out of food

Both are correct, though they don't mean the same thing. The first sentence means he still has some food but will soon have none. The second sentence he has none.

Thanks, Shapeless.

He is running out of food.
He has run out of food.

He is out of food and He has run out of food hold slightly different meanings.

The first tells us that he is in a state of having no food and the second focuses on the action of running out of food.

Maybe the phrase "out of" has a slightly different meaning in the US, (in any case it is rather conversational/informal/colloquial), but in British English being "out of" something can definitely carry the implication that you (now, in the present) do not have something that you (formerly, in the past) used to have, and were accustomed or expected to have.

The shop is out of toilet paper. The bus driver was out of small change. We must drink our tea black because we are out of milk.

I'd say that the meaning is the same, Contrex. Anyway, the situations surrounding this adjective/verb are all conversational/informal/colloquial.

Thanks, fellow members, for your guidance.