Is there an idiom or expression for a situation or a part of time that there's an atmosphere of distrust and every one is cheating each other and no one can be able to find a trusthworthy person?
I think such a world can be described as a dog-eat-dog world. — TFD
This might also be considered as a world where anything goes. — TFD
if a situation is dog eat dog, people will do anything to be successful, even if what they do harms other people.
"In showbusiness it's dog eat dog - one day you're a star, the next you've been replaced by younger talent."
The expression "There is no honor among thieves" can sometimes be used in a figurative sense, where "thieves" is understood to mean any disreputable people.
Every man for himself (and the devil take the hindmost) — TFD
Prov. Everyone has to fight for his or her own survival. (You can use this to describe an extremely competitive situation.)
"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."
"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."
"When the ship began to sink, it was every man for himself."