Study of co-dimension two bifurcation of a prey-predator model with prey refuge and non-linear harvesting on both species

The dynamics of prey-predator system, when one or both the species are harvested non linearly, has become a topic of intense study because of its wide applications in biological control and species conservation. In this paper we have discuss different bifurcation analysis of a two dimensional prey-predator model with Beddington-DeAngelis type functional response in the presence of prey refuge and non-linear harvesting of both species. We have studied the positivity and boundedness of the model system. All the biologically feasible equilibrium points are investigated and their local stability is analyzed in terms of model parameters. The global stability of coexistence equilibrium point has been discussed. Depending on the prey harvesting effort (E-1) and degree of competition among the boats, fishermen and other technology (l(1)) used for prey harvesting, the number of axial and interior equilibrium points may change. The system experiences different type of co-dimension one bifurcations such as transcritical, Hopf, saddle-node bifurcation and co-dimension two Bogdanov-Takens bifurcation. The parameter values at the Bogdanov-Takens bifurcation point are highly sensitive in the sense that the nature of coexistence equilibrium point changes dramatically in the neighbourhood of this point. The feasible region of the bifurcation diagram in the l(1) - E-1 parametric plane divides into nine distinct sub-regions depending on the number and nature of equilibrium points. We carried out some numerical simulations using the Maple and MATLAB software to justify our theoretical findings and finally some conclusions are given.