Impact of fear on a tri-trophic food chain model with supplementary food source

The present article deals with the dynamical complexity of a three-species food chain model with the inclusion of supplementary food source for the predator at the expense of fear into the prey population. Basic preliminaries regarding existence, uniqueness, positivity, boundedness and dissipativity of the model system are provided as essential requirements. The existence of ecologically feasible equilibria of the system along with their stability analyses are carried out. The stable steady state turns out to be reactive if there is a unique interior equilibrium point. When the system grows bistability, the outcomes become complex. One of the two stable coexistence states changes from being non-reactive to reactive. However, it has been found that the disruption brought about by the fear component in the present system modifies the system ephemeral behaviour, making it more stable in every instance compared to the entire Hastings-Powell model system. Additionally, proper consideration is given to the local bifurcation analysis of codimensions 1 and 2 with respect to the significant system parameters. The source of chaotic dynamics (chaotic -> period doubling -> limit cycle -> stable focus) has been intriguingly investigated using the maximum Lyapunov exponent. Through quantitative simulations, certain fascinating dynamical complexity, such as the variation in the number of equilibria and the bifurcation scenario, have also been revealed. Numerical simulations are performed in order to explore the influence of both fear among the prey and supplementary food source for the predator on the interacting population. The numerical results reveal that the impact of both these factors can alter the stability/instability of the model system under consideration. The effects of the fear factor and supplemental food source are carefully examined in regard to the coexistence equilibrium point.