There are two ways to die. One is by the gradual deterioration of bodily function in terminal illness or senescence. The other is by suffering a trauma that causes a sudden collapse in bodily functions. The neomalthusian notion of an ecological collapse of modern civilization follows the latter analogy.
The collapse is deemed to be self-induced, hence the related notion of "overshoot": a society is able to exhaust its resource base at high levels of activity and consumption; then exhaustion is reached, leading to a sudden drop in consumption and population size. The standard reference for this phenomenon is still the 1970s work The Limits to Growth , which has recently been updated.
Economic theory isn't very welcoming of the concept of overshoot-collapse. Rather, the price system would ration out a disappearing (exhaustible) resource. As it gets scarcer, it gets harder to extract, so the cost and price go up, making people skimp on it more. More than that: if conditions of scarcity become certain, such that future prices are sure to increase, owners of long-term rights to the resource would (as rational decisionmakers) hold off on extracting a lot today in anticipation of better prices next year (or decade or...)
Think of oil. It is said that the Saudi's are just extracting the oil as fast as they can, to create the illusion of big reserves. Nonsense. If peak oil has been reached then oil prices are on a long term upward trend. The owners would therefore keep their oil extraction in check. (If you held reserves to a trillion barrels of oil, wouldn't you?)
Now two things can happen: either technological change succeeds (under the whip of high resource prices) in finding abundant substitutes; or it fails, and society lives with escalating prices, converging to what would in practice mean zero extraction of the resource. (Think of petrol at US$ 1,000 per liter.) The latter scenario is the whimper version of society's demise - a long slow adjustment back to near pre-industrial levels of production, consumption, and population. (Hey, nearly two billion people in the planet are already at this standard of living!) Interestingly, the whimper version gives a lot of time for society to undertake the social and technological innovations to deal with tightening resource scarcity. The bang version obviously doesn't.
So how will our civilization end?
Between the conception
And the creation
Between the emotion
And the response
Falls the Shadow
Life is very long
Between the desire
And the spasm
Between the potency
And the existence
Between the essence
And the descent
Falls the Shadow
For Thine is the Kingdom
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