Mathematical modelling

Scenario

A yeast culture in a warm flask is counted every hour. Early on the numbers behave: eight cells, then sixteen, then thirty-two, sixty-four, doubling on the hour like clockwork. Fit the obvious rule, a doubling every hour, and it predicts five hundred and twelve cells by the sixth hour. The count comes back at two hundred and eighty, wrong by nearly half. The rule that fit the early data flawlessly has failed, and the failure is the interesting part. Somewhere in the tidy doubling was an assumption the flask does not honor, and finding it is how the model improves. A model is a deliberately simplified account of a system; George Box put the stance exactly, that all models are wrong but some are useful.

How it works

The modelling loop has a fixed shape. Begin with a real question precise enough to answer. Choose variables for the quantities that matter and, just as importantly, write down the assumptions, every simplification the model rests on, out loud where they can be checked. State the relations among the variables. Work out the consequences the relations predict. Compare those predictions against what the system actually does. Then revise the model where it failed or, if it failed badly enough, reject it and start from a different set of assumptions. The loop runs until the model is good enough for the question, not until it is true, because it will never be true.

Two probes keep the loop honest, and they are different. Parameter sensitivity asks what happens to the answer when a number in the model is nudged: perturb a rate or a coefficient and watch whether the prediction drifts a little or lurches. An answer that swings wildly on a parameter you only know roughly is a warning. Assumption criticality is the deeper probe: delete an assumption entirely and watch whether the model changes class, predicting a qualitatively different behavior. Confusing the two hides where a model is fragile, so this entry keeps them separate: sensitivity varies a value inside the model, criticality removes a commitment the model was built on.

Worked example

Return to the yeast. The doubling rule is an exponential model, and its buried assumption is that resources are unlimited, so every cell always divides on schedule. That is the assumption the flask refuses: sugar runs low, and division slows as the population crowds its food supply. Delete the unlimited-resources assumption and the model changes class, from exponential to logistic, where growth is fast when the population is small and flattens as it approaches a carrying capacity the flask can sustain. Refit the logistic model to the same early data plus the later plateau, and it reproduces the two hundred and eighty and predicts where the population levels off. Now test sensitivity: nudge the carrying capacity up by a tenth and the predicted plateau rises by about a tenth while the early curve barely moves, which tells you the late behavior leans hard on that one number and the early behavior does not. The cooling of a hot drink would show the same loop, but it is the worked territory of the differential-equations entry, so the population case is used here to keep this entry's burden its own.

Limits and boundary conditions

Fitting is not explaining. A model tuned until it matches the data can match for the wrong reasons, and a good fit is evidence, not proof, that the mechanism is right. A model validated in one regime claims nothing outside it: the logistic curve fit to one flask at one temperature does not transfer to a colder flask without new testing. And a model is not the system. The map-territory caution is not decoration here; it is the reason the comparison step exists, to measure the gap between the model and the world rather than to pretend it away.

Common mistakes

Three habits wreck models. The first is adding parameters to rescue a poor fit: enough free knobs will fit anything, including the noise, and the model then predicts nothing new. The second is mistaking elegance for truth, trusting a beautiful equation past the point where the data contradicts it. The third is keeping a refuted model because it is familiar or was expensive to build, which turns the whole loop into theater.

Build with it

Build a model of something you can measure within a week: a plant's height, a cooling cup, the balance of a repaid loan. Name your variables and, in writing, every assumption. State in advance the observation that would make you reject the model, an actual number or behavior you would count as failure. Take the measurement. Then either revise, naming the assumption that failed and the class the model moves to, or record honestly why the model survived and what regime that verdict is limited to. Success is the pre-stated rejection criterion, the measurement taken, and either a named failed assumption or a bounded survival note.