Krismiler's points above are all valid and well-presented. They point towards the shrinkage rather than expansion or domination of the big three.
An additional consideration, however, is the relative importance of the ME3 carriers to their states' economies. They all provide a much greater percentage contribution to the GDP of each state. Furthermore, Dubai has become a highly-developed tourist destination. Qatar and Abu Dhabi are endeavouring to follow suit. The airlines provide frequent direct services to their base states.
For those two additional reasons, it may well be that the ME3 will receive a disproportionate level of state support, both direct and in the form of subsidies, which could give them a competitive edge. It is certain that they will be unwilling to concede hard-won market share without a fight.
At this moment, it is hard to predict what may happen, My recently-acquired knowledge is that EK claim to be ready to start full ops within 48 hours. I find that hard to believe, but I may be wrong in my beliefs. QR are still operating widely, with light loads. EY seem to be in the weakest position of the three.