Control Ecological Model for the Survival of Biological Species which are Directly Affected by Pollutants Emitted from External Sources: A Qualitative Approach

Abstract

In this paper, a non-linear control ecological model is proposed and analyzed to study the survival of biological species which is directly
affected by a toxicant emitted into the environment by external sources. 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 nonlinear 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.It has been found that the control measures (carbon emitter taxes) prove as disincentive to the emitters of the pollutants and its emission at source is checked and reduced and the biological species can be maintained at a desired level. The results are illustrated with the help of a numerical example and computer simulation.