I think what will happen is what happened on the desktop. There was a time when every application you wanted on the desktop had to be installed natively. This was because of the following reasons: web didn't exist yet, web existed but the tools sucked, web existed but networks were slow. Over time the networks got fast enough and the tools got good enough that the vast majority of applications could be built as a website. There are still cases where building a desktop application makes sense such as when performance is critical, but those cases are in the minority now. Building a website also has the distinct advantage that you only need to build it once and it works everywhere (mostly).

When it comes to mobile, I think we are in the "web exists and the tools are almost good enough but the networks suck" phase. When the mobile networks get fast enough, I suspect most applications will just be websites. Of course there will always be some applications that are built for native because of performance or they have some fancy touch-based interaction.