Spatially explicit non-Mendelian diploid model

Abstract

We introduce a spatially explicit model for the competition between type $a$ and type $b$ alleles. Each vertex of the $d$-dimensional integer lattice is occupied by a diploid individual, which is in one of three possible states or genotypes: $aa$, $ab$ or $bb$. We are interested in the long-term behavior of the gene frequencies when Mendel's law of segregation does not hold. This results in a voter type model depending on four parameters; each of these parameters measures the strength of competition between genes during meiosis. We prove that with or without a spatial structure, type $a$ and type $b$ alleles coexist at equilibrium when homozygotes are poor competitors. The inclusion of a spatial structure, however, reduces the parameter region where coexistence occurs.