On the evolution of dominance modifiers I. A nonlinear analysis.

A well known mathematical model of evolution of dominance is subjected to a nonlinear analysis. For the case where the primary locus and the modifying locus are completely linked (r = 0) a global Ljapunov function is given. This proves that selection of dominance modifiers entirely due to their modifying effect is possible. This result is also extended to small recombination fractions r by using the method of Ljapunov functions in a more sophisticated way. For r = 0 and mu = 0 a lower bound for the success of selection of the modifier is given. Furthermore, the influence of the dominance relations between the alleles (measured by the parameters h and k) is investigated. Finally it is shown that differential and difference equations lead to the same results (which need not be the case in general). As a by-product we obtain a new equilibrium point in the classical one locus selection-mutation model.