I believe the bowls come after the trumpets and that the rapture will occur at the last trumpet:
Jesus tells us, "At that time the sign of the Son of Man will appear in the sky, and all the nations of the earth will mourn. They will see the Son of Man coming on the clouds of the sky, with power and great glory. And he will send his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of the heavens to the other." (Matt 24:30-31)
Paul tells us, "Brothers, we do not want you to be ignorant about those who fall asleep, or to grieve like the rest of men, who have no hope. We believe that Jesus died and rose again and so we believe that God will bring with Jesus those who have fallen asleep in him. According to the Lord's own word, we tell you that we who are still alive, who are left till the coming of the Lord, will certainly not precede those who have fallen asleep. For the Lord himself will come down from heaven, with a loud command, with the voice of the archangel and with the trumpet call of God, and the dead in Christ will rise first. After that, we who are still alive and are left will be caught up together with them in the clouds to meet the Lord in the air. And so we will be with the Lord forever." (1 Thess 4:13-17)
So we know there is going to be a shout and a trumpet, but how does that help us? Let's look what Paul tells the Corinthians:
"I declare to you, brothers, that flesh and blood cannot inherit the kingdom of God, nor does the perishable inherit the imperishable. Listen, I tell you a mystery: We will not all sleep, but we will all be changed-- in a flash, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, the dead will be raised imperishable, and we will be changed." (1 Cor. 15:50-52)
So we know now that the rapture will happen at the last trumpet. To learn more let's go to Revelation. In the book of Revelation, John tells us about 7 seal judgments. The Lamb (Jesus) breaks seven seals on a scroll (Rev 6). When the seventh seal is broken, seven angels are given seven trumpets (Rev 8:1-2). Let's take a look at what happens when the seventh angel sounds the seventh trumpet (Rev 11:15-18):
"The seventh angel sounded his trumpet, and there were loud voices in heaven, which said:
"The kingdom of the world has become the kingdom of our Lord and of his Christ,
and he will reign for ever and ever." And the twenty-four elders, who were seated on their thrones before God, fell on their faces and worshiped God, saying:
"We give thanks to you, Lord God Almighty,
the One who is and who was,
because you have taken your great power
and have begun to reign.
The nations were angry; and your wrath has come.
The time has come for judging the dead,
and for rewarding your servants the prophets
and your saints and those who reverence your name,
both small and great--
and for destroying those who destroy the earth."
The last trumpet marks a transition. From this passage, we can see a few things:
The triumph of Christ's Kingdom is announced
God's people will be rewarded
God's enemies will be destroyed
The triumph of Christ's Kingdom is announced. This is the point where He is going to reclaim His people. While the passage does not say that specifically, we can see more detail about numbers 2 and 3 where John talks about a 'harvest' in chapter 14:
"I looked, and there before me was a white cloud, and seated on the cloud was one "like a son of man" with a crown of gold on his head and a sharp sickle in his hand. Then another angel came out of the temple and called in a loud voice to him who was sitting on the cloud, "Take your sickle and reap, because the time to reap has come, for the harvest of the earth is ripe." So he who was seated on the cloud swung his sickle over the earth, and the earth was harvested." (v14-16)
"Another angel came out of the temple in heaven, and he too had a sharp sickle. Still another angel, who had charge of the fire, came from the altar and called in a loud voice to him who had the sharp sickle, "Take your sharp sickle and gather the clusters of grapes from the earth's vine, because its grapes are ripe." The angel swung his sickle on the earth, gathered its grapes and threw them into the great winepress of God's wrath. They were trampled in the winepress outside the city, and blood flowed out of the press, rising as high as the horses' bridles for a distance of 1,600 stadia." (v 17-20)
Clearly Jesus and His angels are claiming His people and preparing the Earth for God's Wrath. This is important for two reasons: 1) It matches what was sung about at the Last Trumpet and 2) It shows God is redeeming His people before He pours His wrath upon the Earth.