Delay-induced patterns in a reaction-diffusion system on complex networks

Pattern formations in reaction-diffusion (RD) systems with time delay constitute a vital class of dynamical mechanisms extensively investigated for biological and chemical processes, where Hopf bifurcation usually occurs. Recent studies show that pattern formations differ significantly between RD systems with large-time and small-time delay. Therefore, in this paper, we aim to explore the exact role of the time delay in RD systems based on complex networks, which would affect the form of patterns. Depicting networked dynamics of the predator-prey system by a set of RD equations, it is found that boundaries of Hopf bifurcation are decided by diffusion coefficients, as well as the Eigen-spectra of networks. We also obtain mathematical expressions of the boundaries in both large-time and small-time delay cases. Through extensive simulations, it is unveiled that the connectivity structures of networks hardly have impact on the trend of evolutionary processes. Compared to large-time delay cases, the oscillation cycle of average prey density becomes shorter red with small-time delay, and the oscillation amplitude also decreases. We finally reveal the evolution process of the prey density and discover the thick-tailed phenomenon in large-time delay cases.