The effects of delays on stability and spatiotemporal dynamics in a diffusive nutrient-phytoplankton model

The control of harmful algal blooms is one of the significant problems in ecology. To investigate the oscillatory nature of blooms, a nutrient-phytoplankton model is considered, in which a discrete growth response delay and a nonlocal nutrient recycling delay are incorporated. The aim of this research is to provide an understanding of the effect of the discrete growth response delay, the nonlocal nutrient recycling delay, as well as the joint effect of two delays on the nutrient-phytoplankton dynamics. For the system with the nonlocal nutrient recycling delay only, the stability of system cannot be changed by the nonlocal delay with high nutrient recycling rate, however, the nonlocal delay may induce Hopf bifurcation and stability switches with low nutrient recycling rate, which means that the nonlocal nutrient recycling delay has a crucial impact on both the formation and the termination of algal blooms. For the system with the discrete growth delay only, the delay may destabilize the positive equilibrium and arise the occurrence of Hopf bifurcation. For the system with both delays, the corresponding characteristic equation incorporates delay-dependent parameters and wave numbers. The stability switching curves are found for the cases for wave number n=0 and n≠0, and the stable region on two-delay plane is determined. It turns out that the stable region is larger for higher level of nutrient recycling rate. On the boundaries of the stable region, double Hopf bifurcation occurs at the intersection of the stability switching curves. Taking two time delays as parameters, the normal form of double Hopf bifurcation is derived, and the parameter plane is divided into six regions near the double Hopf bifurcation point, on which all the corresponding complex dynamics is found, including the existence of an unstable spatially nonhomogeneous quasi-periodic solution and the coexistence of a stable spatially homogeneous periodic solution and a stable spatially nonhomogeneous periodic solution.