PD control-driven regulation of spatiotemporal patterns in a delayed predator–prey system

In this paper, we introduce a Proportional-Derivative (PD) control into the Leslie–Gower predator–prey reaction–diffusion model with time delay to explore its impact on self-organized spatial pattern formation. By incorporating time delay, the model captures realistic ecological constraints arising from predation pressure. Through linear stability analysis and bifurcation theory, we derive the critical conditions for Hopf and Turing bifurcations and demonstrate how PD control modulate these bifurcations to influence pattern formation. Numerical simulations reveal that PD control effectively stabilizes complex ecological patterns, including spiral patterns, spot patterns and irregular patterns, providing a potential approach for ecosystem regulation. Furthermore, we investigate pattern transitions within the Turing instability region using the pattern selection theorem, demonstrating the ability of PD control to guide the system toward desired ecological states.