The spatio-temporal dynamics of interacting genetic incompatibilities. Part I: the case of stacked underdominant clines

We explore the interaction between two genetic incompatibilities (underdominant loci in diploid organisms) in a population occupying a one-dimensional space. We derive a system of partial differential equations describing the dynamics of allele frequencies and linkage disequilibrium between the two loci, and use a quasi-linkage equilibrium approximation in order to reduce the number of variables. We investigate the solutions of this system and demonstrate the existence of a solution in which the two clines in allele frequency remain stacked together. In the case of asymmetric incompatibilities (i.e. when one homozygote is favored over the other at each locus), these stacked clines propagate in the form of a traveling wave. We obtain an approximation for the speed of this wave which, in particular, is decreased by recombination between the two loci but is always larger than the speed of “one cline alone”.