Theoretical study and control optimization of an integrated pest management predator–prey model with power growth rate

• This work presents a pest control scaled prey-predator model, where the yield releases of the predator and chemical control strength are dependent on the pest control level.
• It has been shown the change in system dynamics by the exponent and the existence of order-1 periodic orbit which keeps the pest density below the selected control level.
• It has extracted a stability criterion by geometric analysis method and verified the stability of the order-1 periodic orbit.
• It has formulated an optimization problem and obtained the optimum pest control level.

This work presents a pest control predator–prey model, where rate of change in prey density follows a scaling law with exponent less than one and the control is by an integrated management strategy. The aim is to investigate the change in system dynamics and determine a pest control level with minimum control price. First, the dynamics of the proposed model without control is investigated by taking the exponent as an index parameter. And then, to determine the frequency of spraying chemical pesticide and yield releases of the predator, the existence of the order-1 periodic orbit of the control system is discussed in cases. Furthermore, to ensure a certain robustness of the adopted control, i.e., for an inaccurately detected species density or a deviation, the control system could be stabilized at the order-1 periodic orbit, the stability of the order-1 periodic orbit is verified by an stability criterion for a general semi-continuous dynamical system. In addition, to minimize the total cost input in pest control, an optimization problem is formulated and the optimum pest control level is obtained. At last, the numerical simulations with a specific model are carried out to complement the theoretical results.