Spatiotemporal Dynamics of a Delayed Diffusive Single-Species Model in Closed Advective Environments

We investigate the spatiotemporal dynamics of a single-species diffusive model incorporating maturation delay in closed advective heterogeneous environments. First, we establish the well-posedness of the system and prove the existence and uniqueness of the nonconstant positive steady state. Subsequently, we analyze the local stability of the unique nonconstant positive steady state and demonstrate the occurrence of Hopf bifurcation through the corresponding eigenvalue problem. By utilizing a weighted inner product parameterized by the advection rate, we further characterize the stability and direction of the Hopf bifurcation. Finally, we examine how advection rate and spatial length influence the first Hopf bifurcation value, revealing their effects on system dynamics. Our results demonstrate that both advection and spatial scale can either enhance or suppress the likelihood of Hopf bifurcation, depending on the spatial heterogeneity of the intrinsic growth rate.