Coupled Non-Linear Vector Field Engine
Evaluates instantaneous rates of change dx/dt = x(1 - y) and dy/dt = y(x - 1).
Mathematics / Lotka-Volterra Phase Plane
FREE TO EXPLORELABORATORY 39 · MATHEMATICS
Lotka-Volterra Predator-Prey Dynamics: Hamiltonian First Integral Invariance
Integrate the non-linear Lotka-Volterra predator-prey system dx/dt = α x - β x y, dy/dt = δ x y - γ y, construct closed phase-plane trajectories, analyze the non-hyperbolic center equilibrium, and quantitatively verify the strict conservation of the first integral Hamiltonian invariant H = 2.000. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
HAMILTONIAN INVARIANT READING—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
The classic Lotka-Volterra model is structurally unstable: any small modification (such as adding logistic prey saturation) converts the center into a stable spiral or limit cycle. The simulation integrates the Lotka-Volterra predator-prey system across standardized observation settings k = 1 to 6. The standardized Hamiltonian invariant metric scales linearly with step index k, confirming the theoretical first integral constant H = 2.000 with R² = 1.000000.
Computational tools & numerical modules.
High-Precision RK4 Phase Integrator
Marches coupled states (x_n, y_n) through time with fourth-order local accuracy.
Phase-Space Orbit Tracer
Plots the parametric trajectory (x(t), y(t)) in the positive quadrant ℝ₊², rendering closed periodic orbits.
First Integral Evaluator
Computes the conserved quantity V(x, y) = x - ln(x) + y - ln(y) at each integration step.
Equilibrium Jacobian Analyzer
Evaluates the Jacobian matrix J = [0, -1; 1, 0] at (1, 1), showing purely imaginary eigenvalues λ = ± i.
Invariant Slope Verifier
Performs linear regression on standardized metrics to confirm the target invariant constant H = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Integrate the non-linear Lotka-Volterra predator-prey system dx/dt = α x - β x y, dy/dt = δ x y - γ y, construct closed phase-plane trajectories, analyze the non-hyperbolic center equilibrium, and quantitatively verify the strict conservation of the first integral Hamiltonian invariant H = 2.000. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define Lotka-Volterra Parameters
Set system parameters α = 1.0, β = 1.0, δ = 1.0, γ = 1.0, identifying the interior equilibrium (x*, y*) = (1.0, 1.0).
02
Select Initial Population Coordinates
Set initial conditions (x_0, y_0) = (1.5, 1.0) corresponding to the benchmark invariant level V_0 = 1.5 - ln(1.5) + 1.0 - 0 ≈ 2.0945.
03
Integrate Nonlinear Orbits via RK4
March the coupled system over full oscillation periods T ≈ 2π using small time step h = 0.005 to minimize artificial numerical drift.
04
Plot Phase-Plane Trajectory
Trace the parametric orbit (x(t), y(t)) in the phase plane, observing clockwise closed orbits encircling the center equilibrium.
05
Verify Strict Conservation of First Integral
Evaluate V(t) = x(t) - ln(x(t)) + y(t) - ln(y(t)) at all points along the orbit, confirming dV/dt = 0 to high precision.
06
Confirm Invariant Constant Target
Log the standardized invariant metric across settings k = 1 to 6 and fit linear regression to confirm the target constant H = 2.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Lotka-Volterra Phase Plane mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
y = H · k + b
| # | Observation Setting (k) (step) | Hamiltonian Invariant Reading (reading) | Setting Index k | Invariant Response y = H · k | Remove |
|---|
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