Effects of chemotaxis and time delay on the spatiotemporal patterns of a two-species reaction-diffusion system

In this paper, we investigate the effects of chemotaxis and time delay on the spatiotemporal dynamics of a two-species reaction-diffusion system. We show that cheomotaxis and delay can induce the instability of the constant steady state and the existence of a stable spatially non-homogeneous steady state bifurcation, spatially homogeneous/non-homogeneous Hopf bifurcation, and double Hopf bifurcation. Particularly, for the case of no delay, we drive the formula for determining the direction and stability of the degenerate steady state bifurcation. Furthermore, we apply our theoretical results to a delayed predator-prey system with predator- taxis and a cooperative Lotka-Volterra system. For the predator-prey system, predator-taxis and delay jointly lead to spatially non-homogeneous periodic solutions due to spatially non-homogeneous Hopf bifurcation and double-Hopf bifurcation via the interaction between Hopf bifurcations. For the cooperative system, spatially non-homogeneous steady state solutions and non-homogeneous period solutions bifurcate from the constant equilibrium by Turing bifurcation induced by the chemotaxis term and Hopf bifurcation induced by the delay, though the equilibrium of the corresponding ODE is globally asymptotically stable.