Aphids damage plants by munching on them, ladybugs in turn eat the aphids, saving the plant.
Aphids are a very common plant and crop pest; they are small insects with long skinny mouths that they use to pierce plants' stems and leaves in order to suck out the plant's fluids. The relationship between the plants and aphids is not a mutually beneficial one but rather one in which one benefits and the other is harmed. In the biology world this is known as parasitism. Parasites, although harmful to their host, tend not to kill it because in order to survive their host needs to be alive.
Although the plant is being damaged by the aphids, it has the ability to make "friends" with things that are harmful to aphids, an example, ladybugs. This new relationship between the plants and ladybugs is called mutualism. The ladybugs get a full meal and the plants are rid of the aphids, it's a win-win!
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Flint, M. L. "How to Manage Pests." Aphids Management Guidelines--UC IPM. N.p., n.d. Web. 05 Sept. 2016.