What if foxes became better hunters?

Red foxes are clever hunters. They listen for mice, then pounce!

What if foxes got even better at catching rabbits? Would that be good for the foxes?

A predator is an animal that hunts other animals, its prey. Red foxes hunt by smell, sight and sound: they can wait for the sound of a mouse moving in grass or snow, then pounce.

Now imagine foxes became much better hunters, so each fox caught twice as many of the rabbits around it. More food sounds like great news for foxes. This lab asks what really happens, over many years, to the rabbits and to the foxes themselves.

In predator–prey ecology, a predator's capture efficiency sets how fast prey are removed for a given number of predators and prey. Intuition says better hunters should mean more predators.

This lab tests that intuition in the Lotka–Volterra model by doubling the catch rate β from 0.02 to 0.04 per fox per year, with every other rate fixed. It then asks which of the model's results are robust lessons about real predators and which come from the model's simplifications.

In Olly's meadow, there are rabbits and foxes.

When there are lots of rabbits, the foxes have lots of food. Then there are more foxes. Then the rabbits go down, and so on.

Our meadow starts with 40 rabbits and 9 foxes, and each fox has a catch rate of 0.02. Press play: the rabbits multiply first, climbing to about 345. With so much food the foxes multiply too, peaking at about 172. Then so many rabbits are eaten that they drop to about 12, and the hungry foxes drop as well.

The whole rise and fall repeats about every seven and a half years. The readouts also show the balance point, where the numbers could stay level together: 100 rabbits and 50 foxes.

With α = 1, β = 0.02, δ = 0.01, γ = 1, 40 rabbits and 9 foxes, the model cycles for ever: rabbit peaks of about 345 every 7.6 years, rabbit troughs of about 12, and fox peaks of about 172 roughly 0.9 years after each rabbit peak.

The equilibrium is x* = γ/δ = 100 rabbits and y* = α/β = 50 foxes. The small-swing period 2π/√(αγ) is 6.3 years; this large cycle is longer, 7.6 years, because the populations swing far from the balance point.

Now Olly makes the foxes better hunters.

The rabbits don't get as many. And there end up fewer foxes, not more!

Double the catch rate to 0.04 and watch. The rabbits now peak at about 269 instead of 345, because more of them are caught early. The foxes peak at only about 67 instead of 172.

The balance point changes too. Rabbits at balance stays at 100, but foxes at balance falls from 50 to 25. Better hunters do not need as many foxes to keep the rabbits in check, so the meadow ends up supporting fewer of them. The rabbits also swing less: their lowest point is about 23 instead of 12.

At β = 0.04 the cycle tightens: rabbit peaks fall to about 269, troughs rise to about 23, fox peaks fall to about 67, and the period shortens to 7.0 years.

The equilibrium shows why. x* = γ/δ does not contain β, so it stays at 100 rabbits: that is the rabbit density at which foxes exactly replace themselves. y* = α/β halves to 25, because only half as many foxes are needed to remove rabbits as fast as they are born. Try 0.05: foxes at balance drops to 20. With the food limit on (K = 400) the cycles fade and the meadow settles near 100 rabbits with 38 foxes at β = 0.02, or 19 foxes at β = 0.04.

Foxes need rabbits to eat. If foxes catch rabbits too fast, there are not many rabbits left.

Then there is not enough food for lots of foxes.

Think about what keeps the numbers steady. Foxes stay level only when there are just enough rabbits to feed them; in this model that is 100 rabbits, whatever the catch rate. Rabbits stay level only when foxes catch them exactly as fast as they are born. Better hunters can do that job with fewer foxes, so the steady number of foxes drops.

The model is very simple. It has one kind of prey and one predator, and its foxes never get full. Its rates are chosen to show the behavior clearly; they are not measured from a real meadow.

Setting dy/dt = 0 gives x* = γ/δ: the prey density that lets predators break even, independent of β. Setting dx/dt = 0 gives βy* = α (or α(1 − x*/K) with the limit): predation must balance prey growth, so y* is inversely proportional to β. Improved hunting therefore lowers predator abundance at equilibrium.

The model's assumptions shape this result: a closed meadow with one prey and one predator, a linear functional response (no handling time, so a fox's appetite has no ceiling), smooth population numbers, and no weather, disease or other food. Without the food limit, its cycles are neutral and never damp.

Red foxes eat mice, voles and hares, and berries too.

In the far north, lynx and hares go up and down about every ten years. Some scientists think it is not only about hunting.

Real foxes are not stuck with one food. The National Park Service says red foxes in Alaska eat mostly voles, mice, lemmings, ground squirrels and hares, and berries from July.

The most famous real cycle is the Canada lynx and the snowshoe hare. Trapping records from North American forests, nearly 200 years old, show their numbers rising and falling about every 10 years, with the lynx 1 to 2 years behind. But some researchers think crowding among the hares may help drive the cycle, not only hunting. And prey have defenses: some blend in with their background, and some play dead or travel in large groups, which lowers the risk of being eaten.

Real predators may not depend on one prey: NPS lists voles, mice, lemmings, arctic ground squirrels and hares as the bulk of red fox diet in Alaska for most of the year, with blueberries and crowberries from July. The model's foxes, by contrast, have rabbits and nothing else.

The lynx and snowshoe hare cycle, seen in nearly 200-year-old trapping data from North American forests, lasts about 10 years with the lynx lagging 1–2 years (OpenStax). Some researchers question whether predation alone drives it: one possibility is density-dependent effects in the hares, such as lower fecundity under crowding. Prey also have defenses, from camouflage and mimicry to playing dead and traveling in large groups.