Dependence of mating rate on variance of eclosion time of cicadas (cicadidae)

In many cicadas, a phenomenon is observed in which the peak of emergence of males precedes that of females (protandry). Generally, it is known that the ecology of emergence of insects is closely related to mating, and protandry is said to be the strategy of males to eclose early and to attempt mating with virgin females. However, as for the cicadas, males only call females with songs, and as far as the behavior of males is concerned, the necessity of protandry is not felt. Thus, as to cicadas, little is known about the relationship between emergence ecology and mating. Therefore, investigating the temporal distribution of cicada emergence is an important step in considering cicadas mating, and even the evolution of mating system. Unfortunately, it is known that cicadas have a very long lifetime and are known to be difficult to keep, so it is meaningful to predict results mathematically. In this paper, we investigate the influence of male and female emergence spreading in appearance period on the mating rate by using mathematical (stochastic) model. Furthermore, we propose a new mathematical virtual model and try to explain the results that mating rate depends on sexual difference and variance of eclosion date.