Carrying capacity

Definition

Carrying capacity is the maximum population size of a species that an environment can sustain indefinitely, given the food, habitat, water, and other necessities available in the environment. In differential equations, it is a key parameter in the logistic growth model.

5 Must Know Facts For Your Next Test

1. Carrying capacity is denoted by $K$ in the logistic equation.
2. The logistic equation models population growth as $\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right)$ where $r$ is the intrinsic growth rate and $P$ is the population size.
3. When the population $P$ equals the carrying capacity $K$, the growth rate $\frac{dP}{dt}$ becomes zero because $1 - \frac{P}{K} = 0$.
4. If the population exceeds carrying capacity ($P > K$), the growth rate becomes negative, leading to a decrease in population until it stabilizes at $K$.
5. The concept of carrying capacity introduces a non-linear term into the differential equation, making it different from simple exponential growth models.

Review Questions

• What does carrying capacity represent in the context of population dynamics?
• How does carrying capacity affect the logistic growth model?
• What happens to population growth when it reaches or exceeds carrying capacity?

Related terms

Logistic Equation:

A differential equation used to model population growth with a limiting factor: $\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right)$.

Intrinsic Growth Rate:

$r$, representing how quickly a population increases when there are no constraints on its growth.

Exponential Growth:

$\frac{dP}{dt} = rP$, describing unbounded population increase without any limiting factors.