What if animal populations grew beyond available food?

A field of grass can feed some deer. But not too many!

What if more and more deer came? What would happen when there was not enough to eat?

Every patch of land grows a certain amount of grass, leaves and twigs each year. That amount can feed only so many deer. Scientists call this number the carrying capacity.

A herd with lots of food grows quickly. So what happens when it gets close to the carrying capacity? Does it stop neatly, or can it grow too big? In this lab you can find out by changing how quickly deer react to crowding.

A population with abundant resources grows almost exponentially. As it approaches the carrying capacity K, the number its environment can sustain, food per animal falls and growth should slow.

The catch is timing. Births today reflect conditions when the parents were feeding, months or years earlier. This lab asks what that delay does: can a population sail past K, and what happens after it does?

Olly's land grows food for 500 deer.

The herd starts small. It grows and grows. Then it slows down and stops at 500. Everyone has enough to eat!

In the model, the land grows 50 tonnes of plant food a year, and each deer needs a tenth of a tonne, so the food can feed 500 deer. The herd starts with 20.

When the deer react to crowding straight away, the herd grows fast at first, reaching about 350 after 5 years. Then it slows and creeps up to 500, the carrying capacity, without ever passing it. Every deer gets all the food it needs.

The carrying capacity is K = food ÷ need = 50 ÷ 0.1 = 500 deer. With no delay (τ = 0) the model is the logistic equation dN/dt = rN(1 − N/K), with r = 0.8 per year.

From 20 deer, the solution N(t) = K / (1 + ((K − N₀)/N₀)·e^(−rt)) gives about 347 deer at 5 years, 496 at 10 and 500 by 15. The curve is an S shape that approaches K from below and never overshoots.

Now the deer notice the crowding late. The herd keeps growing, even when food is running short.

Soon there are too many deer for the plants. Some deer go hungry, and fewer babies are born. The herd gets smaller, then grows again.

Now give the deer a two-year breeding delay: the herd reacts to how crowded it was two years ago. It keeps growing long after the food runs short.

In the model, the herd climbs to about 1,020 deer after six and a half years, about twice what the food can feed, so each deer gets only about half the food it needs. Then the herd falls to about 160 and climbs again. It keeps booming and shrinking, over and over.

With τ = 2 years, rτ = 1.6, just above π/2 ≈ 1.57, so the model predicts sustained oscillations. The herd peaks at about 1,023 deer after 6.4 years, roughly 2K, when food share is about 49 percent, then crashes to about 161 at 10.1 years. Later peaks are 919 and 866 deer, and the oscillation is still running after 40 years.

A one-year delay (rτ = 0.8, between 1/e and π/2) gives damped waves instead: a peak of about 564 deer after 6.1 years, a dip to 481, and then settling at 500.

Baby deer start growing before they are born. Their mothers ate when food was plenty.

So more babies come even after food gets short. That is why the herd grows too big.

In real herds, this year's babies come from mothers that fed months earlier, when food was more plentiful. So the herd keeps growing on yesterday's good news. By the time births slow down, there are already too many deer.

The model is simple. It has no predators, no disease and no seasons, and the plants always grow back to the same amount however hard the deer eat them. Its growth rate and food numbers were chosen to show the pattern clearly, not measured from a real herd.

The delay turns crowding into delayed negative feedback. Linearizing near K shows the behavior depends only on rτ: below 1/e the approach is smooth, between 1/e and π/2 it overshoots in damped waves, and above π/2 it settles into a repeating boom and crash.

Key assumptions: the plants regrow fully every year, so overgrazing never lowers K; one lag τ stands in for ages, seasons and breeding times; there are no predators, disease, hunting or migration; and the population is a smooth number that can fall to fractions of an animal and recover, which a real herd cannot. The rates are illustrative, not measured.

Long ago, people brought 29 reindeer to an island in Alaska. There was lots of food, so the herd grew very big.

The reindeer ate up their winter food, and the herd became small again. Parks today keep watch so plants and animals stay in balance.

In 1944, 29 reindeer were brought to St. Matthew Island in Alaska. With rich food, the herd grew to about 6,000 by 1963. The reindeer ate up the lichens they needed in winter, and after a winter of very deep snow only 42 were left by 1966.

White-tailed deer in Rock Creek Park in Washington, D.C., had no natural predators and flourished, reaching nearly 100 per square mile in 2003. Since the park began lowering deer numbers in 2013, its count of tree seedlings has more than doubled.

Klein (1968) described reindeer introduced to St. Matthew Island in 1944 rising from 29 to 6,000 by summer 1963. Lichens had been eliminated from their winter diet and body weights had dropped; in late winter 1963–64, with extreme snow, almost the whole population starved, leaving 42 in 1966. He concluded food supply, interacting with climate, regulated the herd. Unlike our model, where the food always regrows, the lichens themselves were eaten out, and all but one of the 1966 survivors were females: there was no rebound like the model's.

In Rock Creek Park, deer without natural predators reached nearly 100 per square mile in 2003; NPS research shows vegetation damage above 20 per square mile. Since deer reduction began in 2013, tree seedling density has more than doubled.