If food source runs out--the yeast colony will die.
If alcohol levels get too high--the yeast colony will die.
When You add a large amount of yeast to a mixture---C02 production will start rapidly and end rapidly.
When You add a very small amount of yeast to a mixture---C02 production will start slowly and increase over time.
If you add X amount of yeast to Y amount of sugar and Z amount of water--the yeast will die when either the food source is gone, or the alcohol levels gets too high.
If you add the same amount of yeast (X) to the same amount of sugar (Y) to 10x the amount of water (Z)--the yeast will die when the food source runs out: Sugar (Y).
In order for you to accomplish your task---you would need to add food: Sugar. And people do this. But it is generally easier to dump and reload--so that is what most people do from what I can gather. Some reuse the yeast---they dump out some of the water and refill and add sugar.
The questions are: How much sugar, and how often? If you add too much sugar--the yeast population will get too large. Not enough and it will die off. As long as there is food--the yeast will multiply. And they will multiply and multiply and multiply.
An Auto-Air-Doser or a pump-doser could probably be set on a timer to dose a certain amount of a sugar/water solution to work this out in a 5g jug. BUT too much sugar--too many yeast: population control becomes the problem.
But I guess with some thought, time and a bubble counter---this could be worked out.
i don't really have room to test this idea now in a small dorm
Sure you do. 1g jug, fill it with water, add 1/4 cup of sugar and a tiny pinch of yeast, run it into a DIY bubble counter---see how long you can get it to last by just adding sugar, and see how well you can nail down the amount and frequency of food addition by trying to stabilize it at X BPS........