Numerical Solutions of Non-Linear Systems of ODES: Lotka–Volterra Predator–Prey Model

Abstract

This study investigates the dynamic behavior of predator–prey interactions using the classical Lotka–Volterra system of nonlinear ordinary differential equations. We employ both analytical and numerical techniques to analyze population fluctuations of hares and lynx, based on historical Hudson Bay Company records. An approximate analytical solution is derived using the Ritz method, while a numerical solution is obtained via the fourth-order Runge–Kutta (RK4) method. To enhance model realism, we estimate system parameters (α, β, γ, δ) using Simulated Annealing (SA), a global optimization technique well-suited for non-convex landscapes. The model is calibrated against empirical data, and SA-based optimization achieved mean RMSE values of 4.12 for hares and 4.01 for lynx across five independent runs. Comparative plots between observed and predicted populations confirm the model’s ability to capture oscillatory behavior and phase shifts. This work compares analytical and numerical solution methods for the Lotka–Volterra system using parameters estimated via Simulated Annealing, demonstrating the relative strengths of the Ritz and Runge–Kutta approaches in modeling real-world population dynamics.