| Abstract: | In his mathematical treatment of Fisher's ideas on sexual selection (so-called runaway selection) Lande (1981) predicted that males may evolve increasingly elaborate sexual characters despite opposing viability selection as a consequence of the associated costs. Lande thereby assumed that female mate preferences are not subject to selection since (1) females are all inseminated and (2) the quantity and quality of their offspring are independent of the female's mate preferences. Kirkpatrick (1985) removed the latter assumption and investigated the consequences for the mean phenotype with respect to both female and male traits. He also explored the dynamics of the (co)-variance matrix by numerical methods. In this paper we consider a simpler model with just two multi-allelic loci. This enables us to derive explicit expressions for (co)-variances under steady state conditions. Rather than assume natural selection through differential fertility (as in Kirkpatrick, 1985), we take sexual selection on females into account by modelling the preference-dependent risk that females remain unmated. We argue that this wallflower effect is a realistic feature of any mating system, since it merely depends on the existence of (1) variation in mating preferences and (2) a finite mating season. Our approach provided an insight into the dynamic behaviour of the means of the phenotypes. This is because the dynamics of the means depend on the steady state (co)-variance matrix. Thus, an insight into the former requires explicit expressions for the latter. Whereas Lande and Kirkpatrick predicted runaway processes, despite opposing viability selection, our model predicts a globally stable steady state, i.e. no runaway, even without opposing viability selection (under the assumption of an asymptotically stable steady state of the (co)-variances. Admittedly, we have no analytic proof of this stability but only support for it, based on simulations.) The absence of the runaway processes in our model is caused by the wallflower effect, since it imposes constraints on the steady state of the (co)-variance matrix. When mutational input applies to female traits but not to male traits, explicit expressions for the (co)-variances under steady state conditions can be derived, and these show that: (1) both the genetic covariance and the variance of male traits are equal to zero, but (2) the variance of the female trait exceeds to zero. Should there be mutational input influencing the male trait, then these results would suggest that the male-to-female ratio of variances is much smaller than unity. This prediction is of tremendous importance for speciation through founding events. |
