Experiment · Ecosystem
Grass, shrubs and trees, mice, hares and roe deer, and above them foxes, wolves and buzzards. In spring the young are born, in winter food runs short, and fungi and bacteria turn remains into fertile soil. Watch the population cycles and trophic cascades, compare them with the Lotka–Volterra equations, and don't let any species die out.
Click the world to release 12 hares
Phase portrait
Click to choose a start
Over time
For more than a hundred years the Hudson's Bay Company bought furs in Canada. The number of snowshoe hare and Canada lynx pelts rises and falls roughly every 10 years — and lynx peaks usually come a year or two after hare peaks. Measure the period: click two neighboring hare peaks.
approximate / illustrative data
Click a hare peak (or focus the chart and use the arrow keys and Enter).
Grass → hares → foxes. At each level only part of the energy moves up — the rest goes into movement, body heat and life processes. A popular rule of thumb says about 10% (in reality it ranges from a few percent to about 20%).
Example: 10,000 → about 1,000 → about 100 units of energy. That's why there are always fewer predators than prey, and food chains rarely have more than 4–5 links.
Wolves live, among other places, in the Białowieża Forest and the Bieszczady Mountains, where Poland has especially many of them. Research in Białowieża shows that their main prey there is red deer, along with wild boar and roe deer.
But a real forest isn't just two species: deer numbers also depend on winter, food supply, disease and human hunting. That's why nature rarely shows cycles as regular as the equations. The wolf is a protected species in Poland.
In 1995–1996 a few dozen wolves were brought from Canada to Yellowstone National Park (USA). Elk began to decline and changed their behavior, and in some places willows and aspens grew back. Such a domino effect through successive levels of the food chain is a trophic cascade.
Scientists still debate how strong and how widespread this effect was — elk were also affected by bears, cougars, droughts and hunting outside the park.
On Isle Royale in Lake Superior (USA), the world's longest continuous study of a predator and its prey has been running since 1958. The wolves most likely arrived over the ice in the late 1940s and hunt mainly moose.
The numbers of both species fluctuated strongly — because of disease, harsh winters, ticks and inbreeding among the wolves. When only two wolves were left, about a dozen new ones were brought to the island in 2018–2019.
The last wild lowland European bison (żubr) was killed in the Białowieża Forest in 1919. The species survived only in zoos, and all of today's European bison descend from about a dozen animals.
Breeding in Białowieża was rebuilt from 1929, and in 1952 the first bison were released into the wild. Today more than two thousand of them live in Poland.
The Eurasian beaver builds dams that raise the water level and create wetlands. Fish, amphibians, insects and waterbirds benefit, and water stays in the landscape longer.
After World War II there were few beavers left in Poland. Thanks to protection and relocation they again live almost all over the country — although their dams are sometimes a source of conflict with farmers.
Minks escaped (and were released) from fur farms. They are excellent swimmers and divers and eat fish, amphibians, rodents and birds — including the eggs and chicks of birds nesting by the water.
In Poland it is considered an invasive species; in some places it is removed to protect bird colonies. A similar newcomer is the raccoon dog, brought from the Far East to the former USSR.
In Ecosystem mode each animal is a separate “agent” with a position and an energy reserve. The terrain is a grid of cells: on each one grow grass, shrubs, young trees (saplings) and trees, and a stream with a pond flows through the meadow. The rules are simple:
Dead animals, fallen leaves and dead grass go into the soil as dead organic matter. Fungi and bacteria — the decomposers — break it down (fastest in summer, hardly at all in winter) and release mineral nutrients that make the soil more fertile. On fertile soil plants grow faster. Turn on the “Soil layer” to see where the soil is rich and where the decomposers are at work (glowing dots). After a fire, ash and burnt wood briefly enrich the ground, and then comes succession: first grass, then shrubs, finally trees.
In Food web mode the same ecosystem is described by equations — for whole populations, without randomness. The biomass of each consumer Bi changes like this (the Yodzis–Innes bioenergetic model):
The first term is respiration (energy used to stay alive), the second — food turned into the consumer's own body, the third — losses to predators. Fij is the functional response: the more prey, the more a predator eats, but not without limit — eventually it is full. Plants grow logistically, and their growth depends on mineral nutrients from the soil. Remove a species and you'll see a trophic cascade: without wolves roe deer increase, and shrubs and saplings decline. Wolves are only a fraction of a percent of animal biomass, yet they change the whole forest — they are a keystone species.
Plants capture the Sun's energy, but herbivores assimilate only a small part of it, and predators only a small part of the herbivores' energy. The rest dissipates as heat. In nature, usually 5–20% of the energy passes to the next level (about 10% on average — Raymond Lindeman, 1942). That's why energy pyramids, and usually biomass pyramids too, narrow toward the top.
Alfred Lotka (1925) and Vito Volterra (1926) independently described predator and prey with two equations. Let x be the number of hares and y the number of foxes (here in hundreds of animals):
The product xy says how often a fox meets a hare. The populations stop changing at the equilibrium point:
The equilibrium number of hares depends only on the foxes' parameters, and vice versa. When the start is away from equilibrium, the populations circle along a closed orbit, because the quantity
does not change over time. For small oscillations the period is T ≈ 2π/√(αγ), and the fox peak lags by a quarter of a period. If we add a limited amount of grass (capacity K: dx/dt = αx(1 − x/K) − βxy), the orbit turns into a spiral winding in to equilibrium — the system becomes stable.
A real fox can't eat infinitely many hares — it needs a time h to chase and eat each one. This is the type II functional response (Crawford Holling, 1959). The Rosenzweig–MacArthur model:
With a small K the populations settle down at equilibrium. But when we “enrich” the environment — raise K above the threshold KH = 2x* + 1/(ah) — the equilibrium loses stability (a so-called Hopf bifurcation) and a limit cycle appears, with ever larger swings. More food for the prey can thus endanger both species. This is the paradox of enrichment (Michael Rosenzweig, 1971).
Two species that eat the same thing get in each other's way. The Lotka–Volterra competition model:
The coefficient α12 says how strongly one individual of species 2 “takes up room” from species 1. Both species survive when each limits itself more than its rival (α12 < K1/K2 and α21 < K2/K1). Otherwise one displaces the other — this is the competitive exclusion principle. Georgy Gause demonstrated it in 1934 with paramecia grown in test tubes.
The computer moves in small steps h. The 4th-order Runge–Kutta method checks the slope four times within each step and takes a weighted average:
As a result, with the default parameters V changes by less than one billionth of its value after a hundred years. Euler's method with the same step would change it by tens of percent.
Simplifications: all the models are sketches, not forecasts. Time is nominal (in the ecosystem 1 year = 120 steps), animals have no sex or families (apart from wolves sharing their kill), buzzards stand in for all birds of prey, and the fish and amphibians that minks eat are not modeled separately. Biomasses in the food web are nominal kg per hectare. The parameters were chosen so that the models behave sensibly, not to reproduce a particular forest. Real hare and lynx cycles also depend on food, disease and stress — not just on predators.