Modulating spatiotemporal patterns: proportional-derivative control in a networked predator–prey system

Abstract

Spatiotemporal patterns in ecological networks are critical for understanding population dynamics and ecosystem stability. This study investigates the role of a proportional-derivative (PD) controller in modulating Hopf bifurcation-induced spatiotemporal patterns in a networked predator–prey model with Allee effect, group defense, and time delay. Through linear stability analysis, we derive the conditions under which the controlled system undergoes a Hopf bifurcation, revealing that the critical bifurcation threshold depends explicitly on the PD control parameters. Subsequently, we perform numerical simulations on three representative network topologies, regular, small-world, and scale-free networks, to examine pattern formation. Our results demonstrate that by appropriately adjusting the proportional gain ϕp and derivative gain ϕd, the system can effectively suppress complex spatiotemporal patterns, stabilizing the homogeneous steady state even in regimes where the uncontrolled system exhibits sustained oscillations. This work highlights the potential of PD control as a practical strategy for managing spatiotemporal complexity in fragmented ecological landscapes.