Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.