What if there were no predators?

In a meadow, rabbits nibble grass. Foxes catch and eat some rabbits.

What if there were no foxes at all? Would there be more rabbits, or the same?

A predator is an animal that hunts other animals for food. The animals it hunts are its prey. Foxes are predators, and rabbits are one of their prey.

It might seem that a meadow with no foxes would simply be a happier place for rabbits. But what would really happen to the number of rabbits, year after year? In this lab you can take every fox out of a model meadow and find out.

Predators and prey are linked: the prey's numbers set how much food the predators have, and the predators' numbers set how many prey are eaten. Ecologists describe this with coupled equations in which each population's growth depends on the other.

This lab asks what happens when that link is cut. If the predator term disappears, what limits the prey population, and how fast does it grow until something does?

With foxes in the meadow, the rabbits go up and down.

Lots of rabbits means lots of food for foxes. Then there are more foxes, and fewer rabbits. Round and round it goes!

Our model meadow starts with 40 rabbits and 9 foxes. The rabbits have babies quickly, so at first their numbers shoot up. With so much food, more fox cubs grow up, and the foxes catch more and more rabbits.

Soon the rabbits become scarce, the hungry foxes decline, and the rabbits recover. In the model the rabbits reach about 345, fall as low as 12, and climb again, peaking about every seven and a half years. The foxes peak just under a year after the rabbits do.

The model uses dx/dt = αx − βxy for rabbits and dy/dt = δxy − γy for foxes, with α = 1 per year, β = 0.02, δ = 0.01 and γ = 1 per year. From 40 rabbits and 9 foxes, the rabbits peak at about 345 after 2.8 years, the foxes at about 172 after 3.7 years, and the rabbits bottom out near 12.

Without a food limit these equations keep a quantity V = δx − γ ln x + βy − α ln y constant, so the same loop repeats every 7.6 years. Small swings around the balance point (100 rabbits, 50 foxes) would repeat about every 2π/√(αγ) ≈ 6.3 years; this large swing takes longer.

In our pretend meadow, Olly takes all the foxes away. They go to live somewhere else.

Now no one is catching the rabbits. There are more and more and more of them!

Switch the foxes off in the model. With nothing catching them, every rabbit family keeps growing, and the babies grow up and have babies too. The more rabbits there are, the faster their number climbs.

After 5 years there are about 5,900 rabbits. After 10 years there are about 880,000, and the model stops just after that, at a million rabbits. Turn on the food limit and the rabbits stop at 400 instead, because that is all the grass can feed.

With y = 0 the rabbit equation becomes dx/dt = αx, so x = x₀·e^(αt): exponential growth, doubling about every 0.69 years at α = 1. From 40 rabbits the model passes about 5,940 after 5 years and 881,000 after 10, and the run stops at a million after about 10.1 years.

Switching on the carrying capacity K = 400 gives logistic growth, dx/dt = αx(1 − x/K): about 377 rabbits after 5 years and 400, leveling off, by year 10. Something must always limit a population; without predators, it is food and space.

Rabbits have lots of babies. Foxes help keep their numbers from growing too big.

With no foxes, only food can stop the rabbits. When the grass runs out, they cannot keep growing.

Each year, every rabbit in the model adds new rabbits. More rabbits make even more babies, so growth speeds up. Foxes normally take away a share of the rabbits, and when rabbits are plentiful the well-fed foxes multiply, so that share grows. That is what turns the rabbits' climb into a rise and fall.

The model keeps things simple: one meadow, one kind of prey, one kind of predator, no weather and no disease. Its rates were chosen to show the pattern clearly, not measured from real rabbits and foxes.

The predation term βxy grows with both populations, so it acts as negative feedback on the prey, delayed by the time foxes take to multiply. Remove it and the only feedback left is crowding, if the model includes it.

The model's assumptions matter: populations are smooth numbers, so they can fall to a fraction of an animal and recover; a fox's appetite has no upper limit; animals never move in or out; and the plain equations with no food limit let rabbits grow forever, which no real meadow allows. The rates are illustrative values, not measurements.

Long ago, people took the wolves away from Yellowstone, a big park in the United States.

In 1995, people brought wolves back. Scientists watch carefully to learn how the park changes.

In the early 1900s, park managers in Yellowstone killed predators, including wolves, and the last wolf pack there was killed in 1926. In 1995 and 1996, 31 gray wolves from western Canada were brought back. Elk make up about 90 percent of the wolves' winter food.

The National Park Service says wolves have lowered how many elk calves survive and changed where elk feed, but bears, cougars, hunting and weather matter too. In Alaska, snowshoe hares rise and fall over 8 to 11 years, and the lynx that hunt them follow one or two years behind.

Yellowstone removed wolves through predator control in the late 1800s and early 1900s; the last pack was killed in 1926. In 1995–1996, 31 wolves from western Canada were relocated there. The NPS reports that wolves contributed to lower survival of elk calves and older females and changed elk habitat use, but grizzly and black bears, cougars, coyotes, human hunting and winter severity also act, and how much wolves caused the elk decline, and any related willow and aspen recovery, is still being researched.

In Alaska, the Department of Fish and Game describes an 8–11 year snowshoe hare cycle that appears to come from hares interacting with both their food and their predators, with lynx numbers lagging one or two years behind. Real cycles are messier than a two-species model.