Maths IA · Modelling · topic 88 of 97
Lynx and hares: predator-prey cycles · IB Maths IA idea: guide, sample graph and step-by-step plan
two equations, one chase: that's your IAWhen hares are plentiful, lynx thrive. When lynx thrive, hares crash. Then the lynx starve and the hares come back. Can two equations produce that endless chase, and what does the maths say about the point where nothing changes?
- ✓ you like seeing a differential equation come alive on a graph
- ✓ you want a model where two quantities push each other round in a loop
- ✓ you are happy to code a numerical method and test how well it works
The big idea
Let the hares grow on their own and be eaten in proportion to how often they meet a lynx. Let the lynx die off on their own and grow in proportion to the same meetings. That gives two coupled differential equations, with the product of the two populations as the link. Setting both rates to zero gives an equilibrium. Near it, the populations oscillate round it, and in the phase plane each starting point gives a closed loop. You cannot solve the equations in closed form, so you simulate them with Euler's method and test how the step size changes the answer. The marks come from asking what the model gets wrong: real populations are limited by food, weather and chance, and real cycles do not repeat perfectly.
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 →
Set both rates to zero and solve, then check by starting the simulation exactly there and a little to one side.
→ A pair of equations solved by factorising. Say why the lynx level depends on the hares' growth rate, and not the lynx's.Step the two populations forward together and plot both against time and against each other.
→ A coupled numerical method. Halve the step size repeatedly and describe what happens to the loops.Find the time between hare peaks for several starting populations and compare with a small-oscillation estimate.
→ A link from the equations to one number you can test. See whether bigger loops take longer.Give the hares a carrying capacity so their growth is logistic, and rerun.
→ The closed loops become a spiral into the equilibrium. Explain the change in terms of the new term.Divide the two equations to get dy/dx, separate the variables and integrate, then check the quantity stays constant along your simulation.
→ A check on the numerical method that does not need the exact solution. It shows why the loops close.Use published fur-trade counts of lynx and hares, and compare the cycle length and the order of the peaks with your model.
→ A model against messy data. Say what the records measure, and which of the model's assumptions they break.Your Maths IA research question: run the check
which animals? which number?Write your own question first. Then make sure it ticks every box:
- Names the two species (or a labelled made-up pair), the four rates in the model and what each stands for.
- Names the quantity you will find: the equilibrium, the cycle length, or the effect of a changed rate.
- Says how the equations are solved (numerically) and how the numerical method is checked.
- Goes beyond simulating: there is an equilibrium to derive and a conserved quantity or a limit to test.
- States what you leave out: food limits, other prey, weather, immigration and chance.
The model
Two populations linked by how often they meet. Label each arrow with the rate you will use, and mark the point where all the rates cancel.
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.
The species or made-up pair, where the rates come from, and a prediction before calculating: what shape will the cycle have?
Each term justified in words, every symbol and unit defined, and the assumptions listed.
Both rates set to zero and solved, with a check that nothing moves when you start there.
The method for two equations at once, the code or spreadsheet, and a table of the first few steps checked by hand.
Both populations against time and against each other, the cycle length, and how the step size changes the picture.
Published counts beside your curve. A number for the gap in cycle length and a reason for it.
Back to your aim: the equilibrium, the cycle, and how far the model explains the records.
Which assumptions break first, how the numerical method may mislead, and what a better model would add. Say which matters most.
Where every rate, record and method came from.
What your Maths IA graph could look like
plot one against the other too!Made-up values to show the form of a strong graph, not the answer: hare and lynx numbers in thousands, close to an equilibrium. Yours will look different. example only
- Both populations on one set of axes, so the lag between the peaks is visible.
- Axis labels give units: years on the time axis, thousands of animals on the population axis.
- Mark the equilibrium level as a line on each curve, and calculate it first.
- Plot your own data as points: agreement with the curve is your evidence, and the gaps are your evaluation.
- Another graph to consider: lynx against hares, so each starting point gives a loop.
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 10 done
- Read and predictFind out how predator and prey numbers rise and fall in the wild. Before any calculation, sketch what you expect to see for both.
- Choose the systemPick a real pair with published records, or a made-up pair you label as made-up, and note where every rate comes from.
- Build the equationsWrite each term and justify it in words: growth, meetings, feeding, death. Define every symbol and unit.
- Trial run (1 lesson)Code Euler's method for ten steps. Check the first two steps by hand and fix any mismatch now: step size, order of updating, signs.
- Find the equilibriumSet both rates to zero, solve, and test the answer by starting the simulation there.
- Run and measurePlot populations against time and against each other for several starting points. Measure the cycle length for each.
- Test the numerical methodHalve the step size several times. Describe how the loops change and find a step size you can trust.
- Go further, with a reasonAdd a limit to the hares' growth, or derive the conserved quantity and check it. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it says about real animals and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Coupled differential equations – Write and and justify each term from a stated assumption.
- Equilibrium by solving simultaneously – Set both rates to zero and solve to get ; test it by starting the simulation there.
- Euler's method for a system – Update and together by and measure how the step changes the loops.
- Phase-plane plots – Plot against for several starting points and describe the closed loops and their direction.
- A conserved quantity – Separate variables in and integrate to get ; check it stays constant in the simulation.
Key equations
Syllabus links
- 5.1Limits and the derivative as a rate of change
- 5.6Standard derivatives; chain, product and quotient rules
- 5.18First-order differential equations; Euler's methodHL
- 2.6The modelling process
- 5.1Limits and the derivative as a rate of change
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
- 5.14Setting up differential equations; separating variablesHL
- 5.16Euler's method, including coupled systemsHL
- 5.17Phase portraits of coupled linear systemsHL
Read up first
Data and sources to start from
Published fur-trade records of lynx and snowshoe hare from the Hudson's Bay Company, used for the cycle length and order of peaks, plus an Euler simulation of the model.
References to cite
- Elton, C., & Nicholson, M. (1942). The ten-year cycle in numbers of the lynx in Canada. Journal of Animal Ecology, 11(2), 215-244.
- Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature, 118, 558-560.
- Lotka, A. J. (1925). Elements of physical biology. Williams & Wilkins.
- Strogatz, S. H. (2015). Nonlinear dynamics and chaos (2nd ed.). Westview Press.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
big steps make fake spirals- Treating the equations as a prediction. They give a pattern, and real cycles are noisier and shorter or longer.
- Using a large Euler step and calling the growing loops a feature of the model. It is an error in the method.
- Updating one population before the other uses the new value. Both must use the values from the start of the step.
- Forgetting that the equilibrium is a point where nothing changes, not the average of the data.
- Claiming records of fur sold are census counts of the animals.
- Putting the units of the four rates in the wrong places, so the equations do not balance.
More Maths IA ideas like this
Epidemics: does a top-down model match a real outbreak?●●●
Population: exponential or logistic for one country?●●○
Chaos: does a butterfly change the weather?●●●
How many fish are in the pond? Capture-recapture●●○
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.