Predators and prey do not settle at comfortable numbers. They chase each other round in circles, for ever, and the size of the circle is set by four numbers you can turn below. Then follow the energy upwards and see why the chain runs out after four or five links — and why every large predator on Earth is rare.
Plot lynx against hares instead of against time and the wandering lines above turn into a single closed loop that the system goes round for ever, anticlockwise. Hares rise on the right, lynx follow them up, hares crash on the left, lynx starve back down. Nothing damps it: the loop you start on is the loop you stay on, which is why this model never settles.
The two dashed lines cross at the balance point. Sit exactly on that crossing and nothing changes; move one animal off it and you are on a loop again. The loop grows with distance from the crossing, so a mild disturbance gives mild cycles and a violent one gives crashes. Tick hares can run out of plants and the loop stops being closed — it spirals inwards to a resting point, because now something is pushing back.
The Hudson's Bay Company bought pelts across northern Canada and wrote down how many, year by year, from the 1840s onward. When Charles Elton and Mary Nicholson plotted the snowshoe hare and Canada lynx figures in 1942 they found exactly the shape above: two ragged waves, the lynx peak trailing the hare peak by a year or two, repeating with an average period of about 9.6 years. It is the most-reproduced graph in ecology, and the default settings on this page are tuned to match it — a hare birth rate of 0.60 and a lynx death rate of 0.66 give a modelled cycle of 10.0 years.
The records count pelts, not animals. They also count trapper effort, fur prices and how far the trading posts reached, so a spike can be a good market rather than a good year for lynx.
The mechanism is not two species either. When Charles Krebs and colleagues fenced and fed one-square-kilometre blocks of Yukon forest through a whole cycle, extra food roughly doubled hare density and excluding predators roughly doubled it — but doing both together raised it about elevenfold. Food and predation multiply rather than add, and chronically frightened hares breed less even when nothing catches them.
So the two-species model gets the shape and the timing right for the wrong reasons. That is still useful: it tells you cycles need no external driver, no bad winters, no sunspots. Coupling alone is enough.
Only about a tenth of the energy in one trophic level ever reaches the next. The rest is spent staying alive, or lost as heat, or walks away in parts nobody eats. Set how much energy the plants capture and watch what is left by the top.
The energy figures are a real calculation from the two numbers above. The pyramid is drawn with each block's width set by the cube root of its energy, or the top level would be a line too thin to see — tick the box to find out how thin.
In 1957 Howard Odum spent four years measuring every energy flow in Silver Springs, Florida — one of the first complete energy budgets of a whole ecosystem. His numbers, in kilocalories per square metre per year:
Step by step that is 16.7%, then 4.5%, then 9.0%. The ten per cent rule is an average of numbers that scatter widely — but the direction never changes, and by the top carnivores 0.03% of the plants' energy is left.
Divide by ten often enough and there is nothing left to divide. A predator that hunts lions would need roughly ten lions' worth of energy a year for each of itself, spread over a range ten times larger, and there is not enough of anything at that height to make a living from.
The longest food chains are marine — six or seven links in some open-ocean systems — partly because transfer between plankton levels is more efficient than between a cow and a field.
Energy is the classic explanation and it is not the only one. Chain length also tracks how big the ecosystem is and how often it gets disturbed: small ponds and unstable habitats have short chains regardless of how productive they are. The energy where all this starts is the subject of Lab 16, Photosynthesis.
This is a Californian giant-kelp forest, simplified to fifteen species. An arrow runs from something to whatever eats it. Click any species to read about it, then remove it and watch the consequences spread outwards — some things collapse, and some things you never suspected explode.
Removing a species runs a signed cascade over the web: lose your food and you fall, lose your predator and you rise, and each effect passes on to neighbours until it fades out. It tells you which way each species moves, not by how much — a real answer needs a population model for every species and years of counting.
Sea otters have no blubber. They stay warm by eating a quarter of their body weight a day, and a great deal of that is sea urchins. The maritime fur trade cut the North Pacific population from perhaps 150,000–300,000 animals to fewer than 2,000 by 1911, scattered in thirteen remnant colonies.
Urchins, unhunted, grazed the kelp holdfasts through. Whole forests became "urchin barrens": bare rock with a carpet of spines and almost nothing else. Where otters were protected and came back, the kelp came back with them, and the difference between an otter coast and an otterless coast one headland away is visible from a boat.
It happened again for a different reason from 2013, when a wasting disease killed sunflower sea stars from Alaska to Mexico. Northern California, which has no otters, lost its second urchin predator; purple urchins increased about sixtyfold and more than 90% of the bull-kelp canopy went in three years.
Wolves were shot out of Yellowstone by 1926. Thirty-one were released back into it in 1995 and 1996. Elk on the park's northern range fell from around 19,000 in 1994 to roughly 4,000 by 2013, and willow, aspen and cottonwood began growing past browsing height along the streams for the first time in decades. Beavers followed the willow, songbirds followed the trees.
The popular version says the wolves changed the rivers. Ecologists are more careful. Grizzly bears and cougars also recovered, human hunting outside the park took thousands of elk, and a long drought thinned the herds too. Willow recovery has been patchy — strong on some streams, absent on others where the water table had already dropped.
What is not in doubt is the direction: putting one predator back changed the number of elk, where the elk dared to stand, and what grew there. That is a trophic cascade, and its size is still being argued about thirty years later.