What if rabbits had more babies?

A mother rabbit can have five or six babies at once.

She can have babies many times a year. What if rabbits had even more babies? Would there be rabbits everywhere?

A wild European rabbit is pregnant for only about a month and usually has five or six babies in a litter. A mother can be ready to have another litter soon after giving birth, so she can have several litters in a year.

Now imagine every rabbit in a meadow had babies twice as fast. It seems obvious that there would be twice as many rabbits. But the meadow also has foxes, which eat rabbits. This lab asks what really happens when you change only the rabbits' birth rate and leave everything else alone.

European rabbits breed quickly: gestation is about 30 days, the average litter is 5 to 6 young, and because females can conceive again just after giving birth they may have several litters a year, according to the Animal Diversity Web.

This lab asks a population question about that speed. In a meadow shared with a predator, does a faster prey birth rate produce more prey? The Lotka–Volterra model gives a clear and slightly counterintuitive answer, which the lab then sets against what ecologists have measured in real rabbit and predator populations.

Olly's meadow has rabbits and foxes.

First the rabbits go up. Then the foxes go up. Then both go down. Then it starts again.

Our model meadow starts with 40 rabbits and 9 foxes. The rabbit birth rate is 1, which means each rabbit adds about one more rabbit a year if nothing eats it.

Press play. The rabbits climb to about 345, then the well-fed foxes climb to about 172. So many rabbits get eaten that they fall to about 12, and then the hungry foxes fall too. The rise and fall repeats about every seven and a half years. The readouts also show the balance point, where both numbers could stay level: 100 rabbits and 50 foxes.

With α = 1, β = 0.02, δ = 0.01 and γ = 1, starting from 40 rabbits and 9 foxes, the model cycles indefinitely: 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 'time for one rise and fall' readout gives the small-swing period, 2π/√(αγ) = 6.3 years; this large cycle takes longer because the populations swing far from the balance point.

Now Olly makes the rabbits have babies twice as fast.

There are big rabbit booms. But then lots more foxes come. In the end, the foxes gain the most!

Set the rabbit birth rate to 2 and run the meadow again. The rabbits boom much higher, to about 612. But the foxes boom too, to about 406, and they arrive faster, only about half a year after the rabbit peak. Then the rabbits crash all the way down to about one rabbit before recovering.

Now look at the balance point. Rabbits at balance is still 100. Foxes at balance has doubled, from 50 to 100. The extra babies did not give the meadow more rabbits in the long run. They gave it more foxes. Try 1.5 as well: the balance becomes 100 rabbits and 75 foxes.

At α = 2 the cycle is faster and more violent: rabbit peaks of about 612 every 6.4 years, troughs near 1.4, and fox peaks of about 406 about 0.5 years after each rabbit peak. The small-swing period falls to 2π/√2 ≈ 4.4 years.

The equilibrium moves only in y: x* = γ/δ stays at 100, while y* = α/β doubles to 100. At α = 1.5 it is 75. Switch on the food limit (K = 400) and the cycles damp out: the meadow settles at 100 rabbits with 38 foxes when α = 1, and 100 rabbits with 75 foxes when α = 2, since y* = (α/β)(1 − x*/K).

Foxes need rabbits to eat. More rabbit babies means more food for foxes.

More foxes catch more rabbits. So the rabbits do not stay high.

Think about what keeps each animal steady. Foxes stay level only when there are just enough rabbits to feed them. In this model that is 100 rabbits, and the rabbits' birth rate does not change how much food a fox needs.

Rabbits stay level only when foxes catch them as fast as they are born. If rabbits are born twice as fast, it takes twice as many foxes to keep up. So the balance moves to more foxes, not more rabbits. The model is very simple: one kind of prey, one predator, no weather or disease, and foxes that never get full. Its rates were chosen to show the idea clearly, not measured in a real meadow.

Setting dy/dt = 0 gives x* = γ/δ: the prey density at which predators exactly replace themselves, which contains no α. Setting dx/dt = 0 gives βy* = α: predation must remove prey as fast as they are born, so y* rises in proportion to α. Faster prey reproduction is converted into predator biomass.

This follows from the model's assumptions: a closed meadow with one prey and one predator, a linear functional response (a fox's appetite has no ceiling), smooth population numbers, and no weather, disease or other food. The deep troughs at α = 2 expose another limit. The model lets the rabbits fall to about one animal and recover, which a real population that small could not be counted on to do.

Long ago, people took rabbits to Australia. The rabbits had lots of babies and spread far.

Where there are many rabbits, there can be many foxes and cats too.

Real rabbits have shown how fast they can multiply. After people brought them to Australia, wild rabbits numbered an estimated 20 million by 1900, competing with sheep and cattle for food and damaging plants.

Scientists in Australia now see the model's lesson in real life. In southern Australia, plenty of rabbits help feed large numbers of foxes and feral cats. In one experiment at the Arid Recovery reserve, researchers removed about 80 percent of the rabbits from a 37-square-kilometer fenced area, and cat activity fell by 40 percent within about a month. Fewer rabbits meant fewer predators, just as more rabbits feed more.

The Animal Diversity Web notes that European rabbits in Australia numbered an estimated 20 million by 1900, competing with livestock and devastating plant communities, and that myxomatosis later cut some populations by more than 90 percent without eradicating them.

Ecologists in Australia report that abundant rabbits support high densities of feral cats and foxes, letting them breed to their maximum capacity. In a Threatened Species Recovery Hub experiment led by Hugh McGregor (University of Tasmania), removing around 80 percent of rabbits from a 37 km² enclosure cut cat activity and the survival of collared cats by 40 percent, and the cats turned to carrion, reptiles and insects. Real predators switch foods in this way, which the one-prey model cannot.