Teaching objectives
Foxes, rabbits and equations that pulse. Why do predator-prey populations oscillate instead of settling down? And why did fishing less during World War I lead to more sharks? This upper-secondary lab builds the Lotka-Volterra model from the ground up: starting from individual agents that move and reproduce in a meadow, moving through differential equations, all the way to deterministic chaos in three-species systems. Across 17 rooms split into four blocks, students go from watching a simulation to designing and perturbing their own dynamical system.
What you'll learn
- Students observe emergence: an agent-based model (rabbits that move, mate and give birth; foxes that hunt and starve) produces population oscillations without anyone programming an explicit cycle.
- They meet Volterra's paradox: the historical discovery that, between 1915 and 1920, less fishing in the Adriatic led to more sharks — the real-world puzzle that motivated the mathematical model.
- Fixed-rate vs. proportional-rate models are fitted to real data (January and April counts) and compared against the actual simulation.
- Malthus's 1798 prediction is contrasted with the logistic model, which does account for limited resources.
- Students see how discrete update rules converge to differential equations as the time step Δt shrinks to zero: constant rate → straight line, proportional rate → exponential.
- The Lotka-Volterra equations (dx/dt = αx − βxy, dy/dt = −γy + δxy) are introduced with an Euler integrator, including the numerical instability that appears with a large Δt.
- The four parameters (α, β, γ, δ) are isolated one at a time via sliders, so students understand each one's role before combining them in the full model.
- Phase space (x, y) is explored, distinguishing trajectory types: center, stable focus, unstable focus.
- Stochastic noise is introduced, showing how it destabilizes orbits that would otherwise be perfectly cyclic in the deterministic model.
- The model is extended to three species (Hastings-Powell) to demonstrate deterministic chaos: nearly identical initial conditions that diverge completely.
- The three-body problem is shown as a gravitational cousin of the same phenomenon: extreme sensitivity to initial conditions with no randomness at all.
- The lab closes with hawks and doves, an evolutionary game showing how a stable mix of aggressive and peaceful strategies emerges in a population.
Key mathematical ideas
- Emergence: large-scale population behavior (oscillations) arising from simple individual-agent rules, with no explicit design of the pattern.
- The move from discrete to continuous models: as Δt → 0, an iterative update rule turns into a differential equation dc/dt.
- The Lotka-Volterra equations as a coupled nonlinear system of two first-order differential equations, with an equilibrium point at (γ/δ, α/β).
- Numerical integration (Euler's method) and its instability when the time step is too large relative to the system's time scale.
- Phase space as a tool for classifying a dynamical system's qualitative behavior (center, stable/unstable focus) without explicitly solving the equation.
- Sensitivity to initial conditions (deterministic chaos): in systems of three or more coupled nonlinear variables, trajectories starting nearly at the same point can diverge exponentially.
- Replicator dynamics in evolutionary game theory: a strategy's proportion in the population changes according to its relative success (payoff) against the others.
Room-by-room contents
Room 1 · Rabbits and foxes
A meadow full of life simulated with individual agents: rabbits wander, mate and give birth (a pregnant rabbit is marked pink); foxes hunt and reproduce when well-fed, turning gray when they go hungry. There is no chart yet — just free observation of the scene to build intuition about the relationship between the two species.
Student tasks
- Watch the simulation for a while without intervening.
- Note what relationship you notice between the number of rabbits and foxes.
Room 2 · The population chart
The same agent-based simulation, now with a live chart below: blue for rabbits, orange for foxes. The lag between the two curves' peaks makes the predator-prey cycle already sensed in room 1 visible in quantitative form for the first time.
Student tasks
- Watch the peaks of the two curves.
- Determine which of the two populations reaches its maximum first.
Room 3 · Volterra's paradox
Historical context: during World War I (1915-1920) fishing in the Adriatic dropped drastically, yet the proportion of sharks and rays caught rose. D'Ancona asked Volterra for a mathematical model explaining why less fishing benefits the predator — the real origin of the equations built in the next rooms.
Student tasks
- Read the historical context of Volterra's paradox.
- Formulate a hypothesis for why less fishing benefits the predator.
Room 4 · From observation to model
Rabbits are counted in January and April within the agent simulation. From just those two data points, two possible models are built: a fixed-rate one (linear growth) and a proportional-rate one (exponential growth). Students compare both predictions against the simulation's actual evolution.
Student tasks
- Compute both predictions (fixed rate and proportional rate) from the January and April data.
- Determine which of the two models fits the real simulation better.
Room 5 · Malthus and his prediction
In 1798 Thomas Malthus predicted population would grow geometrically while food supply grew only arithmetically, foreseeing catastrophe. The chart compares that exponential prediction with a logistic model that does account for a resource limit, showing why reality resembles the latter curve more closely.
Student tasks
- Visually compare Malthus's exponential curve with the logistic curve.
- Explain what variable was missing from Malthus's model.
