Populational heritability: Extending Punnett Square concepts to evolution at the metapopulation level

In a previous study, using experimental metapopulations of the flour beetle, Tribolium castaneum, we investigated phase III of Wright's shifting balance process (Wade and Griesemer 1998). We experimentally modeled migration of varying amounts from demes of high mean fitness into demes of lower mean fitness (as in Wright's characterization of phase III) as well as the reciprocal (the opposite of phase III). We estimated the "meta-populational heritability" for this level of selection by regression of offspring deme means on the weighted parental deme means. Here we develop a Punnett Square representation of the inheritance of the group mean to place our empirical findings in a conceptual context similar to Mendelian inheritance of individual traits. The comparison of Punnett Squares for individual and group inheritance shows how the latter concept can be rigorously defined and extended despite the lack of explicitly formulated, simple "Mendelian" laws of inheritance at the group level. Whereas Wright's phase III combines both interdemic selection and meta-populational inheritance, our formulation separates the issue of meta-populational heritability from that of interdemic selection. We use this conceptual context to discuss the controversies over the levels of selection and the units of inheritance.