When should faster-moving animals have better visual ability? A computational study of Leuckart's law

Background: Leuckart's law postulates that, among vertebrates, swifter animals should have larger eyes with better visual ability. It was supported for mammals but rejected for birds. Question: We ask what are the conditions in which faster-moving animals evolve to have better visual ability? Methods: Computer simulation studies of an animal moving on a plane containing many food items as well as obstacles. The animal moved at a constant speed but changed its directional angle when it recognized food items or obstacles. We examined the number of food items the animal consumed and the number of obstacles it collided with. Results: We assume that visible distance is in proportion to eye size, and that it is accompanied by a small cost. We obtained the optimal visual distance, which depended on movement speed, and densities of foods and obstacles in the field. Conclusions: The positive correlation between movement speed and optimal visual distance was stronger with more obstacles and fewer food items. In contrast, the correlation was weak (Leuckart's law does not hold) when food was abundant, obstacles were rare, and collision damage minimal.