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Supplementary Information:

Appendix S1. Solutions to equation 4; Stability analysis of equation 4; Invasion of an AB-linked mating type: stability of the equilibrium; Invasion of an AB* mating type that has lost its self-incompatible function: stability of the equilibrium.

Figure S1. Dynamics of linkage disequilibrium in bifactorial population.

Figure S2. Trajectories of mating-type genotypes frequencies when a new AB-linked mating-type genotype arises in bifactorial population when r = 0.001 and α = 0.

Figure S3. Trajectories of linkage disequilibrium when a new AB-linked mating-type genotype arises in a bifactorial population when r = 0.001 and α = 0.

Figure S4. Trajectories of linkage disequilibrium when a new AB-linked mating-type genotype arises in a bifactorial population when r = 0.5 and α = 0.

Figure S5. Trajectories of mating-type genotypes frequencies when a new AB-linked mating-type genotype arises in a bifactorial population. r = 0.5 and α = 0.5.

Figure S6. Trajectories of mating-type genotypes frequencies when a new AB-linked mating-type genotype arises in a bifactorial population. r = 0. 001 and α = 0.5.

Figure S7. Trajectories of the changes in frequency of mating-type genotypes when a new mating type in which incompatibility function is lost at the MAT B locus arises in a bifactorial population and when there is no fitness reduction (α = 0).

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