A petri dish is a short container that looks like an inverted lid to a jar. To culture bacteria, one places a substance called nutrient agar very thinly across the bottom.  In the beginning therefore, is abundant food, and without even attempting to contaminate the dish with a specific bacteria, a few colonies of bacteria and fungi will float onto the dish before one can get the lid back on!  From one or more of these bacteria, the population will increase.  The problem is that they increase exponentially!  1 to 2, 2 to 4, 4 to 8, etc.
    Similarly, humans evolved onto the scene some 4 million years ago.  But it was approximately 3,000 years ago when all heck started breaking loose.  Humans learned how to communicate effectively, and they learned agriculture.  This first step made us more like the bacteria because by using the knowledge gained from each other, and by using agriculture, our survival rates skyrocketed.  Still, human population didn't go past 300 million until the 1700s.  Plagues and drought kept the population down.  Enter antibiotics, mass irrigation, fertilizers, pesticides, and population enters near exponential growth (currently at 6.2 billion).
    But nothing can reproduce successfully at the exponential rate forever.  In bacteria cultures, as with every other living thing, there are natural defenses, and there are wastes.  Bacterial defense comes in the form of toxins (only found in some species of bacteria and fungi) which are suited to kill off other bacteria.  This is why "E coli." (actually strain H1:857), salmonella, and botulism are dangerous to humans).  A buildup of wastes will also kill off bacteria.  And if bacteria cover each square mm of the dish, there is no food left, either.  The petri dish changes color as decay and death set in.
    So what is the critical point for humans?  While we don't secrete toxins, we do make chemical and biological weapons; we produce radiological byproducts more dangerous than any other substance known to man.  As our population has increased, that leaves less arable cropland.  As more humans become wealthier, they eat more meat, and a rising cattle population also puts additional stress on productive land.  These factors combine to make our groundwater aquifers deplete at a rapid rate.  And we are now conducting the largest experiment known to the planet by changing our climate every day that we drive; effects unknown.
    So when does the mass die-off begin, like in the petri dish?  Some might say it already has: Every day, 37,000 children die of starvation, and 87 million more people are born than die.  Others might say it will accelerate as a few individuals within the population feel a sense of "anomie"; as social theorist Emile Durkheim put it, a loss of entity within the community and begin to try and take others down with them.  Many examples can be given for this.  One thing is certain: wisdom is shown in those who live their lives according to the Iroquois, who attempted to base decisions on the impact those decisions might have on the "seventh generation."  Perhaps despite the lack of western scientific knowledge, they knew about the petri dish analogy.