Here's a talk about the mathematics of cities and how it compares to biology. It's fascinating stuff but my impression is that the models ultimately predict collapse because there are finite limitations on 1) resources, which is countered by innovation, and 2) our ability to innovate, which he provides no solution for countering. Our ability to rapidly innovate is staving off collapse from resource depletion, but in order for that to continue we have to continuously increase the speed of innovation. But there is necessarily a limit to the speed of cultural diffusion of innovation. While the internet provides very high speed diffusion of information, there is still a limit to how fast the information can be taken in and effect changes in behavior. Thus, we are destined to see the collapse of cities once that speed limit of cultural diffusion is reached.
Although if we could get a handle on the growth of population then perhaps that could bring about a balance between the need for innovation and the ability to adopt innovations.