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Function of tabanid eye patterns?

Does anyone have any idea why deer flies and horse flies have such boldly patterned eyes?

Hey Charley
Check out this study.

I had found that. Too bad there's not an abstract--I'm not sure I want to pay $35 to read the rest of the article.

the study Andrew found
is relevant, but it more seems to discuss how the colored eyes might change what they see, in other words how they function. They can only speculate the purpose the colored eyes serve. The question is a very interesting one, and little research has been done so I'll do a little summarizing and 'handwaving'

First, let's consider the distribution of eyes with complex color patterns in horse flies. The patterns are generally more pronounced in females than males. This suggesting that while complex patterns might be connected to courtship or mate finding, they probably are more important for hunting hosts for blood meals. The eye patterns tend to be conserved at higher levels. Nearly all Chrysops have swirled eye patterns, nearly all Hybomitra (as pictured above) have four green stripes, nearly all Haematopota have 'zig zags', nearly all Silvius have 'polka dots.' Horse flies with complex eye patterns tend to be more diverse (often used as a proxy for success) than those without; the four most diverse horse fly genera are Chrysops, Haematopota, Hybomitra, and Tabanus. This suggests that eye patterns confer some advantage.

The way they are thought to work is that the colored eyes filter out or alter their respective colors. It's the same effect if you've ever had to wear colored ski goggles, for instance. So for horse flies with green eyes, the green of the background vegetation fades out, making their darker colored prey stand out more. Horse fly eyes with two different colors might be analogous to bifocal lenses, letting the fly choose to blank out certain colors as appropriate. So for the Hybomitra you link, it might rely more on the green ommatidia in brightly lit open areas, and the brown lens for darker areas. This doesn't quite explain the swirls found in the most complex eye patterns.

Many other groups of flies with varied ecologies have these patterns, such as Eristalinus or Limnia. While the patterns may still useful for host finding, my explanation doesn't work as well for them.

Taking hints from the first page . . .
it seems that the striped areas are differentially sensitive to wavelengths in the (fly's) visible spectrum. So the answer would be, the purpose is color vision.

Maybe these flies don't have rods and cones?

no arthropod has rods and/or cones
their eyes are not homologous to vertebrates'
the stripes must be a product of the outer cuticlar layer, while eye pigments responsible for color vision lie much deeper in the ommatidia

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