On–off policy and hysteresis on–off policy control of the herbivore-vegetation dynamics in a semi-arid grazing system

In the grazing management context, an on–off policy as well as a hysteresis on–off policy can be used to avoid the collapse in the vegetation-herbivore dynamical system model. We consider a dynamical system sensitive to soil degradation. The on–off policy changes the dynamics of the vegetation-herbivore system in such a way that a positive stable equilibrium point is achieved, and the hysteresis on–off policy changes the dynamics in such a way that a limit cycle (bounded oscillation) can be achieved, and the multiple stable states that arise in this type of system are avoided. A new model is introduced in which changes in the dynamics that occur on account of switching are explicitly introduced. Policies can be designed by a suitable choice of so called virtual equilibrium points in a simple and intuitive manner.