What you are saying is not a war of ideas, but a war of two different belief systems. Believers of Islam will never buy into The Bible. Also, you're right, It's not about population. Population decline is just another symptom of the liberal self-loathing that has been spewed-out on the west for the last forty plus years.
No, it's a war of ideas. Belief is just the doorway. There is no moral equivalence between Christianity and radical Islam, so to call them both "belief systems" is not accurate. There is an objective truth to one, and not to the other.