Lotka-Volterra Predator-Prey Model
The Lotka-Volterra equations model predator-prey dynamics. dPrey/dt = αPrey − βPrey×Predator. dPredator/dt = δPrey×Predator − γPredator. Classic example: Canadian lynx and snowshoe hare cycles (~10yr period). Key parameters: α (prey growth), β (predation rate), γ (predator death), δ (predator reproduction efficiency).
🌍 Why This Matters for the Planet
Why It Matters
Understanding predator-prey dynamics is fundamental to ecology. The Lotka-Volterra model shows how populations oscillate: when prey are abundant, predators thrive; when predators overhunt, prey crash and predators starve. The cycle repeats. Real examples include lynx-hare, wolf-deer, and many marine systems.
How You Can Help
Enter initial prey and predator populations, plus the four parameters α, β, γ, δ. The calculator simulates the ODEs using Euler method (dt=0.01) over 50 time units. You'll see equilibrium values (Prey*=γ/δ, Pred*=α/β), max/min populations, cycle period, and charts: time series, phase portrait, equilibrium bar, parameter sensitivity.
Key Insights
- ●Equilibrium: Prey* = γ/δ, Predator* = α/β
- ●Cycle period ≈ 2π/√(αγ)
- ●Lynx-hare Hudson Bay data shows ~10-year cycles
- ●Model applies to economics and epidemiology
📋 Quick Examples — Click to Load
For educational and informational purposes only. Verify with a qualified professional.
🌎 Planet Impact Facts
Canadian lynx and snowshoe hare show ~10-year cycles in Hudson Bay fur records
— Hudson Bay Company
Lotka and Volterra developed the model independently in 1925-1926
— History of Science
At equilibrium Prey* = γ/δ and Predator* = α/β
— Ecology
Cycle period is approximately 2π/√(αγ)
— Dynamical Systems
The model has been applied to economics and epidemiology
— Applied Math
The equilibrium is neutrally stable—closed orbits, no damping
— Nonlinear Dynamics
The Lotka-Volterra equations model predator-prey dynamics. dPrey/dt = αPrey − βPrey×Predator. dPredator/dt = δPrey×Predator − γPredator. Classic example: Canadian lynx and snowshoe hare cycles (~10yr period). Key parameters: α (prey growth), β (predation rate), γ (predator death), δ (predator reproduction efficiency). Developed by Lotka (1925) and Volterra (1926).
Key Takeaways
- • Equilibrium: Prey* = γ/δ, Predator* = α/β—populations oscillate around this point
- • Cycle period ≈ 2π/√(αγ)—depends on prey growth and predator death rates
- • Euler method: Prey(t+dt) = Prey(t) + dt×(αPrey − βPrey×Pred), same for predator
- • Lynx-hare fur-trade data from Hudson Bay shows ~10-year cycles matching the model
Did You Know?
How the Equations Work
Prey equation: dPrey/dt = αPrey − βPrey×Predator
Prey grow at rate α when alone; predation removes β×Prey×Predator per unit time.
Predator equation: dPredator/dt = δPrey×Predator − γPredator
Predators gain from eating prey (δ×Prey×Predator) and die at rate γ when prey are scarce.
Euler simulation
Prey(t+dt) = Prey(t) + dt×(αPrey − βPrey×Pred). Pred(t+dt) = Pred(t) + dt×(δPrey×Pred − γPred). With dt=0.01, we simulate over 50 time units.
Expert Tips
Start Near Equilibrium
Use Prey = γ/δ and Predator = α/β to see small oscillations. Try the "Balanced Equilibrium" example.
Compare Species Pairs
Lynx-hare, wolf-deer, fox-rabbit—each has different α, β, γ, δ. Compare cycle periods and amplitudes.
Phase Portrait
The phase portrait (prey vs predator) shows closed orbits. Each orbit is a different initial condition.
Time Lag Effects
Real systems often have time lags (e.g., predator reproduction delay). The basic model ignores these.
Parameter Ranges by System
| System | α | β | γ | δ | Period |
|---|---|---|---|---|---|
| Lynx-Hare | 1.0 | 0.02 | 0.5 | 0.01 | ~12.6 |
| Wolf-Deer | 0.8 | 0.015 | 0.4 | 0.008 | ~12.5 |
| Fox-Rabbit | 1.2 | 0.025 | 0.6 | 0.012 | ~9.1 |
Frequently Asked Questions
What is the Lotka-Volterra model?
The Lotka-Volterra equations model predator-prey dynamics. dPrey/dt = αPrey − βPrey×Predator. dPredator/dt = δPrey×Predator − γPredator. α = prey growth rate, β = predation rate, γ = predator death rate, δ = predator reproduction efficiency. Developed independently by Lotka (1925) and Volterra (1926).
What is the classic lynx-hare cycle?
Canadian lynx and snowshoe hare populations show ~10-year cycles in Hudson Bay fur-trade data. Hare populations boom, lynx follow; when hares crash from predation, lynx starve and decline. The cycle repeats. This is the textbook example of Lotka-Volterra dynamics in nature.
What is the equilibrium point?
At equilibrium, both populations are constant: Prey* = γ/δ and Predator* = α/β. Neither population grows nor shrinks. The system oscillates around this point; the equilibrium is neutrally stable (closed orbits).
What is the cycle period?
The period of oscillation is approximately T ≈ 2π/√(αγ). It depends on prey growth rate (α) and predator death rate (γ). Higher α or γ shortens the period; lower values lengthen it.
Can Lotka-Volterra apply to economics?
Yes. The model has been adapted to economics (e.g., predator = firms, prey = market share), epidemiology (susceptible-infected), and other fields. The same coupled ODE structure describes many oscillating systems.
Why use Euler method for simulation?
Euler method approximates the ODEs with small time steps: Prey(t+dt) = Prey(t) + dt×(αPrey − βPrey×Pred). With dt=0.01, the simulation is stable and accurate enough for visualization. More sophisticated methods (RK4) give smoother results but Euler is sufficient for educational purposes.
Key Statistics
Official Data Sources
⚠️ Disclaimer: The Lotka-Volterra model is a simplified representation of predator-prey dynamics. Real ecosystems have spatial structure, multiple species, stochasticity, and time lags. Use for educational and exploratory purposes. Not a substitute for professional ecological modeling.
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