Spatially explicit non-Mendelian diploid model. (English) Zbl 1195.60125

Lanchier, Nicolas.; Neuhauser, Claudia

Ann. Appl. Probab. 19, No. 5, 1880-1920 (2009).

Summary: 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.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C22	Interacting particle systems in time-dependent statistical mechanics
92D25	Population dynamics (general)

Keywords:

Voter model; annihilating branching process; non-Mendelian segregation

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