Lotka-Volterra Predator-Prey Dynamics — 3D Lab Simulation
Self-regulating cyclic oscillations in predator-prey ecology
Theoretical Foundation & Historical Context
In 1925-1926, Lotka and Volterra independently formulated coupled nonlinear differential equations modeling the population dynamics between two interacting biological species: prey (e.g. rabbits) and predators (e.g. foxes), demonstrating perpetual oscillations around an ecological equilibrium.
Law: Ecological Differential Equations & Closed Phase Orbits | Scientist: Alfred Lotka & Vito Volterra | Year: 1925
Key Learning Objectives & Competencies
- ✓ Analyze the non-linear coupling between prey growth dx/dt = αx - βxy and predator decay dy/dt = δxy - γy.
- ✓ Locate the non-trivial co-existence equilibrium fixed point (x* = γ/δ, y* = α/β).
- ✓ Demonstrate why phase plane trajectories form closed periodic orbits with conserved invariant quantities.
Mathematical Formulations & Differential Equations
dx/dt = α·x - β·x·y (Con mồi) dy/dt = δ·x·y - γ·y (Thú săn)
x represents prey population density, y represents predator population
x* = γ / δ và y* = α / β
Steady state where populations remain constant over time
V(x, y) = δ·x - γ·ln(x) + β·y - α·ln(y) = const
Invariant function proving closed periodic phase trajectories
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| α | Intrinsic growth rate of prey population | — | — |
| β | Predation rate coefficient per interaction | — | — |
| γ | Natural mortality rate of predators | — | — |
| δ | Efficiency of turning consumed prey into new predators | — | — |
Real-World & Industrial Engineering Applications
Wildlife reserve management and setting sustainable harvesting quotas for marine fisheries.
Integrated biological pest management releasing beneficial predatory insects against crop pests.
Macroeconomic models simulating employment rates and capital wage share business cycles.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why is the predator population peak always delayed after the prey population peak?
Predators can only reproduce and increase in number after an abundance of food (prey) has already grown. As predators multiply, they overconsume the prey, causing the prey population to crash first, followed later by a predator decline.
What happens if all predators are completely eradicated from the ecosystem?
In the basic model without carrying capacity, the prey population explodes exponentially without bound (dx/dt = αx). In real ecosystems, prey exceed environmental food supply, causing severe overgrazing and catastrophic famine collapse.
What is the conserved invariant quantity in the standard Lotka-Volterra equations?
The constant of motion is V(x, y) = δx - γ·ln(x) + βy - α·ln(y). Because dV/dt = 0 along all solutions, trajectories form closed continuous orbits around the equilibrium point without ever spiraling inward or outward.
Related Simulations & Experiments
Spatial molecular architecture and complementary base pairing
Compartmental transmission dynamics: Susceptible (S), Infectious (I), Recovered (R)
Nonlinear differential model of ecological interaction generating periodic population cycles
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.