Gompertz equation

Definition

The Gompertz equation is a type of mathematical model for time series, often used to describe growth processes. It is particularly useful in modeling populations, tumors, or other biological systems where growth slows over time.

5 Must Know Facts For Your Next Test

1. The Gompertz equation is given by $\frac{dN}{dt} = rN \ln \left( \frac{K}{N} \right)$, where $N$ is the population size, $r$ is the growth rate, and $K$ is the carrying capacity.
2. It describes a sigmoidal (S-shaped) growth curve that initially resembles exponential growth but eventually levels off as it approaches the carrying capacity.
3. The Gompertz equation can be derived from a differential equation that combines aspects of both exponential and logistic growth models.
4. It has applications in biology, demography, marketing, and reliability engineering due to its ability to model limited growth scenarios realistically.
5. In Calculus II and differential equations contexts, solving the Gompertz equation typically involves separation of variables or integrating factors.

Review Questions

Related terms

Logistic Equation:

A differential equation describing population growth that accounts for environmental limitations: $\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)$.

Exponential Growth:

A process of increase at a constant rate per unit time: often modeled by $\frac{dN}{dt} = rN$, where $r$ is the exponential growth rate.

Carrying Capacity:

The maximum population size that an environment can sustain indefinitely without being degraded.