Investigation of Spatial Pattern in SI Model with PD Control and Cross-Diffusion

Spatial patterns are an important characteristic of ecological epidemic reaction–diffusion models, as they help predict the spatial dynamics of epidemic spread. However, there is currently a significant shortage in research on control strategies for generating and regulating these patterns. In this paper, we establish a Susceptible–Infectious (SI) model with Proportional-Derivative (PD) control and cross-diffusion. We analyze the stability of the model without diffusion and provide the conditions for Turing instability driven by diffusion. Using the cross-diffusion coefficient as the bifurcation parameter, we derive the amplitude equation of the two-dimensional Turing pattern at the Turing bifurcation threshold based on the multiple scale method, which determines the selection and stability of pattern formation. Numerical simulations in two-dimensional space show that the bifurcation parameter varies with changes in control parameters, and the PD control not only effectively alters the structure of Turing patterns but also suppresses Turing instability. Furthermore, it is found that more complex spatial patterns in three-dimensional space emerge under PD control.