Bifurcation and dynamic analysis of prey–predator model with combined nonlinear harvesting

Abstract

Due to the random search of species and from the economic point of view, combined harvesting is more suitable than selective harvesting. Thus, we have developed and analyzed a prey–predator model with the combined effect of nonlinear harvesting in this research paper. Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system. We have examined the local stability analysis of all the equilibrium points. Besides these various types, rich and complex dynamical behaviors such as backward, saddle-node, Hopf and Bogdanov–Takens (BT) bifurcations, homo-clinic loop and limit cycles appear in this model. Furthermore, interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points. Also, we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species. We have observed that the extinction of predator species may not happen due to backward bifurcation, although a stable predator-free equilibrium (PFE) exists. Finally, numerical simulations are discussed using MATLAB to verify all the theoretical results.