# Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions

## TL;DR

This paper explores a variant of the cyclic Lotka-Volterra model with three-agent interactions, analyzing its bifurcation behavior, stochastic effects, and spatial dynamics, including the impact of mobility and nonlinear diffusion in ecological systems.

## Contribution

It introduces a new model with three-agent interactions, analyzes bifurcations and stochastic effects, and studies spatial and mobility influences on cyclic dominance dynamics.

## Key findings

- Degenerate Hopf bifurcation identified in well-mixed environments.
- Stochastic noise influences extinction probabilities near bifurcation.
- Spatial locality eliminates bifurcations, affecting stability and dynamics.

## Abstract

We study a variant of the cyclic Lotka-Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two preys plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects turn neutrally stable orbits into erratically diverging trajectories. In particular, we compare analytic predictions for the extinction probability, derived in the Fokker-Planck approximation, with numerical simulations based on the Gillespie stochastic algorithm. We then extend the analysis of the phase portrait to heterogeneous rates. In a well-mixed environment, we observe a continuum of degenerate Hopf bifurcations, generalizing the above one. Neutral stability ensues from a complex equilibrium between different reactions. Remarkably, on a two-dimensional lattice, all bifurcations disappear as a consequence of the spatial locality of the interactions. In the second part of the paper, we investigate the effects of mobility in a lattice metapopulation model with patches hosting several agents. We find that strategies propagate along the arms of rotating spirals, as they usually do in models of cyclic dominance. We observe propagation instabilities in the regime of large wavelengths. We also examine three-agent interactions inducing nonlinear diffusion.