Hopf Bifurcation Analysis of a Phagocyte–Bacteria Diffusion Model with Delay in Crohn’s Disease

Dysbiosis of the gut microbiota and dysregulated immune responses are key pathological features in both the onset and progression of Crohn’s disease. We propose a phagocyte–bacteria diffusion model with a time delay to explore their dynamic interactions and impact on the progression of Crohn’s disease. We first supplement the proof of the positivity, boundedness, existence, uniqueness, and global stability of the solutions for the ordinary differential system without time delay. Then we examine the stability of the positive equilibrium point and the occurrence of a Hopf bifurcation. By applying normal form and center manifold theory, we determine the direction of the bifurcation and the stability of the bifurcating periodic solution. Numerical simulations are used to verify the theoretical results. We find that the time delay significantly slows the system’s approach to a steady state. With a fixed delay, increased intestinal permeability prolongs the stabilization time. Conversely, with fixed intestinal permeability, a larger delay renders the system more prone to oscillations. Furthermore, a higher maximum engulfment rate by phagocytes reduces bacterial biomass but prolongs stabilization, whereas an increased phagocyte death rate shortens it. Additionally, an elevated bacterial growth rate increases both the bacterial biomass and the stabilization time. These results enhance our understanding of the dynamic equilibrium in immune systems.