Maths IA · Modelling · topic 36 of 40

Population: exponential or logistic for one country? · IB Maths IA idea: guide, sample graph and step-by-step plan

a classic, so make a choice that is yours

Any spreadsheet will draw a curve through population data. The real question is which model deserves to, and what happens when you use it to predict.

Difficulty●●○ Medium
KitPublished population data, a spreadsheet or GDC
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like checking which model deserves to win
  • ✓ you can find and cite real data
  • ✓ you want to predict, not just plot

The big idea

Exponential growth says a population grows by the same percentage every year, for ever. Logistic growth adds a ceiling, the carrying capacity, so growth slows as the population fills it. Both can fit early data well. The test is what each predicts beyond the years you fitted. Choose one country and a clear window of years, fit both models, and keep some data back to test the predictions. Say why you chose the models before the software does. At HL, derive the logistic solution from its differential equation instead of only fitting it.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Choose the country for a reason

A country you know, or one with a famous change (a baby boom, a policy, migration).

→ Gives you a context and a reason for the years you choose. The data may break the model, which is interesting.
Hold back data

Fit on the first 70% of the years and predict the rest.

→ A genuine test of prediction, and an error you can compare across models.
Straighten the exponential

Plot ln P against t. For the logistic, look for a transformation that also gives a line.

→ You see why regression suits one model easily and needs more care for the other.
Add a third model

A straight line, a logistic with a changing K, or a Gompertz curve.

→ A comparison of three with residuals, and more to evaluate.
Derive the logistic

Solve dP/dt = rP(1 − P/K) by separating variables and partial fractions; check against a numerical solution.

→ HL-level maths, and your fitted parameters get an explanation instead of just a value.
Compare two countries

Same models, one country near its ceiling and one far from it.

→ Parameters carry meaning: compare r and K, and discuss why they differ.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • Names one country (or region), the exact years of data and the source.
  • Names the models being compared and how 'better' will be judged (residuals, prediction error), not just a graph that looks right.
  • Says what you will predict or test beyond the fitted data.
  • Is narrow enough to go deep: two models in depth, not five lightly.
  • Has a clear mathematical heart (a derivation, a fit and a check), not only a plotted curve.

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The model

Data points, an exponential curve that keeps climbing and a logistic curve that levels at K. Sketch yours with your country's real shape.

Ptcarrying capacity K: why that value?published data: cite it, say which yearsexponential: fine early,then it runs awaygrowth fastest here

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; why this country; what you expect each model to do and why.
The data~1.5 pages
Source, units, which years and why, and anything unusual (a break, a change of definition).
The two models~2.5 pages
Assumptions, formulas, and the meaning and units of each parameter. HL: derive the logistic solution.
Fitting~2.5 pages
One fit worked by hand (for example by straightening); then the software fit; parameters with units and sensible accuracy.
Testing on held-back data~2 pages
Prediction error for each model; residual plots; what the pattern says.
Comparison and judgement~1.5 pages
Which model do you trust, and what is your evidence? Where does each fail?
Going further~1.5 pages
A third model, a second country or a numerical check of the differential equation.
Conclusion~1 page
Answer the question with numbers and say how far the models can be trusted.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

data AND both models together ✓

Made-up data to show the form of a strong graph, not the answer. Yours will look different. example only

01020304050600102030405060708090100Years since 1960, tPopulation P / millionslogistic model (K = 100)exponential fitted to early yearspublished data (to nearest million)data bends away fromthe exponentialILLUSTRATIVE · MADE-UP DATA
  • Years since a stated start keeps the numbers small: say what t = 0 is.
  • Both models on the same axes as the data, so the reader sees which one keeps up.
  • Units on the axis (millions), not just the word 'population'.
  • Shade or mark the years you held back: they are the test of prediction.
  • Another graph to consider: residuals (data minus model) against t for each model. A pattern in them means the model is wrong, even if the curve looks close.

How far off is each prediction?

a number you can rank!

Fit each model to the earlier years only, then compare what it predicts for a later year. The percentage error is a number you can rank. These numbers are made up; yours will depend on your country.

Exponential67.9%Straight line17%Logistic0.7%which model do you trust, and why?% error in the predicted population at year 60, fitted on years 0 to 40ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

hold some data back: it's your test

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictRead about both models. Sketch what you expect each to do for your country, and where you expect them to disagree.
  2. Find the dataChoose a source you can cite, with years you can defend. Check units and any change in how it was counted.
  3. Trial run (1 lesson)Fit both models in a spreadsheet with the first few years. Does the logistic fit settle on a sensible K, or run off?
  4. Decide the windowChoose the years, and which to hold back. Write down why.
  5. Fit by hand onceDo one fit yourself (straighten the data, or solve for parameters from two or three points) so you know what the software is doing.
  6. Fit with softwareGet parameters with units and sensible accuracy. Say what each one means for your country.
  7. Predict and comparePredict the held-back years with each model and compute the errors. Plot the residuals.
  8. Go deeperSL: justify the models and test the limits. HL: derive the logistic solution from its differential equation, or check it numerically.
  9. Write, then cutWrite in layout order. One worked example, one table of results, and graphs placed next to the text that uses them.
  10. Check before your teacher's draftCheck each symbol is defined once, parameters have units, graphs have titled axes, and reflection runs through the piece.

Maths you'll need

Traps to avoid in this Maths IA

  • Plotting a curve and stopping: examiners want the maths behind the fit and a test, not a software trendline.
  • Letting the software pick the carrying capacity K without saying why it is plausible.
  • Judging the models by R² alone, or only on the data you fitted.
  • Fitting through a break in the data (war, a change of definition) as if it were smooth.
  • Mixing units, such as thousands in one table and millions in another.
  • Calling a model 'accurate' without an error value.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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