Spatiotemporal dynamics near the Turing-Hopf bifurcation in a toxic-phytoplankton-zooplankton model with cross-diffusion

Spatiotemporal dynamics of the toxic-phytoplankton-zooplankton system are studied on the basis of a two-dimensional reaction-diffusion model. We first perform linear stability analysis to determine the conditions for the Turing-Hopf bifurcation where Turing and Hopf modes bifurcate simultaneously. Then the coupled amplitude equations describing the slow time evolution of system are derived by weakly nonlinear analysis. Hence, stability criteria corresponding to specific pattern formations can be obtained, which helps to understand the competition between different modes. Numerical simulations illustrate the theoretical results.