Dynamics of a density-dependent microorganism flocculation model with diffusion

This study comprehensively investigates a microorganism flocculation model that incorporates diffusion processes and density-dependent growth. We analyze the boundary equilibrium, positive equilibria and Turing instability to model the removal, continuous collection and distribution of microorganisms. For the ordinary differential equation (ODE) version, we analytically establish that the equilibrium undergoes forward and backward bifurcations. Meanwhile, we delve into the local and global stability of the microorganism-free state (boundary equilibrium) and coexistence state (positive equilibria), as well as examine the existence of Hopf bifurcations at the positive equilibrium, which indicate periodic oscillatory behaviors. Moreover, for the partial differential equation (PDE) system extended with spatial diffusion, we derive the conditions for Turing instability and establish the conditions leading to Turing–Hopf bifurcations. Finally, we provide numerical studies to illustrate and support our theoretical findings.