Dynamical complexity of a three species food chain model incorporating delays and toxic habitat

The impact of toxins on predator and prey populations has recently gained attention in ecological modeling. In our research, we want to develop and analyse a food chain model that integrates the effects of toxins produced within the ecological system. The model focuses on a three-tier food chain, comprising prey, an intermediate predator, and a top predator, where toxins released by species impact their interactions and survival rates. The growth rates of prey and middle predators are assumed to follow a logistic pattern. By incorporating time delays in growth, toxin accumulation and its effects, we explore how delay and toxin influence together alters the stability of the system. In this model, we have shown that the system exhibits uniform persistence. The necessary and sufficient conditions for stability and Hopf-bifurcation of the coexistence interior equilibrium point have been discussed analytically and numerically both, taking delay as threshold parameter. In addition to that, Lyapunov function has been constructed to study the local asymptotic stability of the coexistence interior equilibrium point. Furthermore, numerical simulations have been carried out with appropriate parameter values to validate the analytical results obtained. Finally, in conclusion, the paper summarizes our findings and highlights potential directions for future research on this topic.