The phenotypic variance within plastic traits under migration-mutation-selection balance

How phenotypic variances of quantitative traits are influenced by the heterogeneity in environment is an important problem in evolutionary biology. In this study, both genetic and environmental variances in a plastic trait under migration-mutation-stabilizing selection are investigated. For this, a linear reaction norm is used to approximate the mapping from genotype to phenotype, and a population of clonal inheritance is assumed to live in a habitat consisting of many patches in which environmental conditions vary among patches and generations. The life cycle is assumed to be selection-reproduction-mutation-migration. Analysis shows that phenotypic plasticity is adaptive if correlations between the optimal phenotype and environment have become established in both space and/or time, and it is thus possible to maintain environmental variance (V-E) in the plastic trait. Under the special situation of no mutation but maximum migration such that separate patches form an effective single-site habitat, the genotype that maximizes the geometric mean fitness will come to fixation and thus genetic variance (V-G) cannot be maintained. With mutation and/or restricted migration, VG can be maintained and it increases with mutation rate but decreases with migration rate; whereas VE is little affected by them. Temporal variation in environmental quality increases VG while its spatial variance decreases VG. Variation in environmental conditions may decrease the environmental variance in the plastic trait.