A Computational Guide to Stability Analysis of Nonlinear Systems: The Lotka-Volterra and SIR Models as Case Studies

Abstract

This paper provides a structured and reproducible guide to analyzing the local stability of nonlinear dynamical systems by systematically combining analytical linearization with computational phase portrait visualization. While these techniques are standard, introductory materials often lack a unified, code-based workflow that connects abstract theory to practical, visual interpretation. This guide bridges that gab using two canonical models from different scientific domains: the Lotka-Volterra predator-prey model and the Susceptible Infected Recovered (SIR) epidemic model. For each system, we identify equilibrium points, compute the Jacobian matrix, and use eigenvalue analysis to determine local stability, and perform a sensitivity analysis to explore how dynamics are affected by key parameter. By comparing the ecological and epidemiological models, we highlight how shared mathematical principles lead to distinct real-world dynamics, such as persistent oscillations versus threshold-based outbreaks. Pythongenerated phase portraits are used throughout to visually validate the analytical results, offering an intuitive complement to the theory. This work serves as a practical toolkit for students and researchers new to nonlinear modeling, emphasizing a clear, step by step process that is essential for both educational setting and applied research.