Time scales and wave formation in non-linear spatial public goods games

The co-evolutionary dynamics of competing populations can be strongly affected by frequency-dependent selection and spatial population structure. As co-evolving populations grow into a spatial domain, their initial spatial arrangement and their growth rate differences are important factors that determine the long-term outcome. We here model producer and free-rider co-evolution in the context of a diffusive public good (PG) that is produced by the producers at a cost but evokes local concentration-dependent growth benefits to all. The benefit of the PG can be non-linearly dependent on public good concentration. We consider the spatial growth dynamics of producers and free-riders in one, two and three dimensions by modeling producer cell, free-rider cell and public good densities in space, driven by the processes of birth, death and diffusion (cell movement and public good distribution). Typically, one population goes extinct, but the time-scale of this process varies with initial conditions and the growth rate functions. We establish that spatial variation is transient regardless of dimensionality, and that structured initial conditions lead to increasing times to get close to an extinction state, called ε-extinction time. Further, we find that uncorrelated initial spatial structures do not influence this ε-extinction time in comparison to a corresponding well-mixed (non-spatial) system. In order to estimate the ε-extinction time of either free-riders or producers we derive a slow manifold solution. For invading populations, i.e. for populations that are initially highly segregated, we observe a traveling wave, whose speed can be calculated. Our results provide quantitative predictions for the transient spatial dynamics of cooperative traits under pressure of extinction.Author summary: Evolutionary public good (PG) games capture the essence of production of growth-beneficial factors that are vulnerable to exploitation by free-riders who do not carry the cost of production. PGs emerge in cellular populations, for example in growing bacteria and cancer cells. We study the eco-evolutionary dynamics of a PG in populations that grow in space. In our model, PG-producer cells and free-rider cells can grow according to their own birth and death rates. Co-evolution occurs due to public good-driven surplus in the intrinsic growth rates at a cost to producers. A net growth rate-benefit to free-riders leads to the well-known tragedy of the commons in which producers go extinct. What is often omitted from discussions is the time scale on which this extinction can occur, especially in spatial populations. Here, we derive analytical estimates of the ε-extinction time in different spatial settings. As we do not consider a stochastic process, the ε-extinction time captures the time needed to approach an extinction state. We identify spatial scenarios in which extinction takes long enough such that the tragedy of the commons never occurs within a meaningful lifetime of the system. Using numerical simulations we analyze the deviations from our analytical predictions.