"It is an obvious truth, which has been taken notice of by many writers,  that population must always be kept down to the level of the means of subsistence; but no writer that the Author recollects has  inquired particularly into the means by which this level is effected..."
  -- Thomas Malthus, 1798 An Essay on the Principle of Population 

Often, there is a cyclical relationship between the populations of predators and their prey.  This keeps the populations of both species in check.

But, what happens when there are no predators?

This issue was addressed in a paper by David Klein,  "The Introduction, Increase and Crash of Reindeer on St. Matthew Island."  Klein reported that in 1944, 29 reindeer were brought to
St. Matthew Island. Initially there were abundant food sources, and the reindeer population increased dramatically. There were no predators to cull the population.

About 20 years after they were first introduced, the reindeer had overshot the food carrying capacity of the island, and there was a sudden, massive die-off.  About 99% of the reindeer died of starvation.

As shown in the graph below, this is an example of a general phenomenon.  All species suffer population collapse or species extinction if they overshoot and degrade the carrying capacity
of their ecology.

This is also the fate that awaits bacteria growing in a Petri dish, as you might remember from your high school biology course.  Imagine a Petri dish with enough nutrients to support a growing
bacteria culture until the dish is completely full of them.  One bacterium is placed inside the dish at 11:00am, and the population of bacteria doubles every minute -- such that the Petri dish will be full by noon.

At what time will the Petri dish be half full of bacteria?

Most people reply incorrectly that the Petri dish will be half full at 11:30am, because we are more familiar with linear, rather  than with exponential, rates of growth.  The correct answer is 11:59am -- which seems rather unintuitive. However, because the rate of growth is exponential (doubling every minute)  the time at which the Petri dish is half full is 11:59am.   With just one more doubling, in the next minute, the Petri dish is completely full, at noon.

Below is another example of a population overshoot and collapse scenario. This is the population graph of yeast cells in a 10% sugar solution. Note that the yeast population first explodes exponentially, and is then followed by population die-off as the finite nutrients are exhausted and their own waste products pollute their environment.

This is how yeast turns grape juice into wine. The next time you say "cheers" over a glass of wine, remember that you are drinking the waste products (alcohol) of a collapsed yeast colony with poor  ecological management skills!

Anyone who perceives a linear rate of growth, but who is actually up against an exponential rate of growth, is likely to be very surprised at how the end comes very quickly and seemingly out of nowhere. They will be completely blindsided.  

For more information about the dangers of exponential growth, I highly recommend the video Arithmetic, Population and Energy, by Prof. Albert Bartlett.  The key point to remember about Professor Bartlett's lecture:  "The greatest shortcoming of the human race is our inability to understand the exponential function."  

Notice how human population has also been on an exponential trend:

Are we humans smarter than yeast?  Or will the graphic above of our population follow the overshoot and collapse graphic of the yeast?  Unlike yeast, do we have any evolved psychological adaptations to help us to identify and avoid ecological overshoot?

The fate of humans on Easter Island may help to provide an answer. When the first humans arrived on the island, there were abundant resources to support the small population.  Just like the yeast and the reindeer, the human population increased dramatically. There were no predators to cull the population. The human population continued to grow until it eventually overshot the island carrying capacity. After overshoot, most of the population starved. Apparently, they even turned on each other, sometimes resorting to cannibalism.

Human ecological exceptionalism?

It will be a race toward either paradise or oblivion, right to the last moment.  
   -- Buckminster Fuller

Human history becomes more and more a race between education and catastrophe.  
  -- H.G. Wells, The Outline of History

At this point, many people refer to human exceptionalism.  Of course we are smarter than yeast or reindeer, and our scientific advances and our technology will save us from ecological overshoot.    We can expand the carrying capacity of the Earth.

Raymond Kurzweil has argued in his 2005 book The Singularity is Near that scientific knowledge, like populations, also grows exponentially. He believes that this will allow us to expand the Earth's ecological carrying capacity, cure disease and aging, and solve problems of energy depletion. He is confident that technology will help us prevent ecological overshoot and population collapse.   

So, we have two opposing, exponentially increasing trends. One exponential trend leads to ecological overshoot and collapse; the other trend could lead to scientific/technological solutions to these problems. Which will arrive first? Ecological overshoot and collapse (Malthus), or a "techno-fix" (Kurzweil)?  No one knows.  But, we probably won't have to wait long to find out. One of these two scenarios will likely occur within the next several decades. But, which one? Generally it is healthy to be optimistic, but optimism can be deadly if it produces a Pollyannaish denial of real problems. We should not ignore ecological problems by assuming "someone else" will take care of it, or that "the free market" or "technological breakthroughs" will always come to the rescue in time. Solutions may not come in time, and we may get quite a rude Malthusian smack down later. To avoid this, one problem we must face is how to make the transition from our finite, depleting oil resources to renewable energy.  Technological civilization depends on cheap, abundant energy.