Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect

The spatiotemporal pattern formation and selection driven by nonlinear cross-diffusion of a toxic-phytoplankton-zooplankton model with Allee effect is investigated in this paper. We first perform the mathematical analysis of the corresponding non-spatial model and spatial model, which give us a complete picture of the global dynamics. Then the linear stability analysis shows that the nonlinear cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, amplitude equations under nonlinear cross-diffusion are derived that describe the spatiotemporal dynamics, which interprets the structural transitions and stability of various forms of Turing patterns. Finally, numerical simulations illustrate the effectiveness of theoretical results. It is shown that the spatiotemporal distribution of the plankton is homogeneous when in the absence of cross-diffusion. However, when the cross-diffusivity is greater than its critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe depending on the nonlinear cross-diffusivity. Simultaneously, the impact of Allee effect and toxin-producing rate of toxic-phytoplankton species on pattern selection is also explored.