## Summary of the Logistic Equation

### Essential Concepts

• When studying population functions, different assumptions—such as exponential growth, logistic growth, or threshold population—lead to different rates of growth.
• The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.
• The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.

## Key Equations

• Logistic differential equation and initial-value problem

$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),P\left(0\right)={P}_{0}$
• Solution to the logistic differential equation/initial-value problem

$P\left(t\right)=\frac{{P}_{0}K{e}^{rt}}{\left(K-{P}_{0}\right)+{P}_{0}{e}^{rt}}$
• Threshold population model

$\frac{dP}{dt}=\text{-}rP\left(1-\frac{P}{K}\right)\left(1-\frac{P}{T}\right)$

## Glossary

carrying capacity
the maximum population of an organism that the environment can sustain indefinitely
growth rate
the constant $r>0$ in the exponential growth function $P\left(t\right)={P}_{0}{e}^{rt}$
initial population
the population at time $t=0$
logistic differential equation
a differential equation that incorporates the carrying capacity $K$ and growth rate $r$ into a population model
phase line
a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions
threshold population
the minimum population that is necessary for a species to survive