Models on butterfly protandry: virgin females are at risk to die.

Current models on protandry in butterflies assume that females are mated instantaneously upon eclosion. However, for most butterfly species this assumption is not realistic. In this paper a model is formulated in which the mating rate depends on both male and female density. Given the female presence curve, protandry is an evolutionarily stable strategy (ESS) for males. The evolutionarily stable amount of protandry decreases with increasing death rate and decreasing encounter rate. Given the male presence curve, protandry also is an ESS for females. However, male and female ESS are not identical; moreover, in the present model a simultaneous ESS does not exist. Protandry critically depends on the assumption that females mate only once, whereas males are capable of multiple mating. If females too are capable of multiple mating, absence of protandry is the ESS for males as well as females. The model predicts that protandry depends on population density: protandry should be more pronounced in populations with high density than in populations with low density. Protandry also depends on sex ratio. It becomes more pronounced when the proportion of males among emerging adults increases.