Search Paths of Fireflies in Two Dimensions

In this study, the investigator asks, "How are the flight patterns of fireflies adapted for finding a mate?" The study compares the flight patterns of two closely related fireflies: Photinus collustrans and Photinus tanytoxus, both of which are active in southern Florida from May to November.
In both species, males fly in search of flightless females, who are responsive to the males during only a short period each evening. Both species are found in relatively open habitats and may be found in the same location, but they are distinguished by their activity at different times of the evening.

P. collustrans begins its activity between 11 and 26 minutes after sunset. Males flash once every 2.2 seconds at 71.6 degrees. Females return a single flash 0.5 seconds after the male flash from their location on the ground near their burrows. After engaging in this activity for about 15 minutes, P. collustrans abruptly goes inactive.
Shortly after P. collustrans has ended its activity for the night, P. tanytoxus becomes active for 35 to 70 minutes. Its flash patterns are nearly identical to P. collustrans, but the primary difference is that the female response time is 1.5 seconds.
Since P. collustrans is active only for a 15 minute period each evening, competition for females is intense. On any given night, a male has only a 13% chance of finding a female. Therefore, male flight patterns must be adapted to be as efficient as possible. And since finding a mate is a two-way process (the male flashes and the female responds) the male must fly in a a pattern that maximizes the chance that a female will see him and respond.
It has been theorized that the most efficient search pattern would be one of a straight line $#151; flying in a series of straight paths from one side of the search area to the other. However, most animals do not search in this way; their paths are influenced by outside factors such as environmental features, the most likely habitat for females, etc. But fireflies, which fly when many of these environmental features are hidden by the dark, show a straighter search pattern than most animals.
Both P. collustrans and P. tanytoxus males show a common pattern of change in their flight patterns throughout their activity periods. In both cases, the males exhibit an increasingly straight flight path as the night wears on. It is easy to understand why this happens with P. collustrans, which begins its flight shortly after sunset, when there is still light to see the environmental features. As it becomes darker, these visual cues become less visible, influencing the flight pattern less and resulting in a straighter path.
Given the influence of available light on flight direction, one would expect the late-emerging P. collustrans and early P. tanytoxus to have very similar flight patterns, since they fly at about the same time. However, the early P. tanytoxus have a less straight-line flight than the late P. collustrans. Clearly, something other than amount of available light is affecting their flight patterns.
One condition that is different for the late-emerging P. collustrans and early P. tanytoxus is the availability of females. Since females are continually being found and mated with during the activity period, there are fewer females available at the end of the activity period. This means that late P. collustrans are searching for more thinly distributed females than early P. tanytoxus. There is some evidence that suggests that the search patterns are affected by the "expected" distribution of females. In areas with few searching males, or during times late in the activity period with fewer females responding to the males, the males tend to fly in a more direct path to increase their chances of finding widely scattered isolated females.