Compactness and bifurcation for nonlocal advection-diffusion single population model with time delay

Nonlocal advection and time delay play important roles on the movement of macroscopic and microscopic substances. In this paper, we study a nonlocal advection-diffusion model with time delay being involved in the advection term. For the proposed model, the compactness of solution operators and global boundedness of solutions are proved. Based on these results, we further investigate the local dynamics near the positive steady state, when the kernel is taken as a triangular function. The results show that the sign and magnitude of the mean of kernel will cause Hopf bifurcation and bifurcation switches, while it has been shown in the literature that the positive steady state is always stable for symmetric kernel, such as top-hat function. Finally, we construct a Lyapunov functional to show the global asymptotic stability of the steady state, when the advection coefficient is less than a critical value.