Hopf–Hopf bifurcation of the memory-based diffusive bacterial infection model

Bacterial infections challenge the immune system, causing inflammation in which leukocytes play an important role in identifying and combating harmful bacteria. These white blood cells, leukocytes, navigate to infection sites by chemotaxis, which is guided by chemical cues from bacteria. The leukocytes then either engulf and destroy the harmful bacteria or release enzymes to neutralize the infection. This phenomenon is crucial for controlling infections and preventing their spread. However, this process is influenced by memory effects, which cause their movement to be affected by previous signals, as well as reaction delays. These variables complicate immune responses, thus understanding their impact on infection dynamics and inflammation is critical for developing better treatments. In this paper, we analyze a diffusive bacterial infection model with a spatial memory effect, taking into account the impact of delay on the movement of leukocytes. Through stability and bifurcation analysis, we obtain the sufficient and necessary conditions for the Hopf bifurcation and stability switches. It is found that in the absence of delay the system remains stable under certain conditions. However, in the presence of time delay, the system will undergo the Hopf bifurcation, when the time delay exceeds a critical threshold, and the stability of the equilibrium point is affected by the memory delay, leading to inhomogeneous spatially periodic oscillations. Moreover, we explore the occurrence of Hopf–Hopf bifurcation and the stability switches. The induced Hopf–Hopf bifurcation is further studied in detail based on normal form theory and the center manifold theorem. Finally, numerical simulations are provided to validate our theoretical findings.