Kenneth ChengMathematical modeling

Interactive teaching examples

Change an assumption. Watch the model answer differently.

Fourteen interactive experiments connect equations to behaviour, inference, and decision. 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.

Research-facingObservation error

Melt-pond topology

Can blur, pixel size, or segmentation-boundary displacement create a spanning threshold that is absent from the latent field?

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Showing all 14 experiments.

01 · Discrete dynamics

Logistic growth and carrying capacity

Pn+1=Pn+rPn(1−PnK)P_{n+1}=P_n+rP_n\left(1-\frac{P_n}{K}\right)

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

dSdt=−βSIN,dIdt=βSIN−γI,dRdt=γI\frac{\mathrm{d}S}{\mathrm{d}t}=-\frac{\beta SI}{N},\qquad \frac{\mathrm{d}I}{\mathrm{d}t}=\frac{\beta SI}{N}-\gamma I,\qquad \frac{\mathrm{d}R}{\mathrm{d}t}=\gamma I

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+v22μgd(v)=vt_r+\frac{v^2}{2\mu g}

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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