How animals maintain their body shape after development to maturity?
Normally, cells do not proliferate without a "command". The "zones of genetic activity" you mentioned are defined by gradients of morphogenic molecules (and cell interactions). In the adult organism the original "map" of concentrations of these molecules is already realized and the pattern they defined before is no longer present.

Another closely linked idea is that most cells in mature tissues are not able to proliferate, because they have already undergone the process of differentiation. In contrast, stem cells maintain their ability to reproduce. [There are still some ways to de-differentiate some types of cells, i.e. to obtain induced stem cells in vitro.] Thus, embrya are composed of embryonic stem cells, while adult organisms have several restricted pools of stem cells (for example, in haematopoetic organs, where new blood cells are produced.)