Rich dynamics of a predator-prey system with state-dependent impulsive controls switching between two means

Abstract

This paper deals with the control of a prey species (as an unwanted species) in a predator-prey system. We consider a scenario where there are two control means available and they are applied in a state-dependent impulsive way, meaning that when the population of the harmful species is lower than a preset threshold, no control measure will be implemented; while when it reaches the threshold, the two control means will be used either in alternating order or random order. We formulate a general mathematical model for this scenario to evaluate the effect of such a control strategy by exploring the dynamics of this model. We define a one-dimensional map (Poincaré map) and by using the properties of this map, we derive sufficient conditions for the existence and global stability of an order-k periodic solution. By using the analogue of Poincaré criterion and bifurcation theory, we also establish sufficient conditions for a transcritical bifurcation near the predator-free periodic solution. Finally, we apply the results for the general model to two particular cases from two distinct fields: (I) integrated pest control and (II) tumour control with a comprehensive therapy. For (I), theoretical and numerical results show that the outbreak period of the pest is longer when two pesticides are applied randomly than when the alternating strategy is used. For (II), we find that the treatment frequency of drug rotation strategy is lower than that of no drug change strategy, and that the higher the control intensity, the lower the treatment frequency.