Logistic growth model

Definition

The logistic growth model is a mathematical function used to describe how a population grows rapidly at first and then levels off as it approaches a maximum sustainable size, known as the carrying capacity. It is often represented by the formula $P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}$, where $P(t)$ is the population at time $t$, $K$ is the carrying capacity, $P_0$ is the initial population size, and $r$ is the growth rate.

5 Must Know Facts For Your Next Test

1. The logistic growth model accounts for limited resources by incorporating a carrying capacity ($K$), which limits exponential growth.
2. The formula for logistic growth is $P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}$.
3. As time approaches infinity, the population size approaches the carrying capacity, making the ratio $\frac{dP}{dt}$ approach zero.
4. In its early stages, logistic growth resembles exponential growth when resources are relatively abundant.
5. Logistic growth models are widely used in biology, ecology, and environmental science to predict population dynamics.

Review Questions

• What does the variable $K$ represent in the logistic growth model?
• How does logistic growth differ from exponential growth?
• Write down and explain the formula for logistic growth.

Related terms

Exponential Growth Model:

A mathematical representation of population increase that assumes unlimited resources, resulting in rapid and continuous acceleration of population size over time. Represented by $P(t) = P_0 e^{rt}$.

Carrying Capacity:

The maximum population size that an environment can sustain indefinitely given available resources such as food, habitat, water, and other necessities.

$e$ (Euler's Number): $e$ is an irrational constant approximately equal to 2.71828. It serves as the base for natural logarithms and appears frequently in mathematical models involving exponential or logarithmic functions.