Spatiotemporal dynamics optimization of a delayed reaction–diffusion mussel–algae model based on PD control strategy

The dynamic control over time evolution of ordinary differential systems has developed rapidly in recent years, but how to control the spatiotemporal evolution dynamics of partial differential systems is still an open question. Turing pattern is a major spatiotemporal evolution behavior in mussel-algal ecosystems and its control can pull the ecosystem back from the borderline of collapse and make it more stable. However, there has been relatively little research on the optimal control of Turing pattern in the mussel-algal system. In this paper, we first attempt to propose a proportional-derivative (PD) control strategy for the reaction–diffusion mussel–algae model under the influence of time delays. Turing instability driven by diffusion occurs in the diffusive mussel–algae model without control. The introduction of PD control can effectively suppress the occurrence of Turing instability and keep the ecological model stable in both time and space. When the time delay exceeds a critical value, the diffusive mussel–algae model will lose stability and undergo a Hopf bifurcation. PD controller can make Hopf bifurcation occur in advance or in hysteresis by changing the threshold point of Hopf bifurcation. The intrinsic mechanism of Hopf bifurcation is analyzed by means of the formal theory and the central manifold theorem. The numerical simulations demonstrate the efficiency and feasibility of the PD control scheme.