Predator-Prey Dynamics Simulation

About this experiment

Fur traders in Canada noticed that the numbers of snowshoe hares and the lynx that hunt them rise and fall in cycles about ten years long. A simple model explains why: lots of hares feed lots of lynx, the lynx eat too many hares, the hungry lynx then die off, and the hares recover. Change the birth, hunting and death rates, watch the meadow fill and empty, and follow the loop that the two populations draw on the phase plot.

Learning goals

Describe how predator and prey populations rise and fall in linked cycles
Explain why the predator peak comes after the prey peak
Interpret a phase plot of predators against prey
Find the balance point where both populations stay constant
Explain how limited food for the prey damps the cycles

How to use it

1Press Play with the default settings. Which population peaks first, and how long after does the other one peak?
2Read the measured cycle length. Compare it with the small-cycle prediction.
3Set the starting numbers close to the balance point: 60 hares and 50 lynx. What happens to the cycles?
4Increase the hunting rate b. How does the balance point change? Which number moves?
5Increase the lynx death rate d. Do the hares benefit at the balance point, or the lynx?
6Switch on Limited food for hares. What happens to the loop on the phase plot? Try a small carrying capacity of 100.

The model

\frac{d H}{d t} = a H - b H L

Hares are born at rate a and eaten when they meet lynx (H hares, L lynx)

\frac{d L}{d t} = c b H L - d L

Lynx gain c new lynx for each hare eaten and die at rate d

H^{*} = \frac{d}{c b}, L^{*} = \frac{a}{b}

Balance point of the classic model, where both rates of change are zero

T \approx \frac{2 π}{a d}

Period of small cycles close to the balance point

\frac{d H}{d t} = a H (1 - \frac{H}{K}) - b H L

With limited food, the hares grow logistically towards the carrying capacity K

Assumptions

Populations are treated as continuous numbers that change smoothly, not as individual animals.
Lynx eat only hares, and the number of hares eaten is proportional to how often hares and lynx meet, b times H times L.
Without limited food, hares would grow without any limit if there were no lynx.
No seasons, migration, disease or random events. The equations are solved with a fourth-order Runge-Kutta method, so the result is the same every time.
The meadow shows one icon for several animals so that large populations still fit on the screen.

Try this in class

Q1Why does the lynx peak always come a little after the hare peak?
Q2Hunters kill some of the lynx each year, which raises d. Look at the balance point: does the average number of lynx go down? What changes instead?
Q3Why do the cycles in the classic model never die away, while they do with limited food?
Q4What would happen to the lynx if a disease killed almost all the hares?
Q5The real hare and lynx cycle is about 10 years long. Which rates would you change to make the model match it?

Frequently asked questions

Is the hare and lynx cycle really caused by lynx alone?

Not entirely. Field studies show that the food supply of the hares and other predators also matter. The model captures the main feedback loop between predator and prey, which is why it is used in ecology lessons, but real cycles have several causes.

Why does the phase plot make a closed loop?

In the classic model a certain combination of H and L stays exactly constant over time, so the system returns to the same point after each cycle. Different starting numbers give different loops around the balance point.

Why do bigger cycles last longer than the prediction?

The formula 2π/√(ad) is exact only for very small cycles. Far from the balance point the populations spend a long time very low before they recover, which stretches each cycle.