Room 6 · From discrete to continuous
Shown here is how, as the time step Δt shrinks, the discrete update rules from earlier rooms converge to differential equations: a constant rate produces dc/dt = r in the limit (a straight line), while a proportional rate produces dc/dt = αc (an exponential). This is the conceptual bridge into the continuous calculus of block C.
Student tasks
- Watch how the discrete curve changes as Δt shrinks.
- Relate the constant-rate case to a straight line and the proportional-rate case to an exponential.
Room 7 · The Lotka-Volterra equations
The two coupled differential equations are presented: dx/dt = αx − βxy for rabbits, dy/dt = −γy + δxy for foxes. An Euler integrator with adjustable Δt (via buttons) lets students observe the system's behavior — and also its numerical instability when the time step is too large.
Student tasks
- Try different Δt values and observe when the simulation becomes unstable.
- Identify the smallest Δt that still produces a stable cycle.
Room 8 · α: rabbit birth rate
With no foxes on scene, only rabbits grow according to their own birth rate α. Students move an α slider and watch on the chart how a positive value produces growth, a negative one produces decline, and α = 0 keeps the population constant.
Student tasks
- Move the α slider and observe the effect on the rabbit population.
- Identify which sign of α corresponds to growth, decline, and equilibrium.
Room 9 · β: the fox effect
Foxes are now added at a fixed count, and students adjust β, the parameter measuring how much each fox reduces rabbit growth. The chart shows how a larger β slows down — or even reverses — the rabbit population's growth more quickly.
Student tasks
- Move the β slider and observe the effect on rabbit growth.
- Explain what β represents in terms of the predator-prey interaction.
Room 10 · γ and δ: fox survival
With the rabbit count fixed, students explore γ (fox natural mortality) and δ (energy each fox gains from hunting a rabbit). The chart reveals a threshold: below a certain number of rabbits, foxes fail to gain enough energy to offset their mortality and go extinct.
Student tasks
- Adjust γ and δ and observe the effect on the fox population.
- Determine the minimum number of rabbits foxes need to survive.
Room 11 · The full model: rabbits and foxes
All four parameters work together for the first time: both species affect each other, and the complete Lotka-Volterra system produces the cyclic oscillations observed in the agent simulation of room 1 — but now governed by explicit, adjustable equations.
Student tasks
- Move all four sliders (α, β, γ, δ) and observe the resulting oscillations.
- Try to keep both species alive for 120 consecutive weeks.
Room 12 · Phase space
Instead of plotting x(t) and y(t) separately, each moment is drawn as a point (x, y): the trajectory traces a closed orbit around the equilibrium point (γ/δ, α/β). Changing the initial conditions x₀, y₀ produces different orbits, but always closed around the same center.
Student tasks
- Change the initial conditions x₀ and y₀ and observe the new orbit.
- Locate the equilibrium point (γ/δ, α/β) in phase space.
Room 13 · Types of trajectories
The same system, with small variations in its formulation, can produce three qualitatively different behaviors in phase space: a center (closed orbits, as in pure Lotka-Volterra), a stable focus (the orbit collapses toward equilibrium) or an unstable focus (the orbit spirals away). Students switch modes and compare the three shapes.
Student tasks
- Select each of the three modes (center, stable focus, unstable focus).
- Describe the visual difference between the three trajectories in phase space.
Room 14 · Noise and destabilisation
Real ecosystems are never perfectly deterministic: multiplicative random noise is added to the equations, and the phase portrait is shown in real time, with a fading trail and an extinction counter. A noise-intensity slider σ lets students see how, as noise grows, it becomes more likely that one of the two species goes extinct even though the pure model would be perfectly cyclic.
Student tasks
- Gradually increase σ and observe the effect on the orbit.
- Explain why noise can drive a species to extinction even though the deterministic model doesn't predict it.
Room 15 · Three species: chaos
The model is extended to a three-link food chain (Hastings-Powell model). With no random element at all, two simulations starting from nearly identical initial conditions (a difference of just 0.001 in one species) end up diverging completely — the signature of deterministic chaos.
Student tasks
- Watch the time series of the three species.
- Compare the two trajectories with nearly equal initial conditions and describe how they diverge over time.
Room 16 · The three-body problem
A gravitational cousin of the chaos from the previous room: two stars and a planet under pure Newtonian gravity, with no randomness at all. Two planets starting from nearly identical positions (a difference of 0.05) end up on completely different orbits, showing that sensitivity to initial conditions isn't exclusive to ecology.
Student tasks
- Watch how the two planets' orbits diverge.
- Relate this behavior to the chaos observed in room 15.
Room 17 · Hawks and doves
The lab closes with evolutionary game theory: an arena of agents playing the role of "hawks" (aggressive) or "doves" (peaceful), governed by a real-time replicator equation. Students adjust the reward V and cost C sliders and watch the proportion of hawks converge to a stable equilibrium value p* = V/C.
Student tasks
- Move the V and C sliders and observe how the equilibrium proportion changes.
- Verify that the final proportion converges to p* = V/C.
Rooms to project
The most striking ones to show and discuss in class.