Bifurcation and pattern formation of a tumor-immune model with time-delay and diffusion

A tumor-immune model with time-delay and diffusion is considered. Firstly, the local stability of equilibria and the existence of Hopf bifurcation are studied. Secondly, the direction and stability of Hopf bifurcation are discussed. Finally, the numerical simulations are used to verify the effectiveness of the theoretical results. It is found that the time-delay can destroy the stability of positive equilibrium and then affect the occurrence of Hopf branch. Specifically, the equilibrium is stable if the model is without delay or with small delay, and so there is no bifurcation; Conversely, when the delay is large, it induces the instability of equilibrium and the Hopf bifurcation occurs, the model then exhibits rich spatiotemporal dynamics. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.