Evolution of environmentally mediated social interactions under isolation by distance

Many social interactions happen indirectly via modifications of environmental variables, e.g. through the depletion of renewable resources or the secretion of functional compounds. Here, we derive the selection gradient on a quantitative trait affecting the dynamics of such environmental vari-ables that feedback on reproduction and survival in a patch-structured population that is finite, of con-stant size, and subject to isolation by distance. Our analysis shows that the selection gradient depends on how a focal individual influences the fitness of all future individuals in the population through modifications of the environmental variables they experience, weighted by the neutral relatedness be-tween recipients and the focal. The evolutionarily relevant trait-driven environmental modifications are formalized as the extended phenotypic effects of an individual, which quantify how a trait change in the individual in the present affects the environmental variables in all patches at all future times. When the trait affects reproduction and survival through some payoff function, the selection gradient can be expressed in terms of extended phenotypic effects weighted by scaled-relatedness coefficients. We show how to compute extended phenotypic effects, relatedness, and scaled-relatedness coefficients using Fourier analysis, allowing us to investigate a broad class of environmentally mediated social in-teractions in a tractable way. We illustrate our approach by studying the evolution of a trait controlling the costly production of some lasting commons (e.g. a common-pool resource or a toxic compound) that can diffuse in space. We show that whether selection favours environmentally mediated altruism or spite depends on the spatial correlation between an individual’s lineage and the commons originat-ing from its patch. The sign of this correlation depends on interactions between dispersal patterns and the commons’ renewal dynamics. More broadly, we suggest that selection can favour a wide range of social behaviours when these are mediated in space and time through environmental feedback.