Zuletzt aktualisiert: 16. Juli 2026
Lotka-Volterra Räuber-Beute-Rechner
Ersteller
Dharmendra SinghPrüfer
Ersteller
Dharmendra SinghPrüfer
Quick Answer
This calculator estimates the theoretical predator-prey equilibrium and simulates the full Lotka-Volterra trajectory with a 4th-order Runge-Kutta method so users can study oscillations, extinction thresholds, and parameter sensitivity.
Use this Lotka-Volterra calculator to find the prey equilibrium x-star equals gamma over delta and the predator equilibrium y-star equals alpha over beta, then simulate predator-prey cycles over time with Runge-Kutta integration.
Wichtige Erkenntnisse
- The Lotka-Volterra model predicts perpetual oscillations between predator and prey populations.
- The stable equilibrium is x* = γ/δ for prey and y* = α/β for predators.
- α (prey birth rate) and γ (predator death rate) drive the equilibrium positions.
- 4th-order Runge-Kutta integration provides high numerical accuracy for the simulation.
- Real ecosystems deviate from ideal Lotka-Volterra behavior due to carrying capacity, migration, and stochastic effects.
Ersteller
Dharmendra SinghPrüfer
Ersteller
Dharmendra SinghPrüfer
Formel
dx/dt = αx − βxy ; dy/dt = δxy − γy
Wobei:
- α=Prey intrinsic growth rate(day⁻¹)
- β=Predation encounter rate(individual⁻¹·day⁻¹)
- γ=Predator mortality rate(day⁻¹)
- δ=Predator conversion efficiency(individual⁻¹·day⁻¹)
Rechenbeispiele
Classic Lotka-Volterra equilibrium
A textbook rabbit-fox style example with moderate prey growth and predation.
- 1Write the equilibrium formulas for the non-zero fixed point: x* = γ/δ and y* = α/β.
- 2Compute prey equilibrium: x* = 0.4 / 0.02 = 20.
- 3Compute predator equilibrium: y* = 0.1 / 0.02 = 5.
- 4Use RK4 simulation to visualize how the populations oscillate around these equilibrium values over 100 days.
Snowshoe hare and lynx scenario
A faster-growing prey population with slower predator conversion, similar to classic boreal predator-prey cycles.
- 1Set x* = γ/δ for prey and y* = α/β for predators.
- 2Compute prey equilibrium: x* = 0.3 / 0.01 = 30.
- 3Compute predator equilibrium: y* = 0.4 / 0.02 = 20.
- 4Run the RK4 integration over 200 days to inspect the lagged predator response to prey abundance.
Rapid oscillation regime
Higher growth and encounter rates create quicker predator-prey cycles and wider swings.
- 1Use the fixed-point equations x* = γ/δ and y* = α/β.
- 2Compute prey equilibrium: x* = 0.5 / 0.02 = 25.
- 3Compute predator equilibrium: y* = 0.6 / 0.04 = 15.
- 4Simulate the system to confirm that larger parameter values lead to faster oscillatory turnover.
Einführung
The Lotka-Volterra predator-prey model is a foundational ecological system of coupled differential equations that describes how prey and predator populations can rise and fall in recurring cycles. This calculator combines the model's closed-form equilibrium relationships with a 4th-order Runge-Kutta (RK4) numerical simulation so you can estimate fixed points, inspect oscillations, and compare theoretical behavior with real ecosystems that rarely behave ideally.
Understanding the Lotka-Volterra Model
Alfred J. Lotka (1925) and Vito Volterra (1926) independently introduced the classic predator-prey equations to explain cyclical abundance in interacting species. The model uses two coupled ordinary differential equations, dx/dt = αx − βxy and dy/dt = δxy − γy, to represent prey growth, predation losses, predator gains, and predator mortality. Its assumptions are deliberately simple: unlimited prey resources, homogeneous mixing, and no age structure or migration. If you need density-limited prey growth, compare it with the carrying capacity calculator. A concise mathematical overview is available from Wolfram MathWorld.
