Where dogs eat dogs and rodents run? (3-4)

I believe the answer is:

rat-race

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'where dogs eat dogs' is the definition.
Although both the answer and definition are singular nouns, I can't understand how one could define the other.

'rodents run?' is the wordplay.
'rodents' becomes 'rat' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'run?' becomes 'race' (synonyms).
'rat'+'race'='RAT-RACE'

'and' acts as a link.

Can you help me to learn more?

(Another definition for rat-race that I've seen is " futile competition".)

I've seen this clue in the Sydney Morning Herald.

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