Effect of intestinal permeability and phagocytes diffusion rate on pattern structure of Crohn’s disease based on the Turing–Hopf bifurcation

The patchy inflammatory pattern in the gastrointestinal tract is one of the typical features of Crohn’s disease, and revealing the evolutionary mechanism of Crohn’s disease is helpful to further understand its pathogenesis. However, the research on the pathogenesis of Crohn’s disease is not comprehensive. To this end, this paper investigates the spatiotemporal dynamics of the phagocyte-bacterium model with diffusion and explores the evolutionary mechanism of Crohn’s disease from the perspective of bifurcation analysis. Theoretically, we use the normal form theory of the reaction-diffusion equation and the neutral manifold theory to derive the normal form of the system near the Turing–Hopf bifurcation. Then, the results are verified by numerical simulations, and it is found that small disturbances in the control parameters led to four different types of dynamic states in the system: uniform state, spatially non-uniform steady state, spatially uniform periodic state, and spatially non-uniform periodic state. At the same time, the effects of small alterations in intestinal permeability and phagocytes diffusion rate near the Turing–Hopf bifurcation point on pattern structure and bacterial biomass of Crohn’s disease were revealed. In addition, we found that there are three different types of pattern transformation in the system. These findings will help to better understand the development process of Crohn’s disease, predict its pattern transformation, and provide theoretical guidance for the research and development of new drugs. Meanwhile, our research findings provide insights into the vulnerability of the mucosal immune system in Crohn’s disease.