How populations change
A population is all the members of one species living in an area. It grows through births and immigration and shrinks through deaths and emigration. When conditions are ideal, a population growsexponentially — the more individuals there are, the faster they multiply. In nature, growth is always limited eventually, by food, space, disease or predators.
Predator–prey cycles
The interaction between a predator and its prey creates a repeating cycle:
- With few predators, the prey population grows quickly.
- Plenty of prey means predators are well fed and their numbers rise.
- Many predators eat prey faster than prey can reproduce, so prey numbers fall.
- With less food, predators starve and decline — and the prey recover.
Because predators respond to prey numbers with a delay, the predator peaks always come after the prey peaks.
Reading the phase plot
The phase plot shows rabbits across and foxes up. Each moment is a single point, so over time the point traces a path. A closed loop means the cycle repeats exactly; a spiral inward means the swings are dying down toward a steady balance; a path that hits an axis means one species has gone extinct.
Carrying capacity
Real prey cannot multiply forever, because grass and burrows run out. The carrying capacity (K) is the largest population the environment can support. With a carrying capacity, prey growth slows as numbers approach K. In the simulator this usually makes the cycles shrink until both populations settle at steady values — and if the foxes disappear, the rabbits level off at K instead of exploding.
The Lotka–Volterra model
Alfred Lotka (1925) and Vito Volterra (1926) independently described predator–prey interactions with two linked equations. With prey H, predators P and time t:
dH/dt = r·H·(1 − H/K) − a·H·PdP/dt = b·a·H·P − m·PHere r is the prey birth rate, a the hunting success, b the fraction of eaten prey turned into new predators (0.2 in the simulator), m the predator death rate, and K the carrying capacity (very large when switched off). The balance point, where both populations stop changing, is at H* = m / (b·a).
Key terms
- Predator and prey
- A predator hunts and eats other animals; the prey is the animal that is eaten.
- Exponential growth
- Growth in which the population increases by a constant percentage each year, producing a J-shaped curve.
- Logistic growth
- Growth that slows as a population approaches its carrying capacity, producing an S-shaped curve.
- Density-dependent factor
- A factor whose effect increases as the population becomes more crowded, such as competition, disease or predation.
- Equilibrium
- A state where populations stop changing because gains and losses balance.
Real-world examples
- Snowshoe hares and lynx in Canada show roughly 10-year cycles, famously recorded in Hudson’s Bay Company fur records. Research shows food supply and predation both play roles.
- Wolves and moose on Isle Royale in Lake Superior have been studied since 1958 — one of the longest predator–prey studies in the world.
- Pest control: farmers release predatory insects such as ladybirds to control aphids, using predator–prey dynamics.
- Fisheries: removing too many predatory fish can let their prey boom and disrupt the whole ecosystem.
Common misconceptions
- “Predators wipe out their prey.” Usually predators decline first when prey become scarce, allowing prey to recover.
- “Removing predators helps the prey.” It can lead to overpopulation, overgrazing and then a crash.
- “Nature always finds a perfect balance.” Many populations fluctuate constantly rather than staying at one level.
What the model simplifies
The model has one predator, one prey, no seasons, no age structure and no randomness. Populations are treated as smooth numbers rather than individual animals, which is why the classic cycles never die out on their own. Real ecosystems contain many interacting species, weather and chance events.
Key takeaways
- Predator and prey populations rise and fall in linked cycles, with predators lagging behind.
- Prey grow exponentially without predators unless limited by a carrying capacity.
- Carrying capacity tends to damp the cycles toward a stable balance.
- The phase plot reveals cycles as loops and stability as an inward spiral.