SINGLE-LOCUS GAMETOPHYTIC INCOMPATIBILITY - THE SYMMETRICAL EQUILIBRIUM IS GLOBALLY ASYMPTOTICALLY STABLE

The deterministic dynamics of the classical single-locus multiple-alle le model of gametophytic incompatibility is analyzed with the intentio n to prove the conjecture that the symmetric state (uniform distributi on of genotypes) is the only polymorphic equilibrium and that this equ ilibrium is globally asymptotically stable in the interior of the freq uency simplex. It is shown that the minimum allelic frequency increase s strictly over the generations as long as a uniform allelic distribut ion is not realized. Hence, the minimum allelic frequency is a Ljapuno v function for the invariant set of genotypic frequencies characterize d by a uniform allelic distribution. Within this set, the uniform geno typic distribution is approached in an exponential fashion, which prov es the assertion. An evolutionary optimization rule associated with th e global convergence to the symmetric state is implied by the fact tha t at this state the overall amount of pollen elimination resulting fro m incompatible crosses is minimized.