Exploring tri-trophic food chain model with toxicity and additional food: Unraveling bifurcation, chaos, density variation and bistability

The prey–predator interaction significantly influences ecosystems. Prey species perform anti-predator activities while foraging and balance their food and safety requirements to improve their survival chances. Several prey species use strategies to mitigate predation risk, including relocating to lower-risk and less lucrative habitats, increasing vigilance and modifying reproductive strategy. Certain prey species use group defenses to diminish the possibility of predator’s predation. This study examines a model involving one prey and two predators, where the prey species exhibits group defense against predator species. While studying we have got several codimension-one bifurcations (Transcritical, Saddle node, Hopf bifurcation) and codimension-two bifurcations (Bogdanov–Takens, generalized Hopf, Cusp bifurcation) exhibited by the proposed system. We have conducted both one-parametric and two-parametric bifurcation analyses of our proposed model. We have investigated the complex dynamical characteristics of the system over several parametric planes using bifurcation and Lyapunov exponent diagrams. We have observed several types of organized periodic structures, including both period-doubling and period-halving routes to chaos. Additionally, we have discussed the variation in the population density of each species. This study examines the influence of toxicity parameters on the dynamics of the system and their roles in the survival of the species.