The system represented by the equations for Xt+1 has two domains of behavior, separated by
a threshold at Xc which may or may not be known with certainty. We assume that when the pol-
lutant level is below Xc, the system is in a desirable stability domain, as for any given natural and
anthropogenic pollutant inputs