Novel Field-Theoretic Framework Unlocks Universal Evolutionary Patterns of Competitive Systems in Nature

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 nature and social systems. However, modeling the nonlinear dynamics and multi-level interactions of such complex has long challenged scientists.

A research team from the Aerospace Information Research Institute of the Chinese Academy of Sciences (AIRCAS), led by DENG Chubo and SUN Xian, has developed a field-theoretic framework to address this issue. Their findings, published in Scientific Reports, reveal that competitive systems universally converge to three distinct evolutionary regimes—Stable Equilibrium, Periodic Oscillations, or Progressive Dominance and Elimination. The discovery offers a robust theoretical tool to explain and predict antagonistic phenomena across disciplines.

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

To overcome these hurdles, the research team turned to mean field theory, a method originally designed for studying 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 key dynamics of competition.

At the heart of the new framework lies a novel class of nonlinear partial differential equations (PDEs) integrated with Dirac δ-source terms, which represent localized resource supply processes—for example, in financial field, fixed price levels where buyers and sellers cluster in markets, or in the field of ecology, specific habitats where prey populations receive nutrient replenishment in an ecosystem.

Through rigorous theoretical derivations and numerical simulation, the team found that all competitive systems, depends on a single hyperparameter representing external energy/resource input. This universal mechanism determines whether systems reach balance, oscillate cyclically, or evolve toward dominance and extinction.

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

The code and simulation tools developed in the study are openly available on GitHub, allowing researchers to adapt and expand the framework for applications in game theory, ecological modeling, social dynamics, and beyond.

This open approach, the team notes, may accelerate discoveries across science and help illuminate the universal laws that govern competition in nature.