Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Abstract

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

Existence of Solution and Self-Exciting Attractor in the Fractional-Order Gyrostat Dynamical System

“…Is called the dynamical system, and a parameter β ∈ R in the velocity vector field is termed as bifurcation parameter if system (1) changes its topological structure with the variation in parameter values, whereas the process of changing in qualitative structures is known as bifurcation. ere are several types of bifurcation including saddle node [2], Hopf [3][4][5][6][7], and zero-Hopf [8][9][10][11]. e bifurcation diagram [12] for the parameter makes it easy for predicting the type of bifurcation and existence of chaos in system (1).…”

Stability and Bifurcation for a Delayed Diffusive Two-Zooplankton One-Phytoplankton Model with Two Different Functions

“…erefore, the influence of spatial diffusion on the phytoplankton-plankton model has been paid more attention by many scholars [35][36][37][38][39][40][41][42][43][44][45]. In the work by Jia et al [46], a three-component plankton model with spatial diffusion and time delay is proposed, which describes the relationship between a zooplankton and two phytoplankton. Rao [47] studied the complex dynamics of a spatial toxicphytoplankton-zooplankton model with the Holling II function and showed that the interaction between toxic phytoplankton and zooplankton in the marine environment may be partially driven by diffusivity or environmental carrying capacity.…”