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

Abstract

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.