# A stochastic analysis of the spatially extended May-Leonard model

## TL;DR

This paper derives a coarse-grained stochastic evolution equation for the May-Leonard model, capturing spatial pattern formation and fluctuations near the Hopf bifurcation, using advanced field-theoretic methods.

## Contribution

It extends deterministic models by incorporating stochastic effects, deriving a complex Ginzburg-Landau equation for the full stochastic dynamics of cyclically competing species.

## Key findings

- Derivation of the complex Ginzburg-Landau equation from stochastic May-Leonard dynamics.
- Identification of parameter constraints for time scale separation.
- Reduction to two slow degrees of freedom in the model.

## Abstract

Numerical studies of the May-Leonard model for cyclically competing species exhibit spontaneous spatial structures in the form of spirals. It is desirable to obtain a simple coarse-grained evolution equation describing spatio-temporal pattern formation in such spatially extended stochastic population dynamics models. Extending earlier work on the corresponding deterministic system, we derive the complex Ginzburg-Landau equation as the effective representation of the fully stochastic dynamics of this paradigmatic model for cyclic dominance near its Hopf bifurcation, and for small fluctuations in the three-species coexistence regime. The internal stochastic reaction noise is accounted for through the Doi-Peliti coherent-state path integral formalism, and subsequent mapping to three coupled non-linear Langevin equations. This analysis provides constraints on the model parameters that allow time scale separation and in consequence a further reduction to just two coarse-grained slow degrees of freedom.