Dynamical System Stability Analysis of the Interaction Nitrogen, Phytoplankton, Macrobenthos, and Sediment with Gradient Variable Method

In the shrimp pond aquaculture system, metabolic waste in the form of unfeed pellets and shrimp 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 analize the stability of the proposed dynamical system. Furthermore, we determine the level of water quality in the aquaculture system based on the interaction between four variables, i.e. nitrogen, abundance of phytoplankton, abundance of macrobenthos, and sediment properties. The Lyapunov stability theory using the Gradient Variable method was used to determine the global stability of the systems. Assumption of this method is determine and for gradient variable function. In the Lyapunov function, 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 simulation, the shrimp pond dynamical system model in BBPBAP Jepara, Indonesia was globally asymptotically stable at equilibrium point.