Turing-Turing bifurcation in an activator-inhibitor system with gene expression time delay

In this paper, the codimension-two Turing-Turing bifurcation of an activator-inhibitor system with gene expression time delay is investigated under the homogeneous Neumann boundary condition. The interaction between two different Turing modes gives rise to the Turing-Turing bifurcation, leading to the emergence of multi-stable and superposed spatial patterns. Rigorous theoretical analysis is first given to study the Turing, Hopf, and Turing-Turing bifurcations of this system. Subsequently, in order to describe the spatiotemporal dynamics resulting from the Turing-Turing bifurcation in more detail, the algorithm for calculating the third-order truncated normal form of the Turing-Turing bifurcation is derived using the center manifold theorem and normal form theory. This algorithm can be applied to analyze the Turing-Turing bifurcation in other diffusive systems with gene expression time delay. The derived normal form enables the theoretical prediction of the spatiotemporal dynamics of this system near the Turing-Turing bifurcation point. Numerical simulations are conducted to support the theoretical analysis, revealing the presence of superposition patterns and quad-stable patterns in particular.