A Control Mathematical Model for the Survival of a Biological Species Affected by Two Toxicants, One Emitted from External Sources as well as Formed by its Precursors and the Other Produced by Species itself

Abstract

In this paper, a control mathematical model is proposed to study the simultaneous effect of two toxicants with different toxicities emitted into the environment on the biological species. One of the toxicants is emitted into the environment by external sources and their concentration is augmented by precursor and the other toxicant is produced by the population itself through its various actions.The model is formulated by using system of non-linear ordinary differential equations. In the analysis, all the feasible equilibrium points of the system have been obtained. The conditions for local and non-linear stability of the non-trivial equilibrium points have been carried out using a suitable Lyapunov function. Also, a region of attraction has been found for non-linear
asymptotic stability of the equilibrium point. . It has been shown that the density of the biological species decreases with the increase in the total emission rate of pollutant in the environment. The analysis of the non linear stability shows that the system settles at much lower density of the biological species when the concentration of the pollutants in the environment and in the uptake phase of the
species is high. But the control measures (environmental taxes) imposed on the pollutant emitting industries and people, control concentration of the pollutants in the environment and due to this, the equilibrium point shifts in such a way that the density of the biological species is more near to the density when eco-system is pollution free.