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

  • Solution to the logistic differential equation/initial-value problem

  • Threshold population model



carrying capacity
the maximum population of an organism that the environment can sustain indefinitely
growth rate
the constant [latex]r>0[/latex] in the exponential growth function [latex]P\left(t\right)={P}_{0}{e}^{rt}[/latex]
initial population
the population at time [latex]t=0[/latex]
logistic differential equation
a differential equation that incorporates the carrying capacity [latex]K[/latex] and growth rate [latex]r[/latex] 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