Reaction-diffusion systems with Turing instabilities have been used to represent several biological phenomena, which may involve growth of organs or tissues. This article studies the influence of growth rate and the spatial scale in the evolution of spatial scale in the evolution of spatial -temporal patterns generated in this kind of systems. A computational study using the finite element method is performed in order to solve a reaction -diffusion system with turing instabilities using a square domain which grows at different rates. it was found that variations in the parameters related to the system and in the growth rate change the shape, heterogeneity and evolution of the generated patterns. The results confirm the robustness of the reaction-diffusion systems, in the terms of the independence regarding the initial conditions, and suggest the existence of a limit growth rate above which heterogeneous spatial-temporal patterns are not generated.