Modeling the Evolution of Population Dynamics Using Ordinary Differential Equations: Mathematical Analysis and Modern Applications
Abstract
Population dynamics modeling plays a critical role in understanding ecological stability, epidemiological spread, and sustainable resource management. Classical models such as the logistic growth and Lotka-Volterra equations offer foundational insights but often overlook environmental stochasticity and multispecies complexity. Existing frameworks frequently simplify nonlinear feedbacks or exclude real-time ecological interactions, limiting predictive capacity in dynamic systems. This study advances population modeling by integrating ordinary differential equations (ODEs) with modern computational tools, including machine learning-enhanced simulations and neural ODE frameworks. Analytical techniques such as linear stability, Lyapunov methods, and bifurcation analysis revealed equilibrium classifications and transitions in logistic, predator-prey, and epidemiological models. Numerical simulations validated theoretical findings, showing that hybrid AI-augmented models achieved higher accuracy (relative error 2.1%) and computational efficiency (98%) than traditional models. A novel contribution lies in embedding neural networks within classical ODE systems to dynamically adjust model parameters using heterogeneous data streams. The developed framework enhances the realism and adaptability of population models, with direct applications in conservation planning, disease control, and ecological forecasting, thus offering a versatile and interdisciplinary tool for addressing real-world biological challenges.
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