Effect of fear with saturated fear cost and harvesting on aquatic food chain model (plankton–fish model) in the presence of nanoparticles

Studying the interplay of phytoplankton–zooplankton–fish (PP–ZP–F) in an aquatic system is crucial for better understanding of nutrient cycling, assessing ecosystem health, predicting and mitigating harmful algal blooms, and managing fisheries in the water bodies. In order to investigate the effectiveness of nanoparticles (NPs), fear, and harvesting, this paper focuses on exploring the dynamics of a food chain model among PP–ZP–F species. We consider the fear of fish on zooplankton species (which reduces the reproduction rate of ZPs) with saturated fear cost in the presence of nanoparticles (NPs) and harvesting in fish. The system dynamics are studied from the viewpoint of proving positivity, boundedness, and uniqueness, followed by analysing the existence and local stability of biologically feasible equilibria. Conditions for the global stability of the interior equilibrium point are also found. Furthermore, we established the transversality conditions for the occurrence of Hopf, transcritical, and saddle–node bifurcations. To validate our theoretical results, we made numerous phase portraits, time-series graphs, tables showing the extinction of species, and bifurcation diagrams. It is numerically observed that increasing the contact rate of NPs with PPs makes the system stable from chaos, and further increase of contact rate may lead to extinction. Chaos at a low contact rate can also be managed by increasing the fear level, and the chaotic behaviour at a low fear level can again be controlled by enhancing the harvesting of fish species. Over-exploitation may result in the extinction of fish, whereas fear may promote coexistence, stability, and long-term survival of the species. Increased saturated fear cost can make the system chaotic from stable dynamics. Therefore, the theoretical as well as numerical findings of our paper may be of great interest in estimating the behaviour of aquatic systems biologically and practically.