Used to describe animals or plants in their natural relationship with insects as a food source. The Venus flytrap is a well-known insectivorous plant.
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Not Human Diet
While it signifies eating insects, it's not commonly used to describe human diets, even if someone eats insects. He eats crickets for protein, but we wouldn't call him insectivorous.
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Scientific Term
Often used in scientific or educational contexts when discussing the diets of certain species. In class we learned that aardvarks are insectivorous mammals.