The very next year, Isaac was born (Gen 21:1-2).  His name ("He laughs") was a constant reminder of the reaction of both parents to God's promise of a son.
       
The birth of Isaac led to the banishment of Ishmael and his mother.  Sarah was not a person who could make peace with her rival.  The fact that God brought joy into her life did not spark generosity and reconciliation in her. She now feared that Ishmael would supplant Isaac as the heir of Abraham, according to the customs of the time.

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        She said to Abraham, "Get rid of that slave woman and her son, for that slave woman's son will never share in the inheritance with my son Isaac" (Gen 21:10).

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One's first response to Sarah's statement is to ask, "Why?"   Why couldn't Ishmael share in the legacy of Abraham, both spiritual and material?  Was there not enough for two sons?  Was Abraham that poor?  Sarah was motivated purely by selfishness, not by any zeal for the honor of God or His purposes on earth. In fact, it is quite possible that the word of God to Hagar about her son --

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        "he will live in hostility toward all his brothers" (Gen 16:12)

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--  was a result not of any inherent character flaw of Ishmael, but primarily the result of getting disowned by his father and cast out by his stepmother.  Of course he was going to be hostile towards his relations -- they had robbed him of his paternity and sent him into the desert to die!

        

The pretext for Ishmael's expulsion occurred on a day of celebration, the weaning of Isaac.  Sarah saw Ishmael "mocking" (or "jesting at") her son, and used this as a reason to pressure Abraham to get rid of both Hagar and Ishmael.  The Hebrew word translated "mocking" is interesting because it is the same word used to describe Sarah's action upon hearing God's promise that she would bear a son: "Therefore Sarah laughed within herself"  (Gen 18:12).  She judged Ishmael for conduct she herself engaged in.  The word does not always have a negative meaning:

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         And Sarah said, God hath made me to laugh, so that all that hear will laugh with me" (Gen 21:6).

  

         Indeed it is from the same root word as the name Isaac itself ("he laughs"). 

       

It is the New Testament that promotes the notion that Ishmael was persecuting the infant Isaac:

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        "At that time the son born in the ordinary way persecuted the son born by the power of the Spirit.   It is the same now" (Gal 4:29). 
    
        Josephus, in The Antiquities, does not accuse Ishmael of doing anything to Isaac, but says that Sarah was afraid that he would do him harm once Abraham died (I.12.3).

       

Abraham, to his credit, resisted his wife's insistence that he expel Hagar and Ishmael.  This shows that he loved his firstborn son.  However, God spoke to him: "Listen to whatever Sarah tells you" (Gen 21:12).  He repeated His promise to make Ishmael into a nation.  It would be easy here to misunderstand God's words as approving of Sarah's selfishness and cruelty.  This is not true.  Her envy just happened to coincide with God's larger purposes for    world history.  He over-rode her motives to accomplish His larger design.  While Sarah could have cared less whether Hagar and Ishmael lived or died, God already had a rescue operation planned out to save them both from death.

       

The actual departure of Ishmael was almost an anticlimax.

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        Early the next morning Abraham took some food and a skin of water and gave them to Hagar (Gen 21:14). 

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At the end of his life, when Abraham sent his six sons by Keturah away from Isaac to the east, he gave them gifts (Gen 25:6).  But to Ishmael, he gave nothing, not even enough food for the journey into the wilderness.   Exeunt Ishmael, stage left.  He was cast out from his home and from his patrimony, but not, like Esau, later, because of his own folly.  If it weren't for the second intervention of God on behalf of Hagar, they both would have died of dehydration in the desert. Yet, for all his losses, Ishmael was granted a consolation prize:
    
         God was with the boy as he grew up. He lived in the desert and became an archer (Gen 21:20).

        

But we must understand that God was with Ishmael in a different way from the way He was with Isaac.  God's protection was with him, God's prosperity was with him in a natural way, but the Essence of the promise to Abraham was not there.  Ishmael's fated role was to play Cain to Isaac's Abel: his offering was not accepted.