Topological Invariance, Phase Inversion, and Boundary Extinction in Coupled Resource-Agent Non-Linear Dynamical Systems

This paper establishes a rigorous non-linear mathematical framework mapping the coupled continuous-time dynamics between an emitting source potential, designated as the Resource Field (x), and an autonomous self-replicating entity, defined as the Agent Population (y). Utilizing a modified Lotka-Volterra topology modulated by a directional parity operator \sigma=\pm1, we analytically derive and simulate both the forward evolution (Growth Phase) and the inverse deconstruction (Dissipation Phase). We prove that while individual trajectories exhibit velocity inversion, the non-trivial equilibrium point---the Dynamic Neutral Point---remains strictly invariant under phase transformations. Furthermore, we define the critical boundary conditions governing system failure through the analytical formulation of both Source Collapse and Agent Extinction states, establishing the global stability landscape of the framework.