Theoretical support for balancing selection and reduced diploid male production in multiple-locus complementary sex determination

            In species where sex is determined through the haplo-diploid mechanism called complementary sex determination (CSD) diploid zygotes heterozygous at sex- determining loci develop into normally-functioning females but homozygous diploids develop into males, which are often sterile or unviable.  In species with a single CSD locus, this form of inbreeding depression results in the maintenance of many CSD alleles through a form of balancing selection.  It is unclear if comparable balancing selection exists in species with two or more CSD loci.  Here we use a combination of theoretical approaches to ask whether balancing selection maintains multiple CSD alleles at multiple loci in populations. We first consider an infinite population with ml-CSD and show that balancing selection acts on all alleles at all loci.  We use a comparable individual-based simulation model to show that in finite populations experiencing drift the long-term fixation of a single allele at one or more CSD loci in a ml-CSD population is unlikely in all but very small populations.  We also show that populations with multiple CSD loci should have lower rates of diploid male production than comparable single-locus CSD populations, suggesting that multiple-locus CSD is potentially adaptive.