# Dynamically generated hierarchies in games of competition

## TL;DR

This paper explores how complex nested spiral patterns can spontaneously form in multi-species predator-prey models, revealing universal mechanisms for pattern generation in non-equilibrium systems.

## Contribution

It demonstrates the emergence of hierarchical spiral patterns in a nine-species predator-prey model and analyzes their dependence on system parameters.

## Key findings

- Spontaneous formation of nested spirals in predator-prey systems
- Parameter dependence of complex pattern formation elucidated
- Analysis of spatial and temporal dynamics through multiple quantitative measures

## Abstract

Spatial many-species predator-prey systems have been shown to yield very rich space-time patterns. This observation begs the question whether there exist universal mechanisms for generating this type of emerging complex patterns in non-equilibrium systems. In this work we investigate the possibility of dynamically generated hierarchies in predator-prey systems. We analyze a nine-species model with competing interactions and show that the studied situation results in the spontaneous formation of spirals within spirals. The parameter dependence of these intriguing nested spirals is elucidated. This is achieved through the numerical investigation of various quantities (correlation lengths, densities of empty sites, Fourier analysis of species densities, interface fluctuations) that allows to gain a rather complete understanding of the spatial arrangements and the temporal evolution of the system. A possible generalization of the interaction scheme yielding dynamically generated hierarchies is discussed. As cyclic interactions occur spontaneously in systems with competing strategies, the mechanism discussed in this work should contribute to our understanding of various social and biological systems.