Parameters and Equilibrium
Each parameter has a biological interpretation. α controls prey growth in the absence of predators, β measures how strongly predator-prey encounters suppress prey, γ is the predator death rate when food is scarce, and δ converts successful predation into predator recruitment. Setting both derivatives to zero gives the non-zero equilibrium point x* = γ/δ and y* = α/β, provided β > 0 and δ > 0. These values are theoretical fixed points: in the ideal model, trajectories often circle around them rather than settling exactly onto them.
4th-Order Runge-Kutta Integration
Because the Lotka-Volterra system generally does not have a simple time-domain closed form for arbitrary starting conditions, the calculator simulates the trajectory numerically. A 4th-order Runge-Kutta method estimates slopes at four points within each time step, then combines them to reduce local truncation error. This gives much better accuracy than a basic Euler step, especially when oscillations are rapid or amplitudes are large. For background on the method itself, see MathWorld's Runge-Kutta entry.
Ecological Applications
The model is often introduced using the snowshoe hare–Canada lynx cycle, but the same predator-prey logic also informs discussions of wolves and moose on Isle Royale and marine predator-fish interactions. In practice, ecologists compare these time-series patterns with richer community metrics such as the Shannon diversity index calculator, because real ecosystems contain many interacting species rather than one predator and one prey. Modern field studies also highlight how climate variation and habitat change alter these cycles, as summarized in Nature research on ecological dynamics.
Limitations and Extensions
The basic model ignores carrying capacity, immigration, stochastic shocks, handling-time saturation, and seasonal forcing, so it is best viewed as a first-principles teaching model rather than a full ecosystem forecast. More realistic extensions include logistic prey growth, Holling functional responses, stochastic differential equations, and spatially explicit metapopulation models. For broader environmental risk framing, you might pair it with the wildfire risk assessment calculator. A rigorous treatment of these extensions appears in Murray's Mathematical Biology.
Kurzreferenzkarte
Lotka-Volterra Quick Reference
Kurzreferenz • Lotka-Volterra Räuber-Beute-Rechner
dx/dt = αx − βxy ; dy/dt = δxy − γyGültiger Bereich: α, β, γ, δ > 0; initial populations > 0
Häufige Werte
⚠ Achtung
- •Equilibrium values are theoretical; actual populations oscillate around them in the ideal model.
- •Very small time steps (dt < 0.01) increase accuracy but slow computation significantly.
- •β and δ must both be > 0 or equilibrium calculations are undefined.
- •The basic model assumes unlimited resources for prey and no carrying capacity—real ecosystems are more complex.
Profi-Tipps
- →Start with x₀ near x* and y₀ near y* to observe small oscillations around the equilibrium.
- →Increase simulationDays until you see at least 2 full cycles to confirm periodic behavior.
- →Compare x* (equilibrium prey) to your initial prey value—large differences create larger oscillation amplitudes.
- →Use time step ≤ 0.1 for accurate RK4 integration; dt = 0.05 is recommended for fast-oscillating systems.
FAQ
What do α, β, γ, and δ represent?
α is the prey growth rate, β is the predation rate, γ is the predator death rate, and δ is the conversion rate that turns prey consumption into predator growth.
What does the equilibrium point mean?
The equilibrium point is the non-zero fixed point where both derivatives equal zero. In this model, prey equilibrium is x* = γ/δ and predator equilibrium is y* = α/β.
Why do populations oscillate?
In the basic Lotka-Volterra system, prey growth feeds predator increases, which then depress prey, causing predator decline and allowing prey to recover. With no damping term, this idealized feedback can produce sustained oscillations.
What is the Runge-Kutta method used for?
The 4th-order Runge-Kutta method numerically integrates the coupled differential equations so the calculator can simulate population trajectories accurately over time.
How do I interpret stable prey/predator values?
They are the theoretical fixed-point populations implied by the parameter ratios. The simulated populations typically move around these values rather than remaining exactly constant.
What happens when β or δ equals zero?
If β = 0 there is no predation pressure on prey, and if δ = 0 predation never increases predator numbers. In both cases the usual non-zero equilibrium formulas break down, so the calculator reports zero for the undefined equilibrium term.