Can we predict about bitcoin network? What will happen to the bitcoin network once all 21M of bitcoins have been mined? The bitcoin network will cease to exist? or the bitcoin network will continue to function the same way with miners being compensated with transaction fees when users spent their bitcoins?
I don't think anyone can predict anything, especially that it's estimated to happen somewhere in 2140 ;)

But I'll share my take on it.

Please note that the number of BTC rewarded will he halved few times until then: https://docs.google.com/spreadsheets/u/1/d/12tR_9WrY0Hj4AQLoJYj9EDBzfA38XIVLQSOOOVePNm0/pubhtml?gid=0&amp;single=true

So the proportion of BTC reward to transaction fees will be less and less - and the network will adjust. 

Also - what is important - that this is self-regulating. If the reward was to disappear, some miners would leave the network (because of the cost bigger than transaction fees earnings). But then the difficulty would decrease, and mining cost to profit should more or less equalize.

So, unless bitcoin network is abandoned for some other reason*, the disappearing of block reward should not have a significant effect on the network itself. 

BTC is like every other money - if people believe it's money, then it's money. Same with Bitcoin - doesn't matter if mining has any extra bonus or not. 

(*) the biggest challenge to bitcoin seems to be the block size, which limits the network throughput. If you wanna read more thoughts on that - check my other answer here https://ethereum.stackexchange.com/questions/39477/why-is-there-a-possibility-to-increase-block-gas-limit/39479#39479