A theoretical perspective on managed rangelands and irreversible states

In spite of a manager's best attempts, a managed rangeland may hit an irreversible state in which it provides neither consumptive nor nonconsumptive services to humans. Therefore, given a particular time-based management regime, it is useful to know how long it takes for a rangeland to hit the irreversible state. The purpose of this paper is to conduct a theoretical analysis of this and related questions. In particular, we first provide a stochastic characterization of a time-based range management regime. Second, we ascertain the expected amount of time it takes for our managed rangeland to hit the irreversible state. Third, we discuss the properties of the above mathematical expectation. Finally, we pose and discuss a simple optimization problem for our range manager.