Modelling effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey

In this paper, a mathematical model is proposed and analyzed to study the effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey population. The model is formulated by using the system of non linear ordinary differential equations. In the model, it is assumed that the growth rate of prey population is affected by organismal toxicant. In this paper, we have introduced distributed delay in the environmental toxicant in the model. The distributed delay differential equations, though simple in structure, possess a rich array of solutions. The models are being analyzed by using variational matrix and Liapunov functions. The conditions for local and global stability of the equilibrium points are obtained. A region of attraction is being found for global asymptotic stability of the equilibrium points. Also, a Hopf bifurcation analysis has been performed with respect to key parameters for non-trivial equilibrium points. Furthermore, we support our analytical findings with numerical simulations. © 2016, Springer International Publishing Switzerland.