Saddles, Arrows, and Spirals: Deterministic Trajectories in Cyclic Competition of Four Species

TL;DR

This paper analyzes the deterministic population trajectories in cyclic competition systems with four or more species, revealing diverse orbit shapes and a key exponential collective variable, with implications for stochastic and universal system behaviors.

Contribution

It introduces a detailed mean field analysis of multi-species cyclic competition, identifying new orbit types and a universal exponential collective variable.

Findings

Discovery of saddle, spiral, and straight-line orbits

Identification of a collective variable evolving exponentially

Insights into universal properties of cyclic competition systems

Abstract

Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here, we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable which evolves simply as an exponential: $\mathcal{Q}% \propto e^{\lambda t}$, where $\lambda$ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for $t\rightarrow -\infty $). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems.