Adult male mandrills develop bare facial skin with a red central stripe over the nose and blue ridged flanges on either side, with the same palette repeated on the rump. The colour is produced by blood flow and pigment in the skin rather than by hair, which is why it can change over weeks rather than over a moult.
Studies of captive and semi-free-ranging mandrills have linked the brightness of that red to circulating testosterone and to dominance rank. Males who rise in the hierarchy brighten; males who are displaced dull down. Females and rival males both attend to the signal, and the brightest males tend to father the most offspring. Body mass follows the same pattern, so colour travels with a broader package of male traits.
What remains debated is how much of the effect is the colour itself and how much is the size, scent and behaviour that accompany it. Mandrills also carry sternal scent glands, and disentangling visual from chemical signalling in a dim forest understorey is difficult.