Lotka-Volterra (Predator-Prey)

Lotka-Volterra Equations

The predator-prey model describes two interacting populations: prey $x$ (rabbits) and predators $y$ (foxes):

$\frac{dx}{dt} = \alpha x - \beta x y quad \text{(prey grow, but die when meeting predators)}$ $\frac{dy}{dt} = \delta x y - \gamma y quad \text{(predators grow by eating prey, die naturally)}$

• $\alpha$: prey birth rate
• $\beta$: predation rate (prey killed per encounter)
• $\delta$: predator growth rate per prey eaten
• $\gamma$: predator death rate

Oscillations

The populations oscillate: when prey are abundant, predators multiply. More predators reduce prey. Fewer prey starve predators. Then prey recover, and the cycle repeats.

Equilibrium

There is a non-trivial equilibrium where both populations are constant:

$x^* = \frac{\gamma}{\delta}, \quad y^* = \frac{\alpha}{\beta}$

Starting exactly at this point, populations stay fixed forever.

Conservation Law

The Lotka-Volterra system has a conserved quantity (a "first integral"):

$V = \delta x - \gamma \ln x + \beta y - \alpha \ln y = \text{constant}$

This means the phase-plane trajectories are closed curves.

Your Task

Implement lotka_volterra(alpha, beta, delta, gamma, x0, y0, t_end, n) using Euler's method. Return (x, y).