At the time of the brit bein habetarim it is stated, that after Avraham took the animals which HaShem commanded and divides them, drove away the birds that came after these carcasses. (Bereshit/Genesis 15:12). What's the importance or meaning behind this verse? After this Avraham slept, couldn't the birds or other animals could have took the meat?
There are two answers to the significance of these birds provided by http://southbrunswickchabad.com/page.asp?pageID=%7B15FCE0CB-8E74-4A3D-8D82-A001D9B6AB87%7D&displayAll=1
1) "Rashi explains that the cut animals represent the non-Jewish nations that keep the Jewish people in Galus. The bird that tried to eat them represents King David, who tried to destroy these nations. The bird was chased away because Hashem didn't allow King David to destroy them. The same applies to other tzadikim throughout the ages who detect a slight weakening in the strength of the non-Jewish nations. When they see this, they try to hasten the coming of Mashiach (symbolized by trying to "eat the pieces"). But they didn't succeed because the time for Redemption had not yet come; they were therefore "chased away." The final success will come with the revelation of Melech HaMashiach."
2) "The bird refers not to King David, but to Mashiach himself. When Melech HaMashiach is revealed, he will "fly down" upon the nations of the world. At that time, Avraham will come to make all the nations do teshuvah and serve Hashem. This is another explanation of the phrase, "Avraham chased them away." The word vayasheiv can be translated "chased away," but can also be translated, "made them do teshuvah." According to this, the phrase is translated, "And Avraham made them do teshuvah." Therefore, when "Mashiach flies down upon the nations," Avraham will make the nations do teshuvah. At that time the prophecy of Tzefaniah (3:9) will be fulfilled: "Then all the nations will be transformed to speak clearly and all call out in the name of Hashem and serve Him together." "
It seems according to both understandings these birds were more symbolic of a future time. Therefore, after the maaseh avos siman l'banim was finished, there would be no need to have the birds return again.