  In this paper, the selective predator harvesting is considered in a predator
prey system. The logistic growth of prey and ratio-dependent functional
response is assumed. The dynamically varying effort is employed for selective
harvesting of predator species using Michaelis-Menten type function The
boundedness, positivity and persistence of the nonlinear dynamical system is
established. The system being non-singular about origin, its dynamics is
explored by transforming it to a regular system. It is observed that the system
may collapse even with positive initial conditions attracted to one of the
boundary points under certain conditions. The local stability of various other
equilibrium states of the dynamic model is investigated. The region of
attraction of interior equilibrium state is obtained using Lyapunov stability.
The occurrence of hopf bifurcation about the coexistence equilibrium point with
respect to the parameter m (fraction of predators available for harvesting) is
established. The sustainable harvesting is possible in two ways firstly in the
form of stable interior equilibrium state and secondly in the form of limit
cycle. Considering the fraction m as a control variable, the optimal harvesting
policy is obtained using Pontraygin's maximum principle. The two-parameter
bifurcation diagram is obtained with respect to c and d (the cost of harvesting
and the death rate of predators respectively). The two bi-stability regions of
parameter space are identified. It is proved that for sustainable harvesting of
predators the ratio of p to c (price to cost) should be greater than a certain
threshold value dependent on predator density. The analytical results are
illustrated numerically with different parametric values.
