Mathematical analysis of the nitrogen and phosphate to phytoplankton, macrobenthos, and sediment in aquaculture system

Abstract

In aquaculture system, metabolic waste in the form of unfeed pellets and fish excretion can reduce water quality and increase both soluble organic matter in waters and organic deposition to sediments. A developed dynamical system was formed based on the interactions that occur between phytoplankton growth, macrobenthos density, and sediment in response to nitrogen and phosphate concentrations. The purpose of this research was to analyze the stability of the proposed dynamical system. Furthermore, we determine the level of water quality in the aquaculture system based on the interaction between five variables, i.e. nitrogen, phosphate, abundance of phytoplankton, abundance of macrobenthos, and sediment properties. The local stability of the system equations model was determined by investigating the eigen values of the Jacobian matrix. The Lyapunov stability theory using the Krasovskii method was used to determine the global stability of the systems. In the Lyapunov method, if the value of the scalar function is definite positive and its first derivative is definite negative, then the system is globally asymptotically stable. Based on mathematical analysis and numerical results, the aquaculture dynamic system model in Jepara was globally asymptotically stable. This indicated that the quality of the waters was still considerably in normal condition.