We study a three-species food chain (prey, intermediate predator, top predator) in a habitat polluted by a constant input of an environmental toxicant. Two features distinguish the model from earlier work: the toxicant concentration amplifies through the prey via a feedback term $c_2 P_1 C$, and the prey-uptake term in the intermediate predator's equation depends on a distributed time delay rather than the instantaneous prey density. Using the linear chain trick we recast the integro-differential system as a five-dimensional ODE. We show that the system admits four biologically meaningful equilibria, give existence conditions for each, and obtain Routh-Hurwitz conditions for local asymptotic stability. A Lyapunov function then yields global asymptotic stability of the coexistence equilibrium $E_4$ inside an explicit attracting region. We derive in closed form the characteristic equation at $E_4$ and the transversality condition for a Hopf bifurcation, treating in turn the toxicant amplification rate $c_2$ and the natural degradation rate $d_3$ as bifurcation parameters. Numerical simulations on a representative parameter set verify the analytical Hopf critical values $c_2^* \approx 0.0327$ and $d_3^* \approx 0.1834$ and exhibit the corresponding limit cycles in the $(P_1, C)$ plane.
Makwana,K and Babu A.,R . (2026). Mathematical modelling the effect of pollutants in a three-species food-chain model by considering distributed delay. (e736781). Mathematics and Computational Sciences, (), e736781 doi: 10.30511/mcs.2026.2062914.1369
MLA
Makwana,K , and Babu A.,R . "Mathematical modelling the effect of pollutants in a three-species food-chain model by considering distributed delay" .e736781 , Mathematics and Computational Sciences, , , 2026, e736781. doi: 10.30511/mcs.2026.2062914.1369
HARVARD
Makwana K, Babu A. R. (2026). 'Mathematical modelling the effect of pollutants in a three-species food-chain model by considering distributed delay', Mathematics and Computational Sciences, (), e736781. doi: 10.30511/mcs.2026.2062914.1369
CHICAGO
K Makwana and R Babu A., "Mathematical modelling the effect of pollutants in a three-species food-chain model by considering distributed delay," Mathematics and Computational Sciences, (2026): e736781, doi: 10.30511/mcs.2026.2062914.1369
VANCOUVER
Makwana K, Babu A. R. Mathematical modelling the effect of pollutants in a three-species food-chain model by considering distributed delay. MCS. 2026;():e736781. doi: 10.30511/mcs.2026.2062914.1369
