  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.
