We investigated a nonlinear model of the interaction between nutrients and plankton, which was addressed using a pair of reaction-advection-diffusion equations. Based on numerical analysis, we studied a model without diffusion and sinking terms, and we found that the phytoplankton density (a stable state) increased with the increase of nutrient density. We analyzed the model using a linear analysis technique and found that the sinking of phytoplankton could affect the system. If the sinking velocity exceeded a certain critical value, the stable state became unstable and the wavelength of phytoplankton increased with the increase of sinking velocity. Furthermore, band patterns were also produced by our model, which was affected by the diffusion and sinking of phytoplankton. Thus, the change in the diffusion and sinking of phytoplankton led to different spatial distributions of phytoplankton. All of these results are expected to be useful in the study of plankton dynamics in aquatic ecosystems.

This study considers the spatiotemporal dynamics of a reaction-diffusion phytoplankton-zooplankton system with a double Allee effect on prey under a homogeneous boundary condition. The qualitative properties are analyzed, including the local stability of all equilibria and the global asymptotic property of the unique positive equilibrium. We also discuss the Hopf bifurcation and the steady state bifurcation of the system. These results are expected to help understand the complexity of the Allee effect and the interaction between phytoplankton and zooplankton.

We investigate a nonlinear model of the interaction between phytoplankton and fish, which uses a pair of semicontinuous systems with biological and artificial control. First, the existence of an order-1 periodic solution to the system is analyzed using a Poincaré map and a geometric method. The stability conditions of the order-1 periodic solution are obtained by a theoretical mathematical analysis. Furthermore, based on previous analysis, we investigate the bifurcation in the order-1 periodic solution and prove that the order-1 periodic solution breaks up an order-1 periodic solution at least. In addition, the transcritical bifurcation of the system is described. Finally, we provide a series of numerical results that illustrate the feasibility of the theoretical results. Based on the theoretical and numerical results, we analyzed the feasibility of biological and artificial control, which showed that biological and artificial methods can control phytoplankton blooms. These results are expected to be useful for the study of phytoplankton dynamics in aquatic ecosystems.

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.