Spatiotemporal Pattern Formation in a Discrete Toxic-Phytoplankton–Zooplankton Model with Cross-Diffusion and Weak Allee Effect

Phytoplankton patterns have been observed widely in aquatic systems. Although pattern formation has been investigated based on many PDEs, discrete models on aquatic systems can provide more complex dynamics. A discrete toxic-phytoplankton–zooplankton model is studied in this paper, with the consideration of Allee effect and cross-diffusion. Focusing on Allee effect coefficient, flip and Neimark–Sacker bifurcation analyses are carried out. And focusing on cross-diffusion coefficient, Turing bifurcation analyses are carried out. Parameter conditions and bifurcation diagram of these bifurcations are obtained correspondingly. Numerical simulations are then performed which are consistent with results of theoretical analysis. Irregular patterns can be formed by flip bifurcation. Spirals can be formed by Neimark–Sacker bifurcation. Spots and stripes can be formed by Turing bifurcation. When Turing and flip, or Turing and Neimark–Sacker bifurcations both occur, special patterns can be obtained.