Modeling alternative strategies of male emergence timing

In many butterfly species, males emerge earlier than females as part of a strategy to maximize male reproductive success. Although behavioral ecological studies using mathematical models have been conducted to explain this phenomenon, certain emergence patterns remain unexplained. In the butterfly species Fabriciana nerippe, some males emerge at the same time as females, in addition to males that emerge earlier than the females. However, it is unclear what emergence patterns occur in populations with male dimorphism, as observed in this species. In this study, we showed the existence of male body size dimorphism in Fabriciana nerippe by conducting a comparative analysis of forewing lengths between males and females. In addition, we developed a comprehensive mathematical model to investigate emergence patterns in the presence of dimorphic males. By introducing a trade-off between large size and early emergence, the model considered a scenario where small early-emerging and large late-emerging males could coexist. Numerical analysis demonstrated the emergence patterns of these two male types with a switch in emergence time. Furthermore, the higher the death rate before emergence, the earlier the emergence switch. These findings suggested that the timing of the switch depends on the death rate and is influenced by environmental factors. This work contributes to ecological and theoretical studies on two types of emergence timing in life-history strategies across a broader range of species.