Evolution of competitive systems in nature

Conflict between systems is ubiquitous in nature and throughout the universe, this study presents a novel field-theoretic framework for modeling competitive systems, which simplifies the modeling of interactions between similar objects by treating them as fields, and employs mathematical models to calculate and solve the evolutionary outcomes. The key contribution lies in developing and solving a novel class of nonlinear partial differential equations that incorporate δ-source terms to characterize resource supply processes, and theoretical demonstrate that system evolution converges to three universal regimes(traveling waves, oscillations or stable equilibrium), with phase transition criteria determined by quantitative relationships between eigenvalues and supply parameters. Rigorous theoretical derivations and numerical results demonstrate that the model's computations possess significant generality and applicability, offering a robust framework to explain various antagonistic phenomena observed in nature.