Consider a non-linear differential equation system that describes the behavior of a population of interacting species in an ecological system. The system consists of multiple coupled equations, each representing the population dynamics of different species. These equations incorporate factors such as competition, predation, and environmental carrying capacity.

Given the initial conditions and parameters of the system, your task is to analyze the long-term behavior of the population dynamics. Specifically, determine whether the system reaches a stable equilibrium, exhibits periodic oscillations, or undergoes chaotic behavior.

To solve this problem, you'll need to employ advanced mathematical techniques such as numerical integration methods, phase plane analysis, stability analysis of equilibrium points, and bifurcation analysis. Additionally, you may need to utilize computer simulations and mathematical software to explore the complex behavior of the system over time.

This problem requires a deep understanding of nonlinear dynamics, differential equations, and ecological principles, making it a highly challenging and intellectually stimulating mathematical endeavor.