Globally stability analysis of the mathematical model in the IMTA system by using the energy-Casimir method

Abstract

Despite its benefits, fish farming has a potential impact on the water environment, such as algae bloom, fish death, and eutrophication. Integrated Multi-Trophic Aquaculture (IMTA) is designed to address environmental problems to reduce the excess of unfed pellets and fish feces under the cage. Mathematical modeling was built to describe a phenomenon in chemical, physical, and biological processes during the operation of IMTA to form a mathematical formula. The phenomenon was explained by the dynamic system, which is a method to describe, model, simulate and analyze dynamical systems. The model analyzed the interactions between nitrogen and phosphate concentrations and phytoplankton during the operation of IMTA. The model was a non-linear system of linear differential equations with three variables. Analysis of global stability is carried out at equilibrium points based on the Lyapunov stability theory using by Energy-Casimir method. Determine equilibrium point and Casimir functions of the dynamical systems, then assume that the Casimir functions are linearly independent. Find the value of the G matrix, then calculate the Lyapunov function with a positive definite value and test the validity of the Lyapunov function.