A Rigorous Sawi Decomposition Framework for the Nonlinear Lotka–Volterra System: Convergence Theory, Stability Analysis, and Numerical Validation

Abstract

In the 1920s, the Lotka-Volterra equations were developed as a foundational model for predator-prey dynamics in mathematical biology. Since these nonlinear equations rarely possess exact analytical solutions, advanced semi-analytical methods are required for their study. This research investigates the Sawi Decomposition Method (SDM), an innovative hybrid approach combining the Sawi integral transform with the Adomian Decomposition Method (ADM). We establish a rigorous mathematical framework for the Sawi transform, detailing its linearity and differential properties. Furthermore, we provide a formal proof of the method's convergence and the uniqueness of its solutions for this class of nonlinear systems. To ensure biological relevance, we conduct a stability analysis of the system's equilibrium points, identifying the coexistence equilibrium as a center with closed orbits. The performance of SDM is benchmarked against the Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), Laplace Decomposition Method (LDM), and the fourth-order Runge-Kutta (RK4) scheme. Utilizing wolf and moose population data from Isle Royale National Park, numerical experiments demonstrate that SDM achieves high accuracy—comparable to LDM—while significantly reducing algebraic complexity and exhibiting faster convergence than HPM and VIM. The results confirm that SDM is a robust and computationally efficient alternative for modeling complex population dynamics.