ABSTRACT
In this paper we discuss a spatial mechanism for the evolution and maintenance of sexual reproduction. We consider three related models in which sexual reproduction is maintained by parasitism despite the twofold reproductive advantage to pathenogenic females. These models take into account that fact that the populations are spatially extended and that the effective local population size is relatively small. They do not rely on the deterministic cycling of genotypes but on the dynamically produced local stochastic genetic variation. The primary model is a probabilistic cellular automaton. In this, for a wide range of parasite mutation rates, the parasites maintain a spatially genetically heterogeneous population of sexuals and this allows the sexuals to overcome the twofold advantage of asexuals because parasites and their adaption are much less effective in a stochastic spatial genetic structure. We also consider the case where the sexuality rate S (the proportion of the time the host breeds sexually) is slowly evolving. With such slow mutation, we find that both sexual (S=1) and asexual (S=0) populations are evolutionarily stable. We examine two other models which allow us to consider the mathematical conditions under which the advantage of this spatial genetic structure overcomes the twofold advantage of asexual reproduction.