Is there any way to tell whether a gene expression pattern would follow constitive or bursting pattern? Is there any database or source where the information regarding expression patterns of the genes is available?
That's a complicated question and depends on several variables. If you're working with cell lines or primary cells, the best way to address your question is to synchronize the cells using mitotic inhibitors and examine kinetic changes in gene expression over time after release from inhibition.
If you're talking about developmental changes in live animals, you're probably better off performing immunohistochemistry on developmentally staged embryos.