Dynamical analysis of a prey–predator model in toxic habitat with weak Allee effect and additional food

The relationship between predator and prey in mathematical ecology is a fundamental natural occurrence that significantly impacts community organization and the preservation of ecological variety. Toxic effects on predator and prey populations are now becoming a popular topic in ecology. In this present work, we have discussed the Gauss-type prey–predator model with a simplified Holling type III functional response, including the weak Allee effect on the prey population. We have introduced toxicity, which impacts both predator and prey species. Furthermore, we have also included resources of additional food for the generalized predator species in order to study its effects on prey–predator interactions and their population size. An analysis of the model parameters has been conducted in order to determine the feasibility and stability criteria of all the equilibrium points. The parametric limitations of at least one interior equilibrium point have been deduced, and the results are shown numerically. It has been shown that the system exhibits local bifurcation such as saddle-node bifurcation, Hopf bifurcation, generalized Hopf bifurcation, Bogdanov–Takens bifurcation, and Cusp bifurcation. Also, we have obtained global bifurcation such as homoclinic bifurcation. We have also discussed the hysteresis loop along with the basin of attraction of the proposed system. Each of these theoretical results are supported by several numerical examples.