Reproductive wastage and the evolution of genetic systems.

An outstanding problem for evolutionary biology is the maintenance of anisogamous sex in the face of the ostensible twofold reproductive advantage accruing to parthenogenetic females that do not produce male offspring (Williams, 1975; Maynard Smith, 1978; Bell, 1982). Conventional solutions are sought in terms of cryptic advantages to sex involving variable environments, genetic load, and DNA damage (Rose & Redfield, 1988). An alternative point of view is that efficient parthenogenesis is not intrinsically deleterious, just difficult to achieve directly by a single mutation (White, 1978, p. 294; Uyenoyama, 1984). In addition, it has been proposed that new parthenogens face the problem of reproductive wastage, in which sexual males fertilize parthenogenetic eggs and thereby induce lethal triploidy (White & Contreras, 1979; Lynch, 1984; Krieber & Rose, 1986). In effect, parthenogenetic variants could be excluded by reproductive wastage extrinsic to their genetic or evolutionary efficiency when free from fertilization. An interesting feature of the analysis of reproductive wastage by Krieber & Rose (1986) was that numerical solutions of the model equations rarely gave polymorphism of genetic systems. In most cases, sex was fixed or parthenogenesis was fixed. However, this was not an analytical result. Hickey & Rose (1988) suggested that population dynamic models like those of Rose (1983) or Tremblay & Rose (1985) might capture the essential features of this evolutionary interaction. Given the absence of genetic exchange between parthenogenetic and sexual forms, their evolutionary interaction becomes like that of two competing populations, complicated by the reproductive wastage that occurs when they mate. Let x represent the density of sexual form and y represent the density of the parthenogenetic derivative. Let r(x) be the density-dependent reproductive output of the sexual form, with q ( y ) the analogous function for the parthenogenetic form. Let cl(y) represent the competitive effect of the parthenogen on the sexual form and c2(x) represent the competitive effect of the sexual form on the parthenogen. Let fl(t) represent the effect of reproductive wastage on the sexual form and f2(x) represent the effect of reproductive wastage on the parthenogen. We assume that r and q are strictly decreasing functions, with r(0)> 0, q(0)> 0, and K~ such that r(K~) = q(K2) = 0. We also take all ci and f to be strictly increasing, taking on zero values when their arguments are zero. With these definitions, our model becomes