The minimal model was proposed by Scheffer in 1991 to reflect the relationship between phytoplankton and zooplankton. In this paper, we analyze a minimal model on a disk with both spatiotemporal distributed delay and discrete delay. By analyzing the stability of the steady state of this model, we obtain the influences of spatiotemporal distributed delay and diffusion coefficient when discrete delay is absent, as well as the influences of discrete delay and diffusion coefficient when discrete delay is present. Through these analyses, the possibility of high codimension bifurcations in the system is revealed. Then, based on the equivariant normal form theory and numerical simulations, we analyzed the equivariant Turing–Hopf bifurcation occurring in the system under appropriate parameters and determined the existence and stability of various spatiotemporal structures, including inhomogeneous periodic solutions, homogeneous periodic solutions, and multiple spatial patterns. • Complex dynamics and periodic solutions on a circular domain are analyzed. • Effects of spatiotemporal and discrete delays on a minimal model are analyzed. • Normal form coefficients for equivariant high-codimension bifurcations are derived. • Interaction dynamics between phytoplankton and zooplankton are investigated.
Xu et al. (Sun,) studied this question.