Mathematical modeling and analysis of the depletion of dissolved oxygen in lake ecosystem

Abstract

A non-linear mathematical model is being proposed to study the depletion of dissolved oxygen caused by the excessive discharge of organic pollutant in water body. The interactions among concentration of nutrients, density of algae, density of detritus, density of zooplankton and concentration of dissolved oxygen are considered in this model. The model consists of five coupled non-linear differential equations. To validate the model, the boundedness of the state variables using the theory of differential inequality and positivity of each state variables have been done in this research. The equilibrium points of the proposed model have been demonstrated. The stability of the equilibrium points has been checked by computing the eigen-value and applying Routh’s Hurwitz criterion. Finally,the characteristics of the state variables with respect to different values of different parameters such as cumulative rate of discharge of nutrients, natural depletion rate of zooplankton, growth rate of algae due to nutrients, natural depletion rate of algae, and depletion rate of dissolved oxygen due to detritus have been discussed both graphically and analytically. It is speculated that detritus uses dissolved oxygen to supplement the total concentration of nutrients in the water body. It has been demonstrated graphically that the density of algae is strongly influenced by the cumulative rate of nutrient discharge from water or wastes itself. Algae engage in self-purification by eating both organic and inorganic contaminants while also making substances that are advantageous to their surroundings. Alarmingly excess detritus reduces the concentration of dissolved oxygen. It has been demonstrated numerically that the density of zooplankton rises when the rate of algal depletion and the rate of nutrients delivery from diverse sources both increase.In this study, it has also been discussed how the concentration of dissolved oxygen decreases as a consequence of the cumulative rate of nutrient discharge, the rate of algae growth due to nutrients, the rate of dissolved oxygen depletion due to detritus, and the rate of zooplankton growth due to algae. In conclusion, the study highlights the intricate interactions between the state variables and emphasizes the delicate balance found within an aquatic ecosystem. The results highlight how crucial it is to comprehend these processes in order to manage and maintain the health of our aquatic ecosystems.