Construction of lyapunov function using gradient method to stability analysis of the nitrogen-phosphate-phytoplankton-sediment interaction model

Abstract

The significant growth in catch fisheries and aquaculture production has enhanced the world's capacity to consume diverse and nutritious food; however, this fast-growing industry has resulted in the environmental disturbance, especially caused by fish farming. The nutrient was generated from fish excretion and unfed pellet in the form of dissolved organic matter, especially as nitrogen and phosphate particulates. Therefore, applying mathematical models is crucial to understand and minimize the impact on the water ecosystem, thus maximizing productivity. This paper aims to research to analyze the global and stability of the equilibrium point the dynamical system that water and sediment in aquaculture Integrated Multi-Trophic Aquaculture (IMTA) system. Analysis stability global has been proved by constructing the Lyapunov function. The gradient method is used to construct the Lyapunov function in the most general form. The method is based on the assumption of a variable gradient function by determining V and ∇V. In the Lyapunov method construction, if the function V(x) would be definite positive, whereas derivative V(x) would be a definite negative. Models of nitrogen, phosphate, phytoplankton, and sediment system exhibited globally asymptotically stable. This implies that fish farming using the IMTA system on Karimunjawa island was still considerably under normal condition.