Is there enough soil to feed a planet of cities?

A law describes the consumption of soil as the population grows. How much longer can the city continue to grow ensuring enough fertile soil to feed the population of the Earth?
Here is a popular science short film recounting this study, and it is animated entirely through programming code for the open source software POV-Ray.

The scientific research entitled “Is there enough soil to feed a planet of growing cities?”, by Roberto D’Autilia and Ilaria D’Ambrosi, analyzes a scaling law for the consumption of agricultural soil by cities. The nonlinear dependence of the size of the city on the number of inhabitants gives rise to an equation for population dynamics. The limit of the solution for this equation is given by the so-called carrying capacity, in terms of number of inhabitants that can be fed. The carrying capacity as a function of the scaling law exponent shows that this exponent must be very small to ensure food sustainability. A bound for the value is suggested, and trying to achieve it is a challenge for future cities.