What is the relationship between the maximal density of a bacterial population during anaerobic and aerobic growth for a specific species - a facultative anaerobe?
Although this can't be generalized, but as you mention one species, still one could think over it. But, relationship of what? Relationship of maximum density and growth is not correct to define in this way.
However, you also mentioned the ratio in heading, it can be answered. But would it make sense?
Anaerobic growth will always be slower for any bacteria for which is question applies or the species which can grow under both conditions. Thus the ratio will always be 1 or less (anaerobic:aerobic).
Maximum density, not rate, is what was asked tho OP offered "during ...growth".
To Dr. Ignatov's question, we might address titers as ratios in context through time points thru expo. growth phase - seems comparison of growth curves. For that I have no answer - look forward to someone who might.
I am interested in understanding the difference in maximum density/yield of facultative anaerobic bacteria (for example, Bacillus or Enterobacteriaceae species) after cultivation in the same medium under aerobic and anaerobic conditions.
Can we obtain the same density of these bacteria in anaerobic growth compared to aerobic growth by making the cultivation time longer or increasing the nutrient concentration in the medium? I don't think so at this point.