Dynamics in the biparametric spaces of a three-species food chain model with vigilance

In nature, vigilance is a common anti-predator strategy prey individuals employ to protect themselves from a possible attack. It gives them sufficient time to take cover or flee. Yet this seemingly profitable strategy, which lowers the number of successful predatory attacks, has a significant impact on the prey's growth rate. Researchers attribute this to reduced foraging time. In this article, we study a three-species food chain model in which both the basal prey and middle predator show active vigilance against their respective predators. Apart from the positivity, boundedness, and local stability analysis of the equilibrium points of the model, we find the conditions for global stability and the occurrence of Hopf bifurcation around the coexisting equilibrium point. Bifurcation diagrams and their corresponding Lyapunov exponent diagrams (largest and second-largest) illustrate diverse dynamical scenarios ranging from stable coexistence to periodic to chaotic oscillations. The isospike diagrams and their associated largest Lyapunov exponent diagrams in the biparametric space of basal prey's and middle predator's vigilance reveal a broader picture of the effects of different levels of vigilance on the dynamics of the system. A great deal of attention has been paid to unwrapping the complex structural beauties like shrimp-shaped periodic structures inside the chaotic sea and multistability between several sets of attractors in the overlapping zones of periodic windows. We observe that multiple coexisting attractors possess fractal basin boundaries. We also explore the dynamics of the system in two other biparametric spaces. The density variation maps at the end show the effect of vigilance on the equilibrium density of all three species. Our results suggest that vigilance promotes regularity and coexistence, but too high levels of vigilance can cause species extinction. For better visualization of the system dynamics, we provide a few high-resolution animations of the phase portraits and the basins of attraction in the supplementary material.