Global dynamics of controlled competitive diffusion in heterogeneous environments

We establish global well-posedness for a two-species competitive reaction-diffusion system in bounded two-dimensional domains under Neumann boundary conditions. The system incorporates spatially heterogeneous growth continuous over the closed domain and time-dependent bounded asymmetric intervention. A sharp threshold condition relating the target species' promotion strength to its natural decay rate is necessary and sufficient to maintain the carrying-capacity constraint globally in time, with the two species' total density not exceeding 1. Meanwhile, the unit square with both species' densities ranging between 0 and 1 remains positively invariant, regardless of the target species' promotion strength. Solutions are classical and unique, uniformly bounded, and Lipschitz-continuous with respect to initial data. Numerical simulations show that asymptotically applied interventions that satisfy the threshold condition result in a near-complete eradication of the competitor species at full efficiency. The promoted species establishes a heterogeneous spatial distribution shaped by growth heterogeneity. This framework extends classical diffusive Lotka-Volterra models and provides rigorous theoretical support for ecological management and misinformation suppression strategies.