It is well known that human cannibals are likely to suffer from a variety of ailments, particularly prion diseases.
However a great many other species, from insect to ape practice cannibalism at much higher rates.
Do they suffer from similar diseases as a result of their cannibalism? If not why not? and do they have evolved mechanisms for dealing with these negative effects?
Some of the processes working against cannibalism:
However, a theoretical argument by Rudolf and Antonovics (2007) shows that
... group cannibalism, i.e. shared consumption of victims, is a necessary condition for disease spread by cannibalism in the absence of alternative transmission modes. Thus, endemic diseases transmitted predominantly by cannibalism are likely to be rare, except in social organisms that share conspecific prey. These results are consistent with a review of the literature showing that diseases transmitted by cannibalism are infrequent in animals, even though both cannibalism and trophic transmission are very common.
That is, kuru and bovine spongiform encephalopathy ("mad cow disease") are anomalies; they spread effectively because tissue from diseased individuals was eaten by many other individuals - in contrast to the more common ecological situation where a single cannibal eats a single diseased conspecific.
First, to be clear, the very act of cannibalism does not cause the development of prions, but the consumption of an animal infected with a compatible form of them, particularly when eating nervous or digestive tissue (i.e. the brain, spine, stomach, or intestines).
To directly answer your question, yes, there are other animals that are susceptible to prions diseases when they consume infected members of their own species, or members of compatible carrier species. In the late 80s and 90s, there was a moderate epidemic of Bovine Spongiform Encephalopathy (Mad Cow Disease) that was concluded to have been caused by including bovine parts in cow feeding stock.
A common defense that I know of against infection from the consumption of infected animals is to simply not consume them at all. I know it's not really the answer you were expecting, but indeed, many animals have an active aversion to corpses, notably perceiving the smell of Cadaverine and Putrescine as repulsive. Evidently, a great defense against disease is to just not eat dead or dying things.
I've never heard of insects or arthropods having neurodegenerative prions, and wasn't able to find anything alluding to them having them, but I wouldn't be too surprised if they had other prion diseases that were infectious between them.
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