Predator–prey population graphs and phase planes
Predator–Prey Cycles graphs a predator and its prey rising and falling out of step, worked out from the Lotka–Volterra model. Pick a pair like the snowshoe hare and Canada lynx, or type your own names, and set where both populations start, what they average and how long one cycle takes. The graph shows the predators peaking after the prey, cycle after cycle, either as both populations over time or as a phase plane, predators against prey, where one cycle is one trip round the loop.
Teachers use it for ecology questions on tests, worksheets and slides. Mark the peaks, the lag from a prey peak to the predator peak after it, and the period, labeled with their lengths or with blank lines for students to fill in. Switch to census counts with some scatter so the graph reads like field data, leave one or both populations off to give students axes to sketch on, and print in black and white, where the predators’ line is dashed.
What you can set
- Pair: snowshoe hare and Canada lynx, rabbit and wolf, rabbit and red fox, moose and wolf (Isle Royale), sea lion and orca, snake and eagle, mouse and owl, aphid and ladybug, or names of your own
- The cycle from populations (both at the start, their averages and the cycle length) or from the model’s four rates, α, β, γ and δ
- How much time is shown, in years, months, weeks or days
- The populations over time, with both drawn, one of them or neither (blank axes), or a phase plane of predators against prey
- Smooth curves or census counts, taken every so often and off by about 0 to 50%, joined with lines or not
- Peaks, the lag and the period marked on the graph over time, labeled with their length, their name only, or a blank line for students
- One y-axis for both or the predators on the right, color or black and white, and the chart and axis titles, axis ranges, gridlines and label size
Copying, printing and sharing
Copy the figure straight into a test, worksheet or slide, or download it as a PNG or SVG. Printing the page prints
just the figure. Share link copies the page’s address with your settings in it, so anyone who opens it sees this
exact figure, and presets save settings you use often in your browser. It’s free, with no sign-up.
Frequently asked questions
Why does the predator population peak after the prey?
Predators need food to grow in number. While prey are plentiful, the predators eat well and their numbers rise, but that takes time, so they keep rising after the prey have peaked. Then the many predators eat the prey down, food runs short, the predators fall, and the prey recover. That delay between a prey peak and the predator peak after it is the lag.
What is the Lotka–Volterra predator–prey model?
A pair of equations for how a prey population x and a predator population y change: dx/dt = αx − βxy and dy/dt = δxy − γy. Prey grow at rate α and are eaten in proportion to how often the two meet (β); predators grow from the prey they eat (δ) and die at rate γ. The populations cycle forever around a balance point, which is also each one’s average over a cycle.
Why do snowshoe hare and lynx populations cycle?
Fur-trade records from Canada show hare and lynx numbers rising and falling about every 10 years, the lynx a year or two behind the hares, the classic example of a predator–prey cycle. Real hare cycles also depend on the hares’ food and other predators, so the Lotka–Volterra model is a simplified picture, and this generator draws the model rather than the historical counts.
What does a predator–prey phase plane show?
It graphs the predators against the prey instead of each against time. Because the populations repeat, the points trace a closed loop, and one trip round it is one full cycle. With prey on the x-axis the loop goes round counterclockwise: prey rise while predators are few, predators rise, prey fall while predators are many, then predators fall. The generator draws arrows on the loop the way it goes.
How do I make a worksheet where students find the lag and period?
Under Marked, turn on Lag and Period and set the labels to Blank for students. Each gets a two-headed arrow between peaks and a blank line to fill in from the time axis. The settings panel shows the cycle length and lag the model works out, for your answer key.
Can I graph data that looks like real field counts?
Yes. Under Data, choose Census counts, set how often the populations are counted and how far off each count is, from 0 to 50%. Each set of counts is the same every time for the same figure, and New counts draws another. The counts can be joined with lines, solid for the prey and dashed for the predators.
Can I use my own animals and numbers?
Yes. Type any names for the prey and predators, then the populations at the start, their averages and the cycle length, and the model works out the rest. Or switch to Model rates to type α, β, γ and δ yourself. The further the start is from the averages, the bigger the swings.
Biology Figures Predator–Prey Cycles