About the Lotka-Volterra equations
The Lotka-Volterra equations, developed independently by Alfred Lotka (1925) and Vito Volterra (1926), are the foundational mathematical model of predator-prey population dynamics in ecology. They describe how the populations of two interacting species — a prey species and a predator species that depends on it for food — change continuously over time. The model is a pair of coupled, nonlinear first-order differential equations that produce the classic cyclical "boom and bust" pattern seen in real predator-prey systems, such as the century of Hudson's Bay Company fur-trapping records for snowshoe hares and Canada lynx.
The equations
Let x(t) be the prey population and y(t) be the predator population at time t. The model is:
- dx/dt = αx − βxy — prey grow exponentially at rate α in the absence of predators, and are removed by predation at a rate proportional to the product of both populations (rate constant β).
- dy/dt = δxy − γy — predators grow in proportion to how much prey they consume (rate constant δ), and die off at a constant per-capita rate γ in the absence of prey.
All four parameters (α, β, γ, δ) must be positive. α is the prey's intrinsic growth rate, β is the predation (attack) rate, γ is the predator's death rate, and δ is the efficiency with which consumed prey are converted into new predators.
Equilibrium point
Setting both derivatives to zero gives a single non-trivial (coexistence) equilibrium: x* = γ/δ (prey) and y* = α/β (predators). At this point neither population changes — but the equilibrium is a "center," not an attractor: starting anywhere else produces a closed periodic orbit around (x*, y*) rather than convergence toward it. The size of the orbit is fixed entirely by the starting populations, which is why this calculator also reports the population trajectory (via numerical integration), not just the fixed point.
Why the populations cycle
When prey are abundant, predators have plenty to eat and their numbers grow (dy/dt > 0). More predators then consume prey faster than prey can reproduce, so the prey population falls. With less prey, predators start to starve and decline (dy/dt < 0). With fewer predators, prey pressure eases and the prey population rebounds — restarting the cycle. This out-of-phase oscillation, with predator peaks lagging prey peaks, is the model's signature prediction and matches the qualitative shape of many real predator-prey time series, even though the basic model ignores carrying capacity, age structure, and environmental noise.
Typical parameter magnitudes
- Textbook demonstration values are often small, e.g. α = 0.6–1.1/year (rabbit-like growth), β = 0.001–0.02 (depends on population units), γ = 0.4–0.9/year, δ = 0.001–0.02.
- Units matter: β and δ scale with how populations are counted (individuals, hundreds, thousands), so the same biological interaction can have very different-looking coefficients depending on the units used to record x and y.
- A conserved quantity, V = δx − γ ln(x) + βy − α ln(y), stays constant along any trajectory of the ideal (undamped) model — a useful check that a numerical simulation is behaving correctly over long time spans.
Limitations of the basic model
The classic Lotka-Volterra system assumes unlimited prey food supply, no carrying-capacity ceiling, no time delays, and a linear ("mass-action") predation term. Real populations are also affected by seasonality, disease, multiple predator or prey species, and demographic randomness at low population sizes. Extended versions — such as the Rosenzweig-MacArthur model with logistic prey growth and saturating (Holling type II) predation — address some of these issues while keeping the same coupled-ODE structure.