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 ϕ p and derivative gain ϕ d , 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.

Hunting cooperation widely exists in biological systems, which increases the possibility of the encounter between prey and predator populations. It is worth noting that predators are often divided into specialists and generalists. Among them, the research on considering cooperation mechanism in the specialist predator-prey model has received extensive attention. However, there is few attentions on the generalist predator-prey model. In this paper, we consider the classical host-generalist parasitoid model to determine the impact of such a cooperation mechanism. Specifically, by studying the existence and stability of the equilibria we find that without consideration of delay factors, the parasitoids’ cooperation does not change the stable coexistence of the two populations. Further, to explore the potential role of the cooperation mechanism in the system, we emphasize the dynamical variation induced by delay. We, therefore, investigate the underlying bifurcation properties under different cases, where the direction of Hopf bifurcation and stability of the periodic solutions are determined by the normal form theory and the center manifold theorem. Interestingly, a series of dynamical changes occur in the population, including periodic oscillation, irregular oscillation, and chaotic attractor when the double delays are introduced in both populations. Then through the numerical experiments, we examine the impacts of cooperation levels on the dynamic properties, which are just consistent with the theoretical findings. Hence, it is conclusive that the cooperation mechanism will inhibit the generation of complex phenomena in the host-generalist parasitoid system, thereby promoting the stability of the biological system. The present study may put insights into the mechanism of predator cooperation in a more straightforward and practical manner.