Logistic differential equation

Definition

A logistic differential equation models population growth by incorporating a carrying capacity, which limits the growth as the population size increases. The general form is $\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right)$, where $P$ is the population size, $r$ is the intrinsic growth rate, and $K$ is the carrying capacity.

5 Must Know Facts For Your Next Test

1. The logistic differential equation can be solved using separation of variables.
2. Its solution describes an S-shaped curve known as a logistic function or sigmoid curve.
3. At small population sizes ($P \approx 0$), the growth is approximately exponential.
4. As $P$ approaches the carrying capacity $K$, the growth rate slows down and eventually stops.
5. The point at which the population grows fastest is at half of the carrying capacity ($P = \frac{K}{2}$).

Review Questions

Related terms

Exponential Growth:

A model of population increase where the rate of change is proportional to the current size, leading to continuous and unbounded growth. It has the form $\frac{dN}{dt} = rN$.

Carrying Capacity:

The maximum population size that an environment can sustain indefinitely. In a logistic model, it is denoted by $K$.

Sigmoid Curve: An S-shaped curve representing logistic growth. Initially exponential, it slows down as it approaches a horizontal asymptote at its carrying capacity.