Stabilising spatiotemporal dynamics of mussel-algae coupled map lattices model via proportional-differential control.

The mussel-algae (M-A) system plays a crucial role in maintaining the balance of marine aquaculture ecosystems. Mussels filter algae from the water as a food source, while algae produce oxygen through photosynthesis and contribute to nutrient cycling. Fluctuations in the density and spatial distribution of algae populations can significantly impact the growth and reproduction of mussels, and conversely, mussels can influence algae dynamics, thereby potentially altering the equilibrium of the system. This study adopts a practical perspective, simultaneously considering the effects of self-diffusion and cross-diffusion, and establishes a spatiotemporally discretised coupled map lattices (CMLs) model for the M-A system. Utilising linear stability analysis, bifurcation theory, and the centre manifold theorem, we explore the stability and classification of fixed points within the CMLs model, as well as the parameter conditions that give rise to flip and Turing bifurcations. Numerical simulations demonstrate the rich temporal dynamics and spatiotemporal patterns induced by five different mechanisms. Notably, we introduce a proportional-differential (PD) control into the CMLs model for the first time. Through numerical simulations, we validate that the PD control can delay the occurrence of the flip bifurcation, thereby preventing multi-period oscillations and chaos in algal population density, which could lead to system instability. Moreover, the PD control can reduce the Turing instability region and adjust the Turing pattern types induced by the five mechanisms, thus ensuring a uniform spatiotemporal distribution of the algal population and contributing to the stability of the ecosystem.