Space mediates coexistence of females and hermaphrodites

In gynodioecious populations of flowering plants females and hermaphrodites coexist. Gynodioecy is widespread and occurs in both asexual and sexual species but does not admit a satisfactory explanation from classical sex ratio theory. In sexual populations male fertility restoring genes have evolved to counter non-nuclear male sterility mutations. In pseudogamous asexual populations pollen retention and increased self-fertilization can make male sterility costly. Both of these mechanisms can promote coexistence. However, it remains unclear how either of these mechanisms could evolve if coexistence was not initially possible. In the absence of these adaptations non-spatial models predict that females either fail to invade hermaphrodite populations or else displace them until pollen shortage drives the population to extinction. We develop a pair approximation to a probabilistic cellular automata model in which females and hermaphrodites interact on a regular lattice. The model features independent pollination and colonization processes which take place on different timescales. The timescale separation is exploited to obtain, with perturbation methods, a more manageable aggregated pair approximation. We present both the mean field model which recreates the classical non-spatial predictions and the pair approximation, which strikingly predicts different invasion criteria and coexistence under a wide range of parameters. The pair approximation is shown to correspond well qualitatively with simulation behaviour.