceptually) different (compartmentalized) way. In other words:
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Do we start with exponential and then convert it into logistic
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Do we start with constants for birth and death rate and then make them into lines
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Do we start with the assumption things aren't density-dependent, and then make it so that it is
And I think my confusion comes this line: "
As we have defined r = b − d , we can define a density dependent r ′ =
b ′ − d ′
from how it seems that $k = (b_0-d_0)P$ as well as $k=(b-d)P$ and so $b-d=b_0 – d_0$. But that only works for small $P$. Not sure how to explain it, but it seems that we're "doubling up" on the same variable and allowing two different things to be the same.
Best Answer
You seem comfortable with the idea that without interaction, or little interaction corresponding to a very small population density, birth and death are proportional to the population size, their rates being constant.
Taking interactions in the population into account, the rates also become variable, functions of $N$ (or better $N/K$ to indicate their lesser variability). The most simple form for $b(N)$ and $d(N)$ is if they are linear functions $b_0-b_1N$ and $d_0+d_1N$, falling for $b$ and increasing for $d$. The linked script then explores the consequences of that model and how to reduce the 4 coefficients to 2 non-redundant constants.
Other functions are as sensible or even more than the linear ones. The birth rate can become negative, which is not very realistic. As example, $b(N)=\frac{b_0}{1+b_1N}$ or $b(N)=\frac{b_0}{(1+b_1N)^2}$ is still falling, but stays positive for all positive $N$. The drawback is just that the manual exploration of the corresponding differential equation is no longer that easy.