Necessary and sufficient conditions for the equivalence of economic and ecological criteria in range management

For rangelands managed with spatial and temporal controls, [a] the long run expected net cost of management operations, [b] the long run rate of moving grazing from one paddock to another, and [c] the long run rate of moving grazing between paddocks because of adverse environmental factors are salient criteria for a range manager. In this paper, we provide the first theoretical analysis of the relationship between these three economic and ecological criteria. We first characterize these criteria mathematically. Then, we provide necessary and sufficient conditions under which criterion [a] (economic criterion) is equivalent to, in turn, criterion [b] (first ecological criterion) and criterion [c] (second ecological criterion).