Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations*

Stochastic reaction-diffusion (SRD) systems serve as versatile models for many complex physical, societal, and ecological systems. The effective coarse-grained reaction rates in continuum descriptions for such systems represent macroscopic parameters that need to be either measured experimentally or determined numerically. In an agent-based Monte Carlo simulation of SRD systems, the control parameters are the prescribed microscopic probabilities for certain events to happen. They ultimately define the large-scale behavior and long-time states of the system, as well as relaxation rates and other relevant time scales such as oscillation frequencies. To match the results of numerical simulations to experiments, a mapping is required between the microscopic probabilities that define a Monte Carlo simulation and the macroscopic reaction rates. This constitutes in general a non-trivial problem, and there exists no systematic method to obtain the functional dependence of the macroscopic rates on the microscopic probabilities and interaction rules. Here we introduce an algorithmic approach using Monte Carlo simulations to evaluate the macroscopic reaction rates by counting how many events occur per simulation timestep. Our technique is first tested on known simple examples such as simple birth reactions, coagulation, and pair annihilation. We then investigate how the microscopic reaction probabilities become coarse-grained into macroscopic rates in more complicated models such as the Lotka-Volterra predator-prey model, the rock-paper-scissors or cyclic Lotka-Volterra model, and the May-Leonard model for cyclic competition of three species. This work aims towards a deeper understanding of coarse-graining in SRD systems with a focus on ecological systems, and improved Monte Carlo simulation techniques to fit experimental or observational data.