No, without owners, if the demand for fish is more than the fish can supply themselves, the number of fish will go down. If the fish have owners, the owners have an incentive to: 1) grow more fish, and 2) not let the number of fish be less than the amount that makes the most money.
A wise man once said something like, "Both jayhawks and humans eat chickens. But if you have more jayhawks, you have less chickens. If you have more humans you have more chickens." This bit of wisdom is absolutely true for chickens...because people OWN chickens. It's NOT true for fish...UNLESS people own the fish.
Interesting comments, Mark. Question... Are fish farms booming? Any data on the rate of growth of the fish-farming business? What with it being (to my mind, anyway) hugely more efficient than at-sea, boat-based fishing, I think it would be "growth industry" defined!