There are a lot of phrases that denote competition for limited resources, but much fewer that specify a single one.
"Winner take all." is the best I can think of that means a singular victor and prize. (There are others but they are not appearing to mind...)
"Cut-throat competition" refers to competitors being extremely serious about beating each other. (literally, it means they'd kill to win) This doesn't mean anything about the number that can win.
"Pigs at a trough" is a related phrase to this, but increased distance from rural life makes it less common anymore. It references the shouldering competition at feeding. That being the image, though, it implies multiple potential 'winners'.
Many terms indicate that there is no arbiter of fairness for the competitors,
"Catch as catch can." means those involved have to struggle to get anything. Can imply that multiple opportunities to progress are available but certainly not guaranteed.
"Every man for himself." means the competitors aren't interested in helping the others. You're on your own.
"First come first served." means whoever claims the stuff first gets theirs. Generally implies no arbitration to keep things fair.
These are often combined: "It's winner-take-all and every man for himself out there." would essentially say the same thing.
You could also just use the phase "Dogs fighting over a bone" and you'd generally be understood.
Just be aware that "A dog on a bone" is a phrase for working on something. It's singular, though, which makes the differences fairly obvious.