Analyzing Turing's Systems via Dynamic Bifurcation Theory

In this paper, we introduce a novel approach to study reaction-diffusion systems -- dynamic transition theory approach developed in Ma and Wang 2015. This approach generalizes Turing's classical result (linear stability analysis) on pattern formation and cast some new insights into Turing's systems. Specifically, we studied the Turing's instability and dynamic transition phenomenon for a Turing's system, and expressions of the critical parameters $L(D_{u},D_{v})$, and $D_{v}^{*}$ are derived. These two simple parameters are sufficient to provide us with enough information on the Turing's instability result as well as the dynamic transition behavior of the system. As an application, based on the method we establish in this paper, we found that the Schnakenberg system has two different transition types: single real eigenvalue transition and double real eigenvalues transition. These transition types are interpreted using phase diagrams.