Why can't organisms in a population follow the exponential growth pattern for a long period of time

Because, even if they sustain a steady birth and death rate, food is bound to grow somewhat shorter at a certain point, and it will decrease. but in the absence of eating as much, the food has a short booming growth rate as well, which raises the organisms growth rate again, and it sustains this cycle of ups and downs.  the same exact thing happens with its predators. another thing that can happen is a deadly disease, or a sudden heat wave or freezing period, or the finding of even more healthy food that create more production.
i hope this helps, and if it does a lot, can you please mark it as brainiest? (i usually don't ask this, but at the moment I'm kinda desperate.) 

Ig national geographic you can search up all kinds of animals and bugs