Lotka-Volterra Equations

The Lotka-Volterra equations (predator-prey model) describe the coupled dynamics of two populations: prey $x$ and predator $y$.

$$\frac{dx}{dt}=ax-bxy$$ $$\frac{dy}{dt}=-cy+dxy$$

where $a,b,c,d>0$ are positive parameters:

• $a$: prey birth rate (prey grow exponentially without predators)
• $b$: predation rate (each encounter reduces prey)
• $c$: predator death rate (predators die without prey)
• $d$: predator growth from predation

Qualitative Behaviour

The populations oscillate periodically. When prey are plentiful, predators grow; when predators are numerous, prey decline; with fewer prey, predators decline; and prey recover. This produces closed orbits in the phase plane $(x,y)$.

Python

Python

import numpy as np import matplotlib.pyplot as plt prey_birth_rate = 1.2 predation_rate = 0.6 predator_death_rate = 0.8 predator_growth_rate = 0.3 def lotka_volterra(t: float, state: np.ndarray) -> np.ndarray: """Compute derivatives for the Lotka-Volterra predator-prey system. Args: t: Current time (not used - autonomous system). state: Array [prey_population, predator_population]. Returns: Array of derivatives [d(prey)/dt, d(predator)/dt]. """ prey_population, predator_population = state d_prey = prey_birth_rate * prey_population - predation_rate * prey_population * predator_population d_predator = -predator_death_rate * predator_population + predator_growth_rate * prey_population * predator_population return np.array([d_prey, d_predator]) dt = 0.001 time_end = 30.0 number_of_steps = int(round(time_end / dt)) time_array = np.zeros(number_of_steps + 1) state_array = np.zeros((2, number_of_steps + 1)) state_array[:, 0] = np.array([2.0, 1.0]) # initial prey=2, predator=1 for step_index in range(number_of_steps): time_array[step_index + 1] = time_array[step_index] + dt state_array[:, step_index + 1] = ( state_array[:, step_index] + dt * lotka_volterra(time_array[step_index], state_array[:, step_index]) ) plt.plot(state_array[0, :], state_array[1, :]) plt.xlabel('Prey population') plt.ylabel('Predator population') plt.title('Predator-prey phase plane (Lotka-Volterra)') plt.tight_layout() plt.show()

Output

Systems of ODEs | Explicit Euler method | Fourth order Runge-Kutta | Runge-Kutta methods