The mechanism of Turing pattern formation in a positive feedback system with cross diffusion

In this paper, we analyze a reaction–diffusion (R–D) system with a double negative feedback loop and find cases where self diffusion alone cannot lead to Turing pattern formation but cross diffusion can. Specifically, we first derive a set of sufficient conditions for Turing instability by performing linear stability analysis, then plot two bifurcation diagrams that specifically identify Turing regions in the parameter phase plane, and finally numerically demonstrate representative Turing patterns according to the theoretical predictions. Our analysis combined with previous studies actually implies an interesting fact that Turing patterns can be generated not only in a class of monostable R–D systems where cross diffusion is not necessary but also in a class of bistable R–D systems where cross diffusion is necessary. In addition, our model would be a good candidate for experimentally testing Turing pattern formation from the viewpoint of synthetic biology. (paper)