Finite size effects in diffusive public goods games

Abstract

A particular challenge in the co-evolutionary dynamics of growing populations is to determine the effects of population assortment on growth advantage of cooperative traits. Here we study the coevolution of cooperators that are public good producers and consumers, and free riders that are only public good consumers. Using an individual-based model and its continuum limit we show that the differential growth rate between producers and free-riders depends on the spatial Fourier decomposition of their respective densities. Further, we derive a finite-size correction, which scales as the cell size, which accurately describes the dynamics of a randomly assorted population. Using this result we are also able to derive an effective benefit to cost relation, which provides a criterion that relates the parameters of the model (e.g. diffusion constant of the public good, decay rate and cost) to the long-term persistence of cooperation. Our findings provide a powerful tool for the analysis of diffusive public goods games, and can explain commonly observed patterns of cooperation, rather than the tragedy of the commons, based on the derived finite-size correction.