Logistic growth
How does density dependence change exponential growth?
Start experiment ↓Interactive teaching examples
Change an assumption. Watch the model answer differently.
Nine interactive experiments connect equations to behaviour, from foundation models to active research questions. Choose by purpose, then change one assumption and inspect what survives.
Model finder
Find the experiment that matches your question.
Use the two filters or scan the research questions. Every result links directly to the corresponding interactive workbench.
SIR intervention
How do intervention timing and strength reshape an epidemic curve?
Start experiment ↓Stopping distance
Why does risk grow faster than speed?
Start experiment ↓Critical transitions
When does gradual pressure trigger an abrupt and path-dependent change?
Start experiment →Equation discovery
Can a governing equation be recovered from noisy observations?
Start experiment →Collective motion
How can local alignment create global order without a leader?
Start experiment →Higher-order contagion
What changes when groups, rather than only pairs, transmit influence?
Start experiment →Operator learning
What must a reusable model learn to map one function into another?
Start experiment →Persistent homology
Which shapes survive a change of observational scale?
Start experiment →Showing all 9 experiments.
01 · Discrete dynamics
Logistic growth and carrying capacity
Pn+1 = Pn + rPn(1 − Pn/K)
The growth term is large when the population is small, then weakens as the state approaches the carrying capacity K. A larger growth rate can create overshoot and alternating corrections.
- Final population
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- Peak population
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- First reaches 90% of K
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Questions to testDoes doubling the initial population halve the time to 90% of K?At what value of r does smooth convergence become alternating convergence?
02 · Coupled differential equations
SIR dynamics with a timed intervention
S′ = −βSI/N · I′ = βSI/N − γI · R′ = γI
The model divides a fixed population of 1,000 into susceptible, infectious, and removed groups. On the selected intervention day, transmission β is reduced by the chosen percentage.
- Initial R₀
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- Peak infectious
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- Peak day
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Questions to testCan a late, strong intervention produce a higher peak than an early, moderate one?Which conclusion changes when the infectious period is uncertain?
This is a deliberately simplified teaching model, not an epidemiological forecast or medical recommendation.
03 · Scaling and decomposition
Why stopping distance grows faster than speed
d(v) = vtr + v²/(2μg)
Reaction distance grows linearly with speed; braking distance grows quadratically. The decomposition shows why a modest increase in speed can create a much larger increase in total distance.
- Reaction distance
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- Braking distance
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- Total distance
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Questions to testDoes doubling speed double total stopping distance?When does reaction time matter more than road friction?
The formula is a classroom baseline on a level road with constant friction. Real stopping distance also depends on gradient, tyres, brakes, weather, perception, and driver behaviour.
From interaction to evidence
A slider movement is a hypothesis, not a conclusion.
Use each example to record one prediction, one parameter change, one observed response, and one limitation. That four-line record turns exploration into a small modeling experiment.
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