The middle east used to be a very diverse place in terms of Religion(at least the monotheistic ones). This has changed quickly over the past half century and we are arriving at a point where there is almost complete Islamic demographic dominance.

Jews were well represented in such countries as Morroco, Egypt, Iraq (and in smaller numbers in other ones). The tensions caused by the creation of the state of Israel and the ensuing wars led them to flee (or be expelled in certain circumstances).

Christians were also well represented. They were the majority in Lebanon, and well represented in Egypt and Palestine. There numbers have been in great decline (especially relative to their muslim cocitizens). The Christians in the middle east had a much lower birth rate and a much higher emigration rate to the west.

It is completly FALSE to say that muslim birthrates are not changing in the middle east. The middle east is undergoing a demographic transition as we speak. It is still a relatively youthful population and so it will increase for the next while. The rate of growth is slowing pretty quickly though.
A good indicator of this is to look at the Muslim birthrate within the state of Israel. This was a group that traditionally had a very high birth rate. Over the past ten years their rate has begun to crash
scroll down to the bottom and look at the rate per 1000