Evolution of eye in vertebrates is likely to be affected by locomotor speed. There is a law that states the relationship between moving speed and eye size named Leuckart’s Law: swifter moving animals should have larger eyes. However, there are only a limited number of experimental studies on Leuckart's Law: Hall and Heesy (2011) concluded that there was no correlation between flying speed and eye size among birds. In contrast, Heard-Booth and Kirk (2012) observed a positive correlation between running speed and eye size among mammals.
We conducted computer simulation studies. We considered an animal moving on a plane, in which there are food items as well as obstacles and examined the number of food items it consumes and the number of obstacles it collides together. Considering a small cost of visual acuity, we obtained the optimal visual acuity for different values of parameters -- visible distance, visible angle, turning ability, moving speed, the densities of foods and obstacles in the field.
The analysis reveals that Leuckart’s Law should hold if animals with similar densities of food items and obstacles are compared. The more there are obstacles or food items, the stronger the positive correlation between moving speed and visual distance is. However Leuckart’s Law will not be supported if those animals with significantly different conditions are compared. The difference in the comparative studies may be explained if birds are more diverse than mammals concerning these conditions.