Population growth

Definition

Population growth describes the change in the number of individuals in a population over time. It can be modeled using exponential functions when considering continuous growth.

5 Must Know Facts For Your Next Test

1. Exponential growth of a population is described by the differential equation $\frac{dP}{dt} = rP$, where $P$ is the population size and $r$ is the growth rate.
2. The general solution to the differential equation for exponential growth is $P(t) = P_0 e^{rt}$, where $P_0$ is the initial population at time $t=0$.
3. When integrating to find population over time, constants of integration are determined using initial conditions.
4. In cases of logistic growth, which accounts for carrying capacity, the differential equation used is $\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right)$ where $K$ is the carrying capacity.
5. The concept of doubling time can be derived from exponential growth models and is given by $T_d = \frac{\ln(2)}{r}$.

Review Questions

• What is the differential equation that describes exponential population growth?
• How do you determine the constant of integration when solving an exponential growth problem?
• Explain how carrying capacity modifies an exponential growth model.

Related terms

Exponential Function: A mathematical function in which an independent variable appears in the exponent; often used to model continuous growth or decay.

Logistic Growth:

A model of population growth that assumes a maximum carrying capacity; described by a sigmoidal curve.

Carrying Capacity:

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