Co-existence in the two-dimensional May-Leonard model with random rates

TL;DR

This study uses Monte Carlo simulations to analyze how mobility and disorder affect species coexistence, pattern formation, and extinction times in a two-dimensional stochastic May-Leonard predator-prey model.

Contribution

It reveals that quenched disorder minimally impacts dynamics, while increased mobility induces a transition from coexistence to extinction, altering extinction time distributions.

Findings

Disorder in reaction or mobility rates has little effect on spiral patterns and extinction times.

Higher mobility rates lead to a transition from exponential to linear extinction time dependence.

Extinction time distributions shift from exponential to Gaussian with increased mobility.

Abstract

We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-)steady state in two-dimensional stochastic May--Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May--Leonard system (for small system sizes): (1) As the mobility rate exceeds a threshold that separates a species coexistence (quasi-)steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ~ e^{cN} / N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.