A Comprehensive Study of Bifurcations in an Interactive Population Model with Food-Limited Growth

In a two-dimensional prey–predator model with a Holling type-II functional response and a food-limited prey growth rate, this study explores the consequences of predator harvesting. There are utmost three coexisting equilibrium points in the system. It has been shown that the prey population’s half-saturation constant has a significant role in boosting the complicated bifurcation structure. To do this, we thoroughly examine the suggested system for codimension-1 bifurcations, including Hopf and saddle-node bifurcations. Calculating the conditions of Sotomayor’s theorem the presence of saddle-node bifurcation is ensured and the signs of first Lyapunov number are computed for the stability of periodic solutions via Hopf bifurcation. With the implementation of saddle-node bifurcation, an unforeseen scenario of species extinction is carried out in the case of three interior equilibria with respect to the half-saturation constant, which gives results apart from the traditional one. Additionally, we executed the continuation of codimension-1 bifurcations for the emergence of codimension-2 bifurcations like generalized-Hopf, cusp, and Bogdanov–Takens bifurcations to understand the role of harvesting the predator population. The work becomes more appealing because it displays topologically different phase diagrams for suitable parameters. Ecologically, the generalized-Hopf bifurcation shows that the system’s behavior is quite sensitive to the prey saturation constant and predator harvesting. In addition, we compare the results of proposed system with the model for a saturated harvesting and linear functional response. Extensive numerical simulations are performed to validate the conclusions for stability and bifurcations.