### Essential Concepts

- When studying population functions, different assumptionsâ€”such as exponential growth, logistic growth, or threshold populationâ€”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**

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

[latex]P\left(t\right)=\frac{{P}_{0}K{e}^{rt}}{\left(K-{P}_{0}\right)+{P}_{0}{e}^{rt}}[/latex]**Threshold population model**

[latex]\frac{dP}{dt}=\text{-}rP\left(1-\frac{P}{K}\right)\left(1-\frac{P}{T}\right)[/latex]

## Glossary

- 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