From the collision of galaxies in the vast universe to the subtle resource competition between ants and anteaters, and even the ebb and flow of market forces in human society, competitive interactions are omnipresent across natural and social systems. However, modeling the nonlinear dynamics and multi-level interactions of such complex systems has long posed a challenge to scientists.

To address this issue, a research team from the Aerospace Information Research Institute (AIR) of the Chinese Academy of Sciences has developed a field-theoretic framework. Their findings, recently published in Scientific Reports, reveal that competitive systems universally converge to three distinct evolutionary regimes: stable equilibrium, periodic oscillations, and progressive dominance and elimination.

Traditional approaches to modeling dynamic systems often rely on particle-based methods, which simulate interactions between individual "particles." However, these methods face drawbacks: as the number of particles increases, computational complexity surges exponentially, and accumulated errors destabilize long-term predictions.

To overcome these hurdles, the team turned to mean field theory, a method originally designed to study complex many-body problems. Instead of tracking individual components, the mean field approach treats competing systems as continuous density fields, simplifying interactions into an "effective field" that captures the system's overall behavior. This approach avoids computational bottlenecks while preserving the key dynamics of competition.

At the core of the new framework is a novel class of nonlinear partial differential equations (PDEs) integrated with Dirac δ-source terms. These terms represent localized resource supply processes: for instance, in finance, they correspond to fixed price levels where buyers and sellers cluster in markets; in ecology, they denote specific habitats where prey populations receive nutrient replenishment in an ecosystem.

Through theoretical derivations and numerical simulations, the team found that all competitive systems depend on a single hyperparameter representing external energy or resource input. This universal mechanism determines whether a system achieves balance, oscillates cyclically, or evolves toward dominance and extinction.

"Competitive systems are everywhere, yet their underlying evolutionary rules have remained elusive until now," said associate Prof. DENG Chubo from AIR. "Our framework provides a universal language to describe these interactions, and its potential to unlock new insights across disciplines is enormous."

The code and simulation tools developed in the study are openly available on GitHub. The team notes that this open approach may accelerate discoveries across scientific fields and help uncover the universal laws governing competition in nature.