Modelling starts by asking what governs the rate of change. Proportional to the amount gives exponential. Proportional to a gap gives cooling. Proportional to both a population and its room to grow gives logistic.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
When two things interact
Predators and prey, susceptible and infected, two connected tanks. Interaction means a system, not a single equation.
When acceleration matters
Anything governed by a force needs a second-order equation, because Newton relates force to the second derivative.
Answer qualitatively first
Equilibria and stability usually answer the real question. Whether the epidemic dies out matters more than the exact curve.
Parameters carry meaning
Every constant should mean something you could measure. A parameter with no interpretation is a warning sign.
Know where the model breaks
Exponential growth is fine early and absurd later. A model is a tool with a working range, not a truth.
The modelling cycle
Translate the situation into an equation, solve or analyse it, interpret the result, and compare with reality. Each stage can fail independently, and the comparison is the one most often skipped.
Assumptions are part of the model
Constant coefficients, no air resistance, unlimited resources. Every model rests on simplifications, and stating them is what allows someone to judge where the conclusions apply.
Three ways to answer
Analytic solution, qualitative analysis, and numerical simulation. Most real equations have no closed form, so the second and third are not fallbacks but the normal tools.
A model must be tested
Compare predictions with data, and revise the assumptions when they disagree. A model that has never been checked against reality is a hypothesis, whatever the quality of the mathematics inside it.
Step 2: Try It Yourself
Tap and try it out.
- Released from(3, -2)
Every arrow points toward the origin, so trajectories settle there. The origin is stable.
Step 3: Watch an Example
One step at a time.
Watch Adaeze Choose a Model
Adaeze studies a fish population in a lake with limited food.
- Step 1
She rules out exponential growth, since unlimited growth contradicts the limited food.
Step 4: Your Turn
Practice makes it stick.
The Choice
Problem 1 of 2
A population with unlimited resources. Enter 1 exponential, 2 logistic, 3 cooling.
The Coffee Again
Problem 2 of 2
An object cooling toward a room at 22 degrees. What is the long-run temperature?
Choose and Interpret
1 of 8
A population limited by food. Enter 1 exponential, 2 logistic, 3 cooling.
2 of 8
A hot object in a cool room. Enter 1 exponential growth, 2 logistic, 3 cooling.
3 of 8
Predators and prey affecting each other. Enter 1 a single equation, 2 a system.
4 of 8
A mass on a spring. What order is the governing equation?
5 of 8
dP/dt = 0.1P(1 − P/700). What is the long-run population?
6 of 8
An exponential growth model run for a thousand years. Is it still realistic? 1 yes, 0 no.
7 of 8
Match each situation to its model.
Tap a card on the left to start.
8 of 8
dT/dt = −k(T − 19). What is the long-run temperature?
Step 5: Quick Check
Show what you know.
Question 1 of 2
dP/dt = 0.3P(1 − P/450). What is the long-run population?
Question 2 of 2
What should you decide first when building a model?
What You Learned
- Choosing a model starts from what governs the rate of change, not from what you know how to solve.
- Interacting quantities need a system; forces need a second-order equation.
- Equilibria and stability usually answer the real question, and every model has a working range.