Network Controllability of Turing Reaction and Diffusion Model

In this paper, the controllability problem of the reaction-diffusion (RD) model, or Turing model is studied. Turing model provides a valuable framework for self-organized system and has been widely used to explain to pattern formation in the real life. With the rapidly development of the biology technology, biologists are trying to control the pattern formation artificially and has achieved some progress. However, the influence that exerted on the pattern formation by the external factors, such as light and temperature, remains to be solved. In this work, The RD model is obtained following the assumptions in Turing's original paper and spatially discretized into square grids. The nodes in the outermost layer are considered as candidates for control. Controllability of the RD system with all such nodes as control is first shown. Then controllability of the RD system with minimal number of control nodes is studied. Our results show that nearly 87.5% control nodes can be saved while the system is still controllable. Numerical simulations are provided to demonstrate the effects of controlling the reaction and diffusion of the morphogens.