Dynamics in a diffusive plankton system with delay and toxic substances effect

• The dynamics of a reaction–diffusion plankton system with delay and toxic substances effect is considered.
• Existence and priori bound of a solution for a model without delay are shown.
• The global asymptotic stability of the axial equilibrium is obtained.
• The stability of the positive equilibrium and the existence of Hopf bifurcation are obtained.
• The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined.

The dynamics of a reaction–diffusion plankton system with delay and toxic substances effect is considered. Existence and priori bound of a solution for a model without delay are shown. Global asymptotic stability of the axial equilibrium is obtained. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of eigenvalues. And the properties of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Some numerical simulations are carried out for illustrating the theoretical results.