Carrying capacity

Definition

Carrying capacity is the maximum population size of a species that an environment can sustain indefinitely, given the available resources such as food, habitat, water, and other necessities. In mathematical models, it is often represented by the constant $K$ in logistic growth equations.

5 Must Know Facts For Your Next Test

1. Carrying capacity ($K$) limits exponential growth and introduces a leveling off at the maximum sustainable population.
2. The logistic growth model is defined by the equation $P(t) = \frac{K}{1 + \frac{K - P_0}{P_0}e^{-rt}}$, where $P(t)$ is the population at time $t$, $r$ is the growth rate, and $P_0$ is the initial population.
3. When population size reaches carrying capacity, growth rate slows down and approaches zero.
4. In exponential models without a carrying capacity term, populations grow without bound as $t \to \infty$.
5. Real-world examples often demonstrate that populations fluctuate around carrying capacity due to environmental changes.

Review Questions

• What does carrying capacity ($K$) represent in a logistic growth model?
• How does including a carrying capacity term change an exponential growth equation?
• Why does population growth slow down as it approaches carrying capacity?

Related terms

Logistic Growth: A model of population growth that incorporates a carrying capacity, resulting in an S-shaped curve.

Exponential Growth: A model of population increase where the rate of change is proportional to the current value, leading to continuous and unlimited growth.

$r$ (Growth Rate): $r$ represents the intrinsic rate of increase in both exponential and logistic growth models.