Origin and persistence of polymorphism in loci targeted by disassortative preference: a general model (Q2105793)

The authors consider an ODE model, \[ z_{i}'(t) = z_{i}(t)\left( b\sum\limits_{j=1}^{k}(1+s_{ij})\frac{z_{j}(t)}{z(t)} -d -cz(t) \right),~~i=1,\ldots,k,~z(t)=\sum\limits_{j=1}^{k}z_{j}(t), \] for a population with a single locus, that has \(k\) possible different alleles. The key assumption about this models is that disassortative crosses are more successful (expressed as the parameters \(s_{ij}\)) than assortative ones. The authors first calculate the unique positive equilibrium, finding the existence conditions for \(k=2,3\) and for a general \(k\), under specific assumptions of the reproduction, i.e., \(s_{ij}\) values. Then, they turn to introducing a mutant into the system and discuss conditions for the co-existence, or extinction of the mutant. Finally, they go further, and model \(s_{ij}\) as a function of a distance between alleles. An allele is considered to be a binary vector of length \(L\), and the distance is a power function of the number of positions on which the two alleles differ. They investigate the stability of systems, where all possible alleles exist, and only the two most different ones; and also the situation when a mutant invades. The study of their ODE system can be done in some cases only numerically.