graze on (something)
1. Of animals, to eat the greens or vegetables in a particular area or location. I put up a fence to try to keep the deer from grazing on my land.
2. Of animals, to eat something in particular. I put up a fence to try to keep the deer from grazing on the vegetables in my garden.
3. Of people, to eat small amounts of food over a period of time, usually instead of full meals. I've been grazing on snacks all day, so I'm not very hungry right now.
4. Of people, to eat a variety of foods in small quantities at one time, instead of a full meal. I'm not too hungry, I think I'll just graze on some snacks at the party.
graze on something
1. [for animals] to browse or forage in a particular location. The cattle are grazing on the neighbor's land. I wish they wouldn't graze on other people's land.
2. [for animals] to browse or forage, eating something in particular. The deer are grazing on my carrots! The cows were grazing on the meadow grasses for weeks.