Stability Analysis of the Plankton Ecosystem Model Considering the Oxygen Deficit

Abstract

Plankton are microscopic organisms that lives suspended on water. In general, the planktons were divided into two groups : group of plants (phytoplankton) and group of animals (zooplankton). The phytoplankton contain chlorophyll which makes this group able to perform photosynthesis activity. The results of photosynthesis is a source of nutrition for other organisms, either directly or indirectly through the food chain system. Therefore, availability of plankton population in the water affect the survival of other organisms. In this study, a mathematical model was proposed that able to describe phenomena occur in the ecosystems plankton. This model is presented in nonlinear differential equations system involving four variables : concentration of nutrients, density of algae, density of zooplankton and concentration of dissolved oxygen. This model is interesting because the growth rate of algae and zooplankton are not only influenced variables, but also by oxygen deficit. The oxygen deficit is defined as the difference between saturation and concentration of oxygen. A high dissolved oxygen concentration will reduce the oxygen deficit, so that the growth of algae and zooplankton increase. If the oxygen concentration decreases, then the oxygen deficit will increase and the growth of algae and zooplankton decreas. In this study, there are four equilibrium points. Obtained, the stability analysis will be done for each equilibrium points. For the first three equilibrium points, the analysis conducted through the eigenvalues of each matrix Jacobi. Different from other equilibrium points, for a fourth equilibrium points, analysis conducted with Lyapunov direct method. The result of analysis showed that the fourth equilibrium points will be stable with certain criteria, while three other equilibrium points are unstable. Then the results of analysis will be examined through numerical simulation by substituting the values of parameters. Those result showed that system will reach a stable condition at the coordinates of fourth equilibrium points. In this section also illustrated the dynamics that occur until the system reaches a stable condition. The results of this study, obtained a condition that makes the plankton population in the water available in stable amount.