I have a problem with the phrase "face with" when it means "encounter" and is used in the sentence containing "which" in the below sentence:
"One of the significant tasks with which biological theorists are faced, is knowing the behavior of real biological systems."
Is the above sentence correct when we use "with " before "which"?
Yes, you can phrase it "with which biological theorists are faced" in lieu of "biological theorists are faced with." The latter is not ungrammatical, but some people have an issue with ending phrases with a preposition because there was a movement in English grammar to make English more like Latin and so not end phrases with prepositions, but that was never a rule and that stopped even being included in English grammar books as a suggestion in the 1960's.
Regardless, the way you've phrased it is also correct. Some might find it stilted, but that is the way you phrase it if you do not want to end with a preposition.
AS AN ASIDE:
The comma after "faced" is in error. The complete subject is "one of the significant tasks with which biological theorists are faced." By placing a comma after "faced," you are improperly separating the subject from its verb. It is tantamount to writing, "The sky, is blue." You would never do that. Neither should you put a comma after "faced" for the same reason.
"One of the significant tasks that the biological theorists face is regarding the behavior of real biological systems."
The sentence can be written as mentioned above.
