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

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)
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### References:

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