Spatiotemporal patterns induced by Turing-Hopf interaction and symmetry on a disk

Turing bifurcation and Hopf bifurcation are two important kinds of transitions giving birth to inhomogeneous solutions, in spatial or temporal ways. On a disk, these two bifurcations may lead to equivariant Turing-Hopf bifurcations whose normal forms are given in three different cases in this paper. In addition, we analyzed the possible solutions for each normal form, which can guide us to find solutions with physical significance in real-world systems, and the breathing, standing wave-like, and rotating wave-like patterns are found in a delayed mussel-algae model.