Summary of the Logistic Equation | Calculus II

Module 4: Differential Equations

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