Dynamics of a guanaco–sheep competitive system with unilateral and bilateral control

In this paper, based on a guanaco–sheep competitive system, we develop and analyze mathematical models with unilateral and bilateral control for the management of overgrazing. We first analyze the dynamics of the uncontrolled system. It then follows the analysis of the system with impulsive control by differential equation geometry theory. We mainly prove the existence and stability of order-1 periodic solution for unilateral control system and order-2 periodic solution for bilateral control system. Some numerical simulations including the bifurcation diagrams of periodic solution are carried out, which not only verify the validity of the theoretical results, but also reveal some special dynamic phenomena, such as the appearance of higher-order periodic solutions and the existence of parameter intervals with the order change of periodic solution. Comparing the unilateral and bilateral control strategy, we encourage bilateral control rather than unilateral control for the management of sheep species.