Basins of attraction in a modified ratio-dependent predator-prey model with prey refugee

In this paper, we analyze a modified ratio-dependent predator-prey model with a strong Allee effect and linear prey refugee. The model exhibits rich dynamics with the existence of separatrices in the phase plane in-between basins of attraction associated with oscillation, coexistence, and extinction of the interacting populations. We prove that if the initial values are positive, all solutions are bounded and stay in the interior of the first quadrant. We show that the system undergoes several bifurcations such as transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Consequently, a homoclinic bifurcation curve exists generating an unstable periodic orbit. Moreover, we find that the Bogdanov-Takens bifurcation acts as an organizing center for the scenario of surviving or extinction of both interacting species. Topologically different phase portraits with all possible trajectories and equilibria are depicted illustrating the behavior of the system.