A guide for converting exponential growth to logistic growth

1. Simple (Exponential) Growth

Hence, per capita net growth rate $r$ (i.e., “births minus deaths” per individual) is: $r = b - d.$

Let $N(t)$ be the population size at time $t$. Then exponential growth is modeled by:

\[\frac{dN}{dt} = r \, N.\]

Interpretation: The larger the population, the faster it grows, with no upper bound.

In nature, birth and death rates tend to change with population size:

Per capita birth rate declines as resources become limited, so we assume:
\[b' = b - a N \quad (\text{where } a \text{ is a positive constant})\]
- When $N$ is small, $b’ \approx b$.
- As $N$ gets larger, the term $-\,aN$ makes the birth rate go down.
Per capita death rate increases as crowding intensifies, so we assume:
\[d' = d + c N \quad (\text{where } c \text{ is a positive constant})\]
- When $N$ is small, $d’ \approx d$.
- As $N$ gets larger, the term $+\,cN$ makes the death rate go up.

From these two modified rates: $\text{net per capita rate} = b' - d' = (b - aN) \;-\; (d + cN).$

Simplify this: $b' - d' = (b - d) \;-\; (a + c)\,N.$

Recalling that $r = b - d$, define $\alpha = a + c$. Then: $b' - d' = r - \alpha \, N.$

Hence, the net growth rate for the whole population $N$ is:

\[\frac{dN}{dt} = \left[r - \alpha N\right] \, N.\]

Let’s rearrange this into a more familiar form. Factor out $r$:

\[\frac{dN}{dt} = rN - \alpha N^2 = rN\bigl(1 - \tfrac{\alpha}{r}N \bigr).\]

Define the carrying capacity $K$ by

\[K = \frac{r}{\alpha}.\]

Then

\[\frac{dN}{dt} = rN\Bigl(1 - \frac{N}{K}\Bigr).\]

This is the logistic growth equation.

$r$: intrinsic rate of increase (as in exponential growth).
$K$: carrying capacity. Once $N$ is near $K$, growth slows and eventually stops.

Initial Exponential Phase: When $N$ is small, $\frac{N}{K}$ is close to 0, so growth looks nearly exponential ($\frac{dN}{dt} \approx rN$).
Resource Limitation: As $N$ grows, limited resources and increased crowding reduce births, increase deaths, and slow the growth.
Equilibrium: In the long run, the population stabilizes around $N = K$, where births and deaths balance out.

Start with a simple exponential model, $\frac{dN}{dt} = rN$.
Acknowledge limitations: constant birth and death rates ignore environmental constraints.
Incorporate density-dependence:
- Birth rates decrease with $N$.
- Death rates increase with $N$.
Combine these rates to find a net per capita rate of $r - \alpha N$.
Obtain $\frac{dN}{dt} = rN - \alpha N^2$.
Rewrite in logistic form: $\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)$.

That’s it! Understanding each step helps clarify why real populations don’t grow forever but instead level off at a carrying capacity.