Abstract:

This paper investigates a time-delayed reaction-diffusion model with chemotaxis for phytoplankton. The stability of the positive steady states is obtained by analyzing the related characteristic equations. With the help of Crandall- Rabinowitz's local partial bifurcation theory, the chemotaxis sensitivity coefficient and delay are taken as bifurcation parameters respectively to investigate the existence of Turing bifurcation and Hopf bifurcation. Then, the direction and stability of Hopf bifurcation are studied by using the central manifold theorem and the normal form method. Finally, numerical simulations are presented to show the influence of chemotaxis and time delay on the bifurcation and pattern formation of the system. Our results show that: in the system without time delay, when the chemotaxis sensitivity coefficient exceeds a critical value, the positive steady-state solution of the system will change from stable to unstable (Turing instability); in the system with delay, when the chemotaxis sensitivity coefficient is less than the critical value, if the delay is below a certain value, the positive steady-state solution is locally asymptotically stable; if the time delay exceeds a certain value, then the system will undergo Hopf bifurcation at the positive steady-state solution, and a stable spatially homogeneous periodic solution will be bifurcated from the positive steady-state solution.

Key words: Toxin-phytoplankton-zooplanktonmodel, Prey-taxis, Time-delayed, Bifurcation, Pattern formation