Concept preview

What you will be able to do

1. 01Write and read the logistic differential equation $\dfrac{dP}{dt}=kP\left(1-\dfrac{P}{M}\right)$, identifying $k$ as the natural growth rate and $M$ as the carrying capacity.
2. 02Explain the shape of the S-curve: near-exponential growth when $P$ is small, slowing growth as $P\to M$, equilibria at $P=0$ and $P=M$, and fastest growth at the inflection point $P=M/2$.
3. 03Determine long-term behaviour $\lim_{t\to\infty}P$ from the sign of $\dfrac{dP}{dt}$, and compute the growth rate or the fastest-growth population for a logistic model in context.

A population growing logistically obeys $\frac{dP}{dt}=kP\left(1-\frac{P}{M}\right).$ Here $P=P(t)$ is the population at time $t$, the constant $k>0$ is the natural (or intrinsic) growth rate, and $M>0$ is the carrying capacity — the maximum population the environment can sustain. Compare it with the exponential law $\frac{dP}{dt}=kP$. The logistic equation is the exponential rate $kP$ multiplied by the extra factor $\left(1-\frac{P}{M}\right)$. That factor is the whole story: it is close to $1$ when $P$ is tiny compared with $M$, and it shrinks toward $0$ as $P$ climbs toward $M$.