We are running out of the very small selection of species of fish that we want to buy. The wast majority of fish belong to those that we don't want to buy, which either get thrown away or sold for almost nothing in order to be smashed into animal feed.
To say that we are running out of fish (generally) is problematic since there is overpopulation problems in lakes near where I live. The problem is that the fish isn't the kind that people want to buy, so fishermen goes where the money is.