Turing–Hopf bifurcation in a diffusive temperature-phytoplankton system with delay

Abstract

This paper studies the dynamics of a one-dimensional temperature-phytoplankton model. We introduce spatial diffusion and time-delay terms to describe the diffusion of temperature and phytoplankton and the growth cycle of phytoplankton, respectively. First, in the delay-free case with diffusion only, the thermal inertia \(\lambda \) and diffusion coefficient \(d_1\) are chosen as bifurcation parameters for Hopf and Turing bifurcations, and the resulting Turing–Hopf bifurcation is identified. Second, when time delay is included, the delay \(\tau \) and \(d_1\) serve as bifurcation parameters; based on the analysis of Hopf and Turing bifurcations, the parameter-induced Turing–Hopf bifurcation is identified. For both cases, the corresponding normal forms and bifurcation diagrams of the Turing–Hopf bifurcation are obtained using center manifold theory. Finally, numerical simulations reveal typical spatiotemporal patterns, including spatially homogeneous periodic solutions, spatially inhomogeneous steady-state solutions, and spatially inhomogeneous quasi-periodic solutions. The results demonstrate that the coupling of diffusion and delay significantly enriches the system’s spatiotemporal dynamics.