Turing and Hopf Bifurcation for a Diffusive Nutrient–Microorganism System in Sediment

The distribution of nutrients and microorganisms in sediments is not uniform, as it results from the complex interactions between chemical, biological, and physical transport processes. This study investigates, through a simplified model, the dynamics of a microbial population and its nutrients, accounting for the active metabolic state of the bacteria. Using the nutrient diffusion coefficient d2 as a bifurcation parameter, we demonstrate that a Turing bifurcation, leading to spatial patterning, occurs at a critical value. Sufficient conditions for the stability of the resulting pattern are also established. Furthermore, by treating time delay as a bifurcation parameter, we prove the occurrence of Hopf bifurcation near the positive constant equilibrium solutions at a sequence of critical values, showing that time delay can induce spatially homogeneous and inhomogeneous periodic oscillatory patterns. Numerical simulations and solutions are provided to illustrate the theoretical findings.