Diffusion and pattern formation in spatial games

TL;DR

This paper explores how diffusion influences pattern formation and cooperation in spatial games, specifically using a cellular automaton model of the noisy spatial prisoner's dilemma, revealing that diffusion can promote large-scale structures and cooperation.

Contribution

It introduces a mean-field model capturing predation dynamics and demonstrates how diffusion induces pattern formation that fosters cooperation in stochastic spatial systems.

Findings

Diffusion enables large-scale pattern formation in the CA model.

Patterns promote cooperation even at high error rates.

Coupling diffusion with non-linear dynamics fosters structure and cooperation.

Abstract

Diffusion plays an important role in a wide variety of phenomena, from bacterial quorum sensing to the dynamics of traffic flow. While it generally tends to level out gradients and inhomogeneities, diffusion has nonetheless been shown to promote pattern formation in certain classes of systems. Formation of stable structures often serves as a key factor in promoting the emergence and persistence of cooperative behavior in otherwise competitive environments, however an in-depth analysis on the impact of diffusion on such systems is lacking. We therefore investigate the effects of diffusion on cooperative behavior using a cellular automaton (CA) model of the noisy spatial iterated prisoner's dilemma (IPD), physical extension and stochasticity being unavoidable characteristics of several natural phenomena. We further derive a mean-field (MF) model that captures the 3-species predation dynamics from the CA model and highlight how pattern formation arises in this new model, then characterize how including diffusion by interchange similarly enables the emergence of large scale structures in the CA model as well. We investigate how these emerging patterns favors cooperative behavior for parameter space regions where IPD error rates classically forbid such dynamics. We thus demonstrate how the coupling of diffusion with non-linear dynamics can, counter-intuitively, promote large scale structure formation and in return establish new grounds for cooperation to take hold in stochastic spatial systems.