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I hope this is the appropiate place for this question, asked from a math/physics perspective with little to no training in biology.

Bacteria are very small and live in huge colonies, ants are bigger (but also very small) and live in smaller colonies (but also very big). Ignoring the human species, I can't think of any big animal living in big colonies

Is there any name for this relation? I want to study it from two perspectives: first, why? (this may be easy to explain in terms of resource supply or energy available) and second, what is the precise relation? (clearly inverse, but linear, logarithmic...?)

Thanks!

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    $\begingroup$ You should know the word "colony" means different things when referring to ants and bacteria. Bacterial colonies arise because bacteria divide faster than they can move (if they can even move). An ant colony is more of a social structure. Not really comparable. $\endgroup$
    – user40950
    Oct 1, 2019 at 21:40

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It is not trivial to define a "colony/society". Sure, you can say "this bacterial blob is a colony", or "this ant nest is a colony", and so on, but these would be your own definitions. These systems you mention don't have definite boundaries (temporal or spatial). Although physicists like to do gross approximations to things in nature (and I'm pretty sure there is already some paper 'answering' your question), I'm not sure anyone can get a "precise relation" for a concept that even biologists struggle with (that of a "colony"). This is because at core, what you are really asking, is what is a biological individual and how can I measure it? (the individual ant, the individual colony, etc); and that's something we have no answer for. For what is worth, I recommend you read about the problem of biological individuality, which will give you no answer to your question but at least can tell you why you can't really answer it: https://link.springer.com/article/10.1162/BIOT_a_00068

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