Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

IA idea · Differential equations & dynamics

Can the Lotka–Volterra model explain the lynx and hare cycles?

AI HLAA HL Ambitious Also in: Environment, Health & biology

Research question

How well does a Lotka–Volterra predator–prey model with fitted parameters reproduce the roughly ten-year cycles in the Hudson's Bay Company lynx and hare records, and what does the model get wrong?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

The fur-trade records are the most famous predator–prey data in ecology, and they do not quite behave as the textbook model predicts. Finding where and why is excellent Reflection.

The mathematics you'll need

Coupled nonlinear differential equations
Equilibrium points and their meaning
Euler's method or technology to solve the system
Phase-plane (hare vs lynx) plots
Comparing periods and amplitudes with data

Where the data comes from

Use the published pelt series (1845–1935); cite the version you use.

Hudson's Bay Company lynx and hare pelt records (1845–1935) — Annual pelt counts (thousands) used as a proxy for population; bundled with the R package astsa and reproduced in many ecology texts. Several versions of this dataset exist — cite the one you use.

A possible outline

Explain the model and each parameter.
Estimate parameters from the data (period, equilibrium levels).
Solve and compare time series and phase plots.
Investigate the famous anomaly where hares seem to “eat” lynx in some years.
Reflect on using pelts as a proxy for populations.

Pitfalls that cost marks

Treating pelt counts as population counts without comment.
Parameters chosen by eye with no reasoning.
Ignoring that the model's cycles depend on initial conditions.

Showing personal engagement

Explain the ecology in your own words.
Change one assumption (hare carrying capacity) and see if the fit improves.
Relate to a predator–prey system you have seen.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Coupled nonlinear differential equations; Equilibrium points and their meaning
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Coupled nonlinear differential equations; Equilibrium points and their meaning

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: treating pelt counts as population counts without comment; and parameters chosen by eye with no reasoning. Say how each one affects your answer.

Use of mathematics (E)

HL: coupled nonlinear differential equations; equilibrium points and their meaning; euler's method or technology to solve the system; phase-plane (hare vs lynx) plots; comparing periods and amplitudes with data, each explained in your own words and carried out correctly.

How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Add a logistic term for the prey and compare the phase portrait (spiral instead of closed loops).

Extending it for HL

Add an HL technique the question genuinely needs: what an HL differential equations exploration adds.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI HL and AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Coupled nonlinear differential equations, equilibrium points and their meaning, euler's method or technology to solve the system, phase-plane (hare vs lynx) plots and comparing periods and amplitudes with data.

Where can I get data for this IA?

Use the published pelt series (1845–1935); cite the version you use. Checked sources and tools: Hudson's Bay Company lynx and hare pelt records (1845–1935).

Turn this idea into your IA

The planner opens with “Can the Lotka–Volterra model explain the lynx and hare cycles?” filled in: the question, the maths and the data sources.

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