A Circular Spatial-diffusion Mathematical Model to Analysis Hopf-Turing Bifurcation in Plankton Population Under the Toxin Control Variation in 2D

In a mathematical model of a system with two reaction-diffusion equations with Neumann-Dirichlet boundary conditions, we formulated zooplankton-phytoplankton in the aquatic environment on the circular domain. The attention has been focused on the toxin producing role of the space in explaining heterogeneity, the distribution of the species and the influence of the spatial structure on their abundance. The key idea of the model formulation is based on a nonlinear equations systems version with Holling II functional response. We base our mathematical analysis on the search for local and global solution with spatial diffusion. We present some mathematical results concerning the solution existence, the stability of the model equilibria. We have obtained important mathematical results for model equilibria stability at long time. Under certain mathematical conditions, the model without diffusion is locally asymptotically stable. Mathematical analysis also shows that the Hopf bifurcation breaks the time symmetry of the system and leads to uniform oscillations in space and periodic oscillations. The Turing bifurcation breaks the space symmetry and leads to the formation of stationary patterns in time and oscillatory patterns in space. A series of structured numerical simulations highlighted the formation of patterns and allowed to identify critical threshold of toxin released by phytoplankton leading to phytoplankton blooms.