F. Population Dynamics · Population DynamicsInvestigation: Modelling Predator-Prey and Host-Parasite Populations
F2.3 F2.2 F3.1 A1.1 A1.5 A1.6 A1.8 A1.10 A1.11About 75 minutes
Learning goals- I can explain each term of the Lotka-Volterra predator-prey model and find its equilibrium.
- I can run a predator-prey simulation step by step (by hand or in a spreadsheet) and describe the resulting cycles.
- I can use a host-parasitoid model to predict how two linked populations change from generation to generation.
- I can evaluate models against laboratory and field results (Gause, Huffaker, lynx and hare) and identify their limitations.
Words to know
model
A simplified representation (equations, a simulation or a physical set-up) used to explain and predict a real system.
Lotka-Volterra model
A pair of equations describing linked predator and prey populations: dH/dt = rH − aHP and dP/dt = baHP − mP.
equilibrium
Population sizes at which neither population changes (both growth rates are zero).
phase plot
A graph of predator numbers against prey numbers; linked cycles appear as loops.
host-parasitoid model
A model (such as the Nicholson-Bailey model) of hosts and the parasitoids that develop in them, one generation at a time.
simulation
Running a model step by step (often on a computer) to see how a system changes over time.
refuge
A place or condition where some prey are safe from predators, which can help both populations persist.
The question
The curriculum asks you to determine, through laboratory inquiry or computer simulation, the characteristics of the growth of two different populations that interact, such as predator and prey or host and parasite. Real predator-prey experiments take months, so in this lesson you will run the classic mathematical models in a spreadsheet, compare them with famous laboratory and field studies, and write up your investigation.
Testable question (A1.1): How does a predator population change over time when it depends on one prey species, and how does the prey population respond?
Hypothesis: If predators depend on one prey species, then both populations will cycle, with predator peaks lagging behind prey peaks, because predators increase only after prey become abundant and prey decline when predators become abundant.
Model 1: the Lotka-Volterra predator-prey equations
In the 1920s, Alfred Lotka and Vito Volterra independently wrote the same pair of equations. Let H be the prey (think hares) and P the predators (lynx).
dH/dt = rH − aHP
- rH: prey grow exponentially without predators (r = prey per capita growth rate).
- aHP: prey killed by predators. The number of meetings is proportional to H × P; a is the attack rate.
dP/dt = baHP − mP
- baHP: new predators produced from prey eaten (b is how efficiently eaten prey are converted into new predators).
- mP: predators die at per capita rate m without food.
Equilibrium: set both rates to zero. From the prey equation, r = aP, so P∗ = r ÷ a. From the predator equation, baH = m, so H∗ = m ÷ (ba). Away from equilibrium, the model predicts endless cycles around these values.
Worked exampleLet's find the equilibrium and take one simulation step
Use r = 1.0 per year, a = 0.01, b = 0.1 and m = 0.5 per year. (a) Find the equilibrium. (b) Starting at H = 800 and P = 60, calculate H and P after one time step of Δt = 0.05 year, using ΔH = (rH − aHP)Δt and ΔP = (baHP − mP)Δt.
- (a) P∗ = r ÷ a = 1.0 ÷ 0.01 = 100 predators. H∗ = m ÷ (ba) = 0.5 ÷ (0.1 × 0.01) = 0.5 ÷ 0.001 = 500 prey.
- (b) Prey: rH = 1.0 × 800 = 800. aHP = 0.01 × 800 × 60 = 480. ΔH = (800 − 480) × 0.05 = +16, so H = 816.
- Predators: baHP = 0.1 × 480 = 48. mP = 0.5 × 60 = 30. ΔP = (48 − 30) × 0.05 = +0.9, so P = 60.9.
- Interpret: H is above H∗ = 500, so there is plenty of food and predators increase. P is below P∗ = 100, so predation is light and prey also increase. This matches the start of a cycle: both rise until predators pass 100, when prey start to fall.
(a) Equilibrium: H∗ = 500, P∗ = 100. (b) After one step: H = 816, P = 60.9.
Hands-on activityRun the predator-prey model in a spreadsheet
You need: a computer, tablet or phone with a free spreadsheet (Google Sheets, LibreOffice Calc or Excel), and about 30 minutes.
- In row 1, type headings: t, H, P. In cells F1 to F5 type the parameter values: r = 1.0 (F1), a = 0.01 (F2), b = 0.1 (F3), m = 0.5 (F4), Δt = 0.05 (F5).
- Row 2: t = 0, H = 800, P = 60.
- Row 3, type these formulas:
- A3: =A2+$F$5
- B3:
=B2+($F$1B2-$F$2B2C2)$F$5 - C3:
=C2+($F$3$F$2B2C2-$F$4C2)*$F$5
(Each ∗ is the ordinary multiplication key, Shift + 8 on most keyboards.)
- Copy row 3 down to row 402 (that is 400 steps, or 20 years).
- Make a line graph of H and P against t. (Tip: put P on a secondary axis or graph 5 × P so both are visible.) Make a second scatter graph of P against H (a phase plot).
- Record: the time of each prey peak and each predator peak, the maximum and minimum of each, and the cycle length.
- Experiment: change one parameter at a time (for example double a, or halve m). Predict first, then run it and explain the result.
What to look for: row 3 should show H = 816 and P = 60.9 (matching the example). Over 20 years you should see repeating cycles about 9.5 years long. Prey peak at about year 1.2 (about 1060) and again at about year 10.6 (about 1140); predators peak about 1.5 years after each prey peak (about 176 at year 2.8 and 187 at year 12.2). At year 10, H ≈ 1051 and P ≈ 73. The phase plot traces loops around (500, 100). The loops slowly grow because the simple step-by-step method adds a small error each step; using a smaller Δt (for example 0.01) reduces it.
Output of the spreadsheet model (r = 1.0, a = 0.01, b = 0.1, m = 0.5, starting at 800 prey and 60 predators). Predator numbers (shown × 5 to fit the scale) peak about one and a half years after prey numbers.The same run as a phase plot. Each loop is one cycle around the equilibrium (500 prey, 100 predators). The loops widen slowly because of the step-by-step calculation method.Model 2: a host-parasitoid model
Many insects have one generation per year, so a discrete model fits better. In 1935, A. J. Nicholson and V. A. Bailey modelled hosts (H) and parasitoids (P) that attack them:
Hₜ₊₁ = λ Hₜ e^(−aPₜ)
Pₜ₊₁ = c Hₜ (1 − e^(−aPₜ))
- e^(−aPₜ) is the fraction of hosts that escape being parasitized (it gets smaller as parasitoids increase).
- λ is the host's geometric growth rate (from Geometric and Exponential Growth Models).
- 1 − e^(−aPₜ) is the fraction parasitized; each parasitized host produces c new parasitoids next year.
Worked exampleLet's run two generations of a host-parasitoid model
Use λ = 2, a = 0.02, c = 1, with H₀ = 40 hosts and P₀ = 30 parasitoids. Find H and P for generations 1 and 2.
- Generation 1: escape fraction = e^(−0.02 × 30) = e^(−0.6) = 0.5488.
- H₁ = 2 × 40 × 0.5488 = 43.9; P₁ = 1 × 40 × (1 − 0.5488) = 40 × 0.4512 = 18.0.
- Generation 2: escape fraction = e^(−0.02 × 18.0) = e^(−0.36) = 0.6977.
- H₂ = 2 × 43.9 × 0.6977 = 61.3; P₂ = 43.9 × (1 − 0.6977) = 43.9 × 0.3023 = 13.3.
- Interpret: with few parasitoids, most hosts escape and the hosts grow; the parasitoids fall at first, but will rise again once hosts are plentiful.
Generation 1: H = 43.9, P = 18.0. Generation 2: H ≈ 61.3, P ≈ 13.3.
Comparing models with real experiments
Gause's predator-prey tubes (1930s). G. F. Gause put the predatory protist Didinium with its prey Paramecium in a simple test tube. Didinium ate every Paramecium, then starved: both went extinct. When he added sediment where some Paramecium could hide (a refuge), the prey survived but the predators died out. Only when he regularly added a few individuals of each (immigration) did he get repeated cycles.
Huffaker's mites (1958). Carl Huffaker studied a plant-eating mite and a predatory mite on oranges arranged in trays. In simple set-ups the predator wiped out its prey and then died. When he made the habitat complex (spreading the food over many oranges, adding barriers, and giving the prey sticks to "parachute" from on silk to new oranges), the two species persisted with cycles for months. Patchiness and dispersal let prey escape and recolonize.
Snowshoe hares and lynx. In the field, the roughly 10-year cycle has predator peaks lagging prey peaks, as the model predicts. But as you learned in Predation, Herbivory and Defence Mechanisms, hare food supply matters too, and several predators are involved.
Limitations of the models (A1.8, A1.10):
- Prey have no carrying capacity in the basic Lotka-Volterra model (they would grow forever without predators).
- Predators never get full (no handling time).
- Every individual is identical, there is no age structure, no refuges, no immigration, no chance events.
- The Nicholson-Bailey model is unstable: its oscillations grow until one population nearly disappears, which is not what many real host-parasitoid systems do. Real systems persist because of patchiness, refuges and density dependence in the host.
Models are still valuable: they reveal the basic mechanism (lagged feedback) and tell us which extra factors are needed to explain real data.
Writing up your investigation (A1.11)
Use the standard format: title, question, hypothesis, model and parameters, method (including the spreadsheet formulas), results (labelled graphs with units and a data table of peaks), analysis (cycle length, lag, effect of each parameter change), evaluation (limitations, sources of error such as the step size Δt), and a conclusion that answers the question and says whether the hypothesis was supported.
Just for funThe Lotka-Volterra model has no hares that get bored, no lynx that sleep in, and no biologists who forget to collect data. It is a perfect world, which is exactly why ecologists keep checking it against the real one.
Your turn
1In dH/dt = rH − aHP, what does the term aHP represent?
The rate at which prey are killed by predators; encounters are proportional to the product of prey and predator numbers.
2For r = 0.8, a = 0.02, b = 0.2 and m = 0.4, find the equilibrium.
P∗ = r ÷ a = 0.8 ÷ 0.02 = 40. H∗ = m ÷ (ba) = 0.4 ÷ (0.2 × 0.02) = 0.4 ÷ 0.004 = 100.
3Using r = 1.0, a = 0.01, b = 0.1, m = 0.5 and Δt = 0.05, find H and P one step after H = 500, P = 100.
ΔH = (500 − 0.01 × 500 × 100) × 0.05 = (500 − 500) × 0.05 = 0. ΔP = (0.1 × 500 − 50) × 0.05 = (50 − 50) × 0.05 = 0. H = 500, P = 100: this is the equilibrium, so nothing changes.
4Why do predator peaks lag behind prey peaks in the model?
Predators increase only while prey are above H∗; they keep increasing until prey fall below H∗, which happens after the prey peak, so predator numbers peak later.
5In Gause's simple tube, what happened to Didinium and Paramecium, and why?
Didinium ate all the Paramecium, then starved. With no refuge and no immigration, the prey had nowhere to escape, so both populations died out.
6What did Huffaker's mite experiments show about habitat complexity?
In a complex, patchy habitat, prey could disperse to new patches and escape predators for a while, so both species persisted with cycles; in simple habitats both went extinct.
7Give two limitations of the Lotka-Volterra model.
Any two: no carrying capacity for prey; predators never get full; no age structure, refuges or immigration; no random events; all individuals identical.
More practice: warm-up
1What are the two populations in a host-parasitoid model?
Hosts and the parasitoids that develop in them.
A graph of predator numbers against prey numbers; cycles appear as loops.
3What is the formula for the predator equilibrium P∗ in the Lotka-Volterra model?
A place or condition where some prey are safe from predators.
5About how long is the snowshoe hare-lynx cycle?
More practice: on your own
1In the spreadsheet model, predict what happens to the predator equilibrium P∗ if the prey growth rate r doubles. Explain.
P∗ = r ÷ a, so P∗ doubles (to 200). Faster-growing prey can support more predators at equilibrium. (Interestingly, H∗ = m ÷ (ba) does not change.)
2If predators become better hunters (a doubles from 0.01 to 0.02, with r = 1.0, b = 0.1, m = 0.5), find the new equilibrium and explain the result.
P∗ = 1.0 ÷ 0.02 = 50; H∗ = 0.5 ÷ (0.1 × 0.02) = 250. Better hunters keep the prey lower, and because there are fewer prey, fewer predators can be supported.
3Run one more host-parasitoid generation from H₂ = 61.3, P₂ = 13.3 (λ = 2, a = 0.02, c = 1).
Escape fraction = e^(−0.266) = 0.7664. H₃ = 2 × 61.3 × 0.7664 = 94.0. P₃ = 61.3 × (1 − 0.7664) = 61.3 × 0.2336 = 14.3.
4Why does a smaller Δt make the spreadsheet model more accurate?
Each step assumes the rates stay constant during Δt, but they really change continuously. A smaller Δt means smaller changes per step, so this approximation error is smaller and the loops stop widening artificially.
5Identify the independent, dependent and controlled variables if you change the attack rate a in the spreadsheet and record the cycle length.
Independent: the attack rate a. Dependent: the cycle length (and peak sizes). Controlled: r, b, m, Δt, the starting H and P, and the number of steps.
Challenge
1Add a carrying capacity to the prey in the spreadsheet: change the prey formula to ΔH = [rH(1 − H/K) − aHP]Δt with K = 2000. Predict and explain how the cycles change. (If you have a spreadsheet, test it.)
With a prey carrying capacity, prey growth slows at high H, so prey cannot overshoot as far. The cycles become damped: each oscillation is smaller, and the populations spiral in toward a stable equilibrium. (With K = 2000, the new equilibrium is still P∗ where r(1 − H∗/K) = aP∗, with H∗ = 500 from the predator equation: P∗ = 1.0 × (1 − 500/2000) ÷ 0.01 = 75.) This is closer to many real systems, which are more stable than the basic model.
2Explain, using the Nicholson-Bailey simulation (H: 40, 44, 61, 94, 141, 177, 124, 25, 5, 7; P: 30, 18, 13, 14, 23, 53, 115, 111, 22, 2), why the model is called unstable, and suggest two features of real systems that could stabilize it.
The oscillations grow each cycle: hosts rise to 177, then crash to 5, and parasitoids crash from 115 to about 2. With such low numbers, chance would likely wipe out one or both populations. Stabilizing features: refuges (some hosts in places parasitoids cannot reach), patchy habitat with dispersal (like Huffaker's oranges), density-dependent limits on the host (a carrying capacity), or parasitoids that interfere with each other at high density.
3Write a short evaluation (4 to 6 sentences) of how well the Lotka-Volterra model explains the snowshoe hare-lynx cycle, using evidence from this unit.
Sample: The model correctly predicts linked cycles with predator peaks lagging prey peaks, which matches the Hudson's Bay Company records and field counts. However, the model assumes prey grow without limit and that one predator controls them. Field experiments at Kluane Lake showed that food supply and several predators together drive the hare cycle, so the model leaves out important factors. The model also ignores weather, disease and the movement of animals, and real cycle lengths and peak sizes vary more than the model's regular loops. So the model is a useful starting explanation of the mechanism (lagged feedback) but not a complete one.
Quiz
1. In the Lotka-Volterra model, the prey equilibrium H∗ equals:
2. With r = 0.6 and a = 0.02, the predator equilibrium P∗ is:
3. In a phase plot of predators against prey, linked cycles appear as:
4. In Gause's simple test tube with Didinium and Paramecium:
5. In the Nicholson-Bailey model, e^(−aPₜ) represents:
6. Which change would make a spreadsheet predator-prey simulation more accurate?
Big idea (F2.3, F2.2, A1): linked populations can be investigated with models. The Lotka-Volterra equations produce predator-prey cycles with predator peaks lagging prey peaks; the Nicholson-Bailey model describes hosts and parasitoids generation by generation. Laboratory studies (Gause, Huffaker) and field data (snowshoe hare and lynx) show that refuges, patchy habitats, food supply and immigration help real populations persist.
How to help: sit with your teen while they set up the spreadsheet; the first formula is the hardest. Check that row 3 shows H = 816 and P = 60.9 before they copy it down. Then ask them to predict the effect of a parameter change before running it, which is the heart of a good investigation.
Common difficulties:
- Spreadsheet formulas: forgetting the $ signs that lock the parameter cells.
- Mixing up which equilibrium formula goes with which population.
- Treating the model as reality: the evaluation section matters as much as the graph.
Sources: Lotka (1925) and Volterra (1926); Nicholson and Bailey (1935); Gause, The Struggle for Existence (1934); Huffaker (1958), Hilgardia. Model output in this lesson was calculated with the parameters given.