One case study of an organism that initially don't compete with each other significantly during large, sudden phases of food abundance is the hagfish, which generally lives in a food-poor environment on the ocean floor. During a whalefall, however, they have access to a near-unlimited amount of food which outweighs them by millions of times. Even in such cases of abundant food, the hagfish often compete with each other for food, especially when the food becomes depleted (video) after long periods of feeding. Food sources are finite, and given enough time there would be sufficiently little food left behind for competition to begin.
Competition is fundamental to biology. Any biological organism has a certain fitness, and organisms that have lower fitness would be outcompeted by higher fitness ones through the processes of natural selection.
In fact, Thomas Malthus described the fundamental instability of such initially competition-free systems in the 18th century. A Malthusian catastrophe occurs when population growth (a fundamentally exponential process) outpaces the growth of available food resources (a fundamentally linear process). Malthus wrote:
Famine seems to be the last, the most dreadful resource of nature. The
power of population is so superior to the power of the earth to
produce subsistence for man, that premature death must in some shape
or other visit the human race. The vices of mankind are active and
able ministers of depopulation. They are the precursors in the great
army of destruction, and often finish the dreadful work themselves.
But should they fail in this war of extermination, sickly seasons,
epidemics, pestilence, and plague advance in terrific array, and sweep
off their thousands and tens of thousands. Should success be still
incomplete, gigantic inevitable famine stalks in the rear, and with
one mighty blow levels the population with the food of the world.
Famine would inevitably result when exponential population growth results in there being insufficient food to feed all of the organisms.
It may be possible for some organisms to evolve birth control, but these organisms would have a lower birth rate and therefore a lower fitness. They would be quickly outcompeted by organisms which do not utilise birth control and maximally utilise the food sources to reproduce. The result is a tragedy of the commons, where the organisms breed themselves into starvation.