This article develops mathematical theory for the population dynamics of microbiomes with their hosts and for holobiont evolution caused by holobiont selection. The objective is to account for the formation of microbiome-host integration. Microbial population-dynamic parameters must mesh with the host's for coexistence. A horizontally transmitted microbiome is a genetic system with "collective inheritance". The microbial source pool in the environment corresponds to the gamete pool for nuclear genes. Poisson sampling of the microbial source pool corresponds to binomial sampling of the gamete pool. However, holobiont selection on the microbiome does not lead to a counterpart of the Hardy-Weinberg Law nor to directional selection that always fixes microbial genes conferring the highest holobiont fitness. A microbe might strike an optimal fitness balance between lowering its within-host fitness while increasing holobiont fitness. Such microbes are replaced by otherwise identical microbes that contribute nothing to holobiont fitness. This replacement can be reversed by hosts that initiate immune responses to non-helpful microbes. This discrimination leads to microbial species sorting. Host-orchestrated species sorting (HOSS) followed by microbial competition, rather than co-evolution or multi-level selection, is predicted to be the cause of microbiome-host integration.
