Does competitive divergence occur if assortative mating is costly?

Most models of sympatric speciation have assumed that assortative mating has no costs. A few studies, however, have shown that the costs for being choosy can prevent such speciation. Here, we investigate the role of the strength of assortment and of the costs for being choosy for a simple genetic model of a single ('magic') trait that mediates both intraspecific competition for a continuum of resources and assortative mating, which is induced by choosy females who preferentially mate with males of similar phenotype. Choosiness may be costly if it is difficult to find a mating partner. Such magic trait models are considered to be most conducive of sympatric speciation. We consider a sexually reproducing population of haploid individuals that is density regulated. The trait is determined by a single locus with multiple alleles. The strength of stabilizing selection (caused by a unimodal resource distribution), the strength of competition, the degree of assortment and the costs for being choosy are independent parameters. We investigate analytically and numerically how these parameters determine the equilibrium and stability structure. In particular, we identify conditions under which no polymorphism at all is maintained as well as conditions under which strong competitive divergence occurs, or the population even splits into two reproductively isolated classes of highly diverse phenotypes. If costs are absent or moderate, genetic variability tends to be minimized at intermediate strengths of assortment, and reproductively isolated classes of phenotypes are a likely result of evolution only for intermediate or strong competition and for very strong assortment. The likelihood of divergence depends relatively weakly on the costs as long as they are not high. With high costs, however, increasingly strong assortment rapidly depletes all genetic variation, and strong competitive divergence is prevented.