Maths IA · Modelling · topic 86 of 97
Will the population keep ageing? Leslie matrices · IB Maths IA idea: guide, sample graph and step-by-step plan
one matrix, repeated: that's the modelSome countries have fewer than one birth per woman. Nothing dramatic happens in year one, but a matrix can show you what the age structure looks like in fifty years, and why it cannot be fixed quickly.
- ✓ you like matrices and want to see them do real work
- ✓ you want to ask what a very low birth rate does to a country
- ✓ you are happy to build the model and then question it
The big idea
A Leslie matrix tracks a population by age group. One row says how many births each age group produces. The other entries say how many survive into the next group. Multiply the population vector by the matrix to move one step forward in time. Repeat, and the age structure settles into a stable shape, while the total grows or shrinks by a fixed factor each step. That factor is the dominant eigenvalue, and the stable shape is its eigenvector. Choose rates from real data, project, and ask what fertility would give a factor of exactly one. The depth comes from asking why the structure settles, not only that it does, and from testing how much the answer depends on your rates.
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 →
Multiply a starting age vector by the matrix for several steps and watch the shares of young, working-age and old.
→ Matrix powers applied to a real situation. Report how the dependency ratio changes, with the arithmetic shown.Find the dominant eigenvalue and its eigenvector, and compare them with what the projection settles to.
→ Eigenvalue theory tested against repeated multiplication. Explain why the projection converges.Find the fertility at which the dominant eigenvalue equals one, and compare it with the usual 2.1.
→ An equation in the fertility, solved by technology or algebra. Say why the answer depends on survival.Raise fertility, or add migration of working-age adults, and compare the effect on the old-age share after fifty years.
→ Two ways of changing the model compared on one measure. A migration term turns the model into x(t+1) = Lx(t) + m.Rebuild the model with 5-year groups or with 20-year groups and compare the eigenvalues.
→ A convergence question: how much of your answer comes from the grouping you chose?Start from a real age structure and compare your projection with an official projection.
→ Your model against a professional one. Explain every gap in terms of what your model leaves out.Your Maths IA research question: run the check
women only? which step?Write your own question first. Then make sure it ticks every box:
- Names the population, the age groups and the time step, and says whether the rates are real or stylised.
- Names the quantity you will find: the dominant eigenvalue, the replacement fertility or a future age share.
- Says where fertility and survival rates come from, or how you will choose them and test them.
- Compares at least two scenarios, such as two fertility levels, so there is something to evaluate.
- States what the model leaves out: migration, changing rates, males and females treated together.
The model
Age groups in a row. Each group survives into the next, and the groups of childbearing age send births back to the first. Draw yours with your own group sizes and mark where births come from.
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.
Which population and why. What you expect the age structure to do, and what you think replacement fertility is.
Every symbol defined. How each entry of the matrix is built from a birth rate or a survival rate, and the matrix written out.
The source of each rate, how you converted it to your time step, and what you simplified.
The age structure over several decades, shown in a table and a graph, read in words.
The dominant eigenvalue and eigenvector, checked against the projection, and the fertility at which the eigenvalue is 1.
Back to your aim: what happens to the population and its ageing, with numbers.
Constant rates, one sex, no migration, grouping. Say which assumption matters most for your conclusion.
The source of every rate and every claim about the population.
What your Maths IA graph could look like
mark where growth = 0Made-up numbers to show the form of a strong graph, not the answer: the long-run growth per year against fertility. Yours will come from your own matrix. example only
- Put growth on the vertical axis and fertility on the horizontal, with units on both.
- Mark the point where growth is zero, and say what fertility that is.
- Plot a few of your own runs as points, so the curve is tied to calculation.
- Show the replacement level you found beside the usual 2.1 and explain any gap.
- Another graph to consider: the share of people over 65 against the number of decades.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 9 done
- Read and predictFind out how a Leslie matrix works and what fertility rates look like in a few places. Predict the replacement level before computing anything.
- Choose the populationPick one country or region, a time step and age groups you can justify. Decide whether to model women only.
- Build the matrixTurn published birth and survival rates into a matrix. Define every entry and say how you converted rates to your time step.
- Trial run (1 lesson)Project one step by hand and check it against your spreadsheet. Fix any mismatch now, such as which group the newborns join.
- Project the structureMultiply forward for fifty years. Show the age shares in a table and a graph, and describe the trend in words.
- Find the eigenvalueFind the dominant eigenvalue and eigenvector with technology and compare them with the projection. Explain why they agree.
- Go further, with a reasonFind the replacement fertility, or test a migration term. HL: this is where the rigour for criterion E comes from, so show the algebra.
- Reflect as you goAfter every result, write one line on what it means for the real population and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes and units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Matrix model – Build from fertility and survival and project with .
- Eigenvalues and eigenvectors – Solve for the dominant eigenvalue and the stable age distribution .
- Matrix powers – Show for large and compare with the projection.
- Replacement condition – Show exactly when the net reproduction number , and solve for the fertility.
- Modelling and sensitivity – Change survival or fertility slightly and measure the change in ; add a migration vector .
Key equations
Syllabus links
- 1.16Systems of up to three linear equationsHL
- 1.3Geometric sequences and series
- 1.14Matrices, determinants and inversesHL
- 1.15Eigenvalues, eigenvectors and matrix powersHL
- 3.15Adjacency matrices, walks and transition matricesHL
- 4.19Markov chains and steady statesHL
- 2.6The modelling process
- 1.3Geometric sequences and series
- 4.12Designing data collection; reliability and validityHL
Read up first
Data and sources to start from
Age-specific fertility and survival rates from a national statistics office or a life table, and a real starting age structure if you compare with projections.
- United Nations: World Population Prospects
- Human Mortality Database
- Singapore Department of Statistics
References to cite
- Leslie, P. H. (1945). On the use of matrices in certain population mathematics. Biometrika, 33(3), 183-212.
- Caswell, H. (2001). Matrix population models: Construction, analysis, and interpretation (2nd ed.). Sinauer Associates.
- United Nations, Department of Economic and Social Affairs, Population Division. (n.d.). World population prospects. https://population.un.org/wpp/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
step size and rate must match- Treating the total fertility rate as the entry in the matrix, without dividing by the number of time steps in a woman's fertile years.
- Counting men and women together while using a women-only fertility rate.
- Using one-year survival rates in a ten-year step, or the other way round.
- Stating the eigenvalue without checking it against a projection.
- Confusing the growth per step with the growth per year.
- Presenting a projection with constant rates as a forecast of what will happen.
- Listing the matrix without saying what each entry means.
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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.