Extinction times in diffusive public good population dynamics

The co-evolutionary dynamics of competing populations can be strongly affected by frequency-dependent selection and population structure in space. As co-evolving populations grow into a spatial domain, their initial spatial arrangement, as well as their growth rate differences determine the dynamics. Here, we are interested in the dynamics of producers and free-rider co-evolution in the context of an ecological public good that is produced by a sub-population but evokes growth benefits to all individuals. We consider the spatial growth dynamics in one, two and three dimensions by modeling producer cell, free-rider cell and public good densities in space, driven by birth, death and diffusion. Typically, one population goes extinct. We find that uncorrelated initial spatial structures do not influence the time to extinction in comparison to the well-mixed system. We derive a slow manifold solution in order to estimate the time to extinction of either free-riders or producers. For invading populations, i.e. for populations that are initially highly segregated, we observe a traveling wave, whose speed can be calculated to improve the extinction time estimate by a simple superposition of the two times. Our results show that local effects of spatial dynamics evolve independently of the dynamics of the mean populations. Our considerations provide quantitative predictions for the transient dynamics of cooperative traits under pressure of extinction, and a potential experiment to derive elusive details of the fitness function of an ecological public goods game through extinction time observations. Author Summary Ecological public goods (PG) relationships emerge in growing cellular populations, for example between bacteria and cancer cells. We study the eco-evolutionary dynamics of a PG in populations that grow in space. In our model, public good-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 and 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. We derive analytical estimates of the time to extinction in different spatial settings, and identify spatial scenarios in which extinction takes long enough such that the tragedy of the commons never occurs within the lifetime of the populations. Using numerical simulations we analyze the deviations from analytical predictions. Our results have direct implications for inferring ecological public good game properties from in vitro and in vivo experimental observations.