What factors affect the rate of siderophore production by microbes?
I think that you are interested in how bacteria regulate the expression of genes that produce siderophores. For a technical reference you can see this paper studying Vibrio, or this paper studying Pseudomonas. For a more general review see here.

Note that this is a diverse piece of metabolism and therefore there are likely to be a diversity of mechanisms by which siderophore production occurs, though superficially it appears that there are some general principles you can draw. I recommend looking at those papers and trying to understand the commonalities.