every man for himself

(It's) every man for himself (and the devil take the hindmost). {букв. каждый за себя, и (пусть) дьявол схватит последнего}
Also
(The) devil take the hindmost.
(everyone has to fight for his or her own survival; each person here should do what is best for themselves) ≈ Каждый за себя (— и к чертям отстающих.); Прежде всего нужно думать о своей шкуре, нужно спасать свою шкуру, действовать в своих интересах.

Often used to describe an extremely competitive situation.

Example: When the ship began to sink, it was every man for himself. (McGraw-Hill Dictionary of American Idioms and Phrasal Verbs)

Example: In the world of politics, it's every man for himself. (Cambridge Dictionary of American Idioms)

Example: At first we tried to help each other study for the exam, but soon it was every man for himself, and the devil take the hindmost. (thefreedictionary.com)

Example: The inventors tried to collaborate, agreeing to share the profits from their invention, but they grew so suspicious of each other that each began to work separately, and devil take the hindmost. (thefreedictionary.com)

Related vocabulary:
[look after number one]
look/be only out for oneself
[шкурный интерес]
[своя рубашка ближе к телу]
каждый за себя