Yes and no.................You could potentially keep that batch of microbes going for months long as you add some type of food (sugar). People in the food industry have kept sour dough or grease for almost a hundred years. All they do is keep feeding the population.
If you were to buy a 'sourdough culture' (i.e. wild yeast culture) from a source like Sourdo.com which carries these specific cultures from Europe, the Middle East, Asia, et al. and activate them, what you'll find is that for the first and maybe the 2nd round of feeding and fermenting, you will lose the culture from the point of origin and those wild yeasts floating around in the air will replace the original culture.
IOW, buying the famous San Francisco sourdough culture will give you that specific taste for the first round and maybe the 2nd round - after that it's no longer San Francisco sourdough but a sourdough based on cultures in your local area.
CC
A member of the B.B.G.A (Bread Bakers Guild of America) for over 4 years and I'm certified in artisan bread processes.
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