There is evidence to suggest this is what the Mayans did, well the ones that survived did.

http://en.wikipedia.org/wiki/Classic_Maya_collapse

I guess the time constant for any civilisation is based on the rate of unsustainability. Technological advances may delay the abandonment of certain overpopulated areas somewhat but you if you cannot square the circle then disaster is sure to arise, it just takes time.

The abandonment of a city like Los Angeles, would be just as quick as say New Orleans was if the taps ran dry. i.e. about a week.