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Sarcoptic Mange
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Mange is caused by mites of the Sarcoptes species. In rabbits, the disease starts around the borders of the ears, edges of the eyelids and in the nose, mouth and toes. The crusts are whitish or beige, often having a musky smell, especially in the ears. The medicine used to treat mange is called selamectin (usually sold under the brand name Revolution) or ivermectin.
Fur Mites
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Cheyletiella parasitivorax is a fur mite which causes fur to fall out, leaving bald or tufty patches. Fur loss is generally around the face, back and neck. Fur mites can be eliminated with selamectin (Revolution) or ivermectin.
Dental Problems
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Fur loss restricted to the area just under the chin, the dewlap (the flap of skin under the chin) and chest may be caused by molar spurs or abscess. Dental problems such as these cause the bunny to drool. As saliva is caustic, it burns the skin and creates a wet, itchy, sore area, causing the fur to fall out. If you suspect dental issues, take your bunny to a rabbit-savvy veterinarian immediately.
Overgrooming
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If you have bonded bunnies, the problem could be that one bunny is over-zealously grooming the other. Bunnies often groom each other around the eyes and on the head. Grooming is normal and healthy, but over-grooming is usually a sign of boredom. Your rabbits may need more out-of-cage time, or more variety in their toys. They may need a more spacious living area, and enough space and hiding places that the rabbits can get away from one another when they so desire.
Fungal Infection
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This is most typically encountered in outdoor "hutch" rabbits in the high humidity of summer, when a wet face or dewlap from drinking from a water bowl causes chronic moist dermatitis. The resulting fungal or bacterial infection may cause loss of fur on face, dewlap and chin. Iodine or a prescribed antifungal cream like Miconazole Nitrate applied to affected areas will stop the infection.
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Hair Loss on the Face of a Bunny
There are several reasons why a bunny may be losing fur on its face, and most of them require medical treatment. There are different kinds of mites that can affect bunnies, or the problem could be an infection or related to dental issues or behavioral problems. If your rabbit is losing fur on its face (or anywhere else) it is up to you, the rabbit's keeper, to fix the problem as soon as possible.