Dynamics of a diffusive nutrient-microorganism model with time delay in the sediment

This study focuses on the dynamics of a diffusive nutrient-microorganism model subject to Neumann boundary conditions. An in-depth analysis is conducted to explore how time delay affects the stability of the constant steady state. Analytical results reveal that time delay can destabilize the system, thereby inducing unstable oscillations in originally stable regions. Specifically, under certain conditions, the equilibrium state undergoes stability switches and gives rise to Hopf bifurcation as the time delay pa- rameter varies. Furthermore, under the synergistic effects of diffusion and time delay, the system exhibits spatially inhomogeneous periodic solutions, which elucidates the dynamical mechanism underlying the alternating distribution of nutrients patches and microbial enrichment zone in sediments. The direction and stability of Hopf bifurcation are determined by applying the center manifold theory and the normal form method for partial functional differential equations. Finally, numerical simulations are carried out to verify the theoretical findings.