Logistic Growth
Growth limited by carrying capacity.
Key idea
Growth limited by carrying capacity.
In depth
y′ = ry(1 − y/K) grows almost exponentially when y is small and levels off at carrying capacity K. The solution is an S-shaped curve y = K/(1 + Ae^(−rt)). The fastest growth occurs at y = K/2.
Worked example
With K = 1000, r = 0.5 and y₀ = 10, the population approaches 1000 over time.
Formula
Real-world uses
- Predicting how a new app or species grows toward a limit set by resources or market size
Common mistake
Treating K as a maximum that can never be exceeded; it is an equilibrium.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Logistic Growth?
Growth limited by carrying capacity. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Logistic Growth?
The key formula is dP/dt = rP(1−P/K).
Where is Logistic Growth used in real life?
Real-world examples: Predicting how a new app or species grows toward a limit set by resources or market size.
What should I learn before Logistic Growth?
Make sure you are comfortable with Exponential Growth and Decay Models, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Logistic Growth?
Treating K as a maximum that can never be exceeded; it is an equilibrium.
How can I practise Logistic Growth?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Logistic Growth?
Move on to Newton’s Law of Cooling next.
Related advanced topics in Differential Equations, Abstract Algebra & Applied Math
- What Is a Differential Equation? – An equation relating a function and its derivatives.
- Order and Linearity of ODEs – Classify equations by highest derivative and linearity.
- Direction Fields – Slope fields show solution curves visually.
- Separable Equations – Separate variables and integrate both sides.
- First-Order Linear Equations – Use an integrating factor.
- Exponential Growth and Decay Models – dy/dt = ky gives y = y₀eᵏᵗ.