Predators and prey: rabbits and foxes

Trapping records from North American forests show snowshoe hares and lynx rising and falling in a cycle of about 10 years, with the lynx 1 to 2 years behind.

A cycle in the fur records

The best-known example of predators and prey rising and falling together comes from trapping records kept in the forests of North America for nearly 200 years. OpenStax describes the pattern. When snowshoe hares are plentiful, lynx have plenty to eat and their numbers rise. Once there are many lynx, they kill so many hares that the hares decline, and soon afterwards the lynx decline too, for lack of food. With few lynx about, the hares recover and the cycle starts again. One full cycle lasts about 10 years, and the lynx peak comes 1 to 2 years after the hare peak.

OpenStax adds a caution. Some researchers think predation alone does not explain the cycle; crowding may lower the hares' breeding when they become too dense, and the lynx may simply follow their main food. Real cycles have several causes. A model is useful because it lets us see what predation alone would do.

The Lotka-Volterra model

The model has two rules for change each year. Rabbits are born at a rate a, and they are eaten at a rate that grows with the number of meetings between rabbits and foxes, b × rabbits × foxes. Foxes die at a rate g, and they gain new foxes from the rabbits they eat, d × rabbits × foxes. In the lab, a = 0.8, b = 0.08, g = 0.5 and d = 0.0125, with 60 rabbits and 6 foxes per square kilometre at the start.

There is a balance point where neither number changes. Foxes stop changing when rabbits equal g ÷ d = 0.5 ÷ 0.0125 = 40. Rabbits stop changing when foxes equal a ÷ b = 0.8 ÷ 0.08 = 10. Start anywhere else and the two numbers circle around this point forever. In the lab the rabbits peak at about 77 in year 1.3, the foxes peak at about 17 in year 3.2, and the pattern repeats every 10.3 years. The fox peak always follows the rabbit peak, by about 1.9 years here, just as the lynx follow the hares.

A surprising result hides in those formulas. Raise the rabbit birth rate a, and the balance number of rabbits stays at 40, while the balance number of foxes goes up. Making life easier for the prey ends up feeding more predators. Try it with the slider: at a = 1.6 the fox balance becomes 20.

Adding a food limit

The plain model assumes the rabbits never run short of grass. Choose Grass limited and the rabbits must now eat grass that regrows at its own pace. The swings shrink each cycle and the three numbers spiral in towards a steady balance of 40 rabbits, about 9.2 foxes and about 46.7 units of grass. Now choose the event that removes all foxes in year 20. With unlimited grass the rabbits grow without end, which no real population can do. With limited grass they boom, eat the grass down, and then settle near what the grass can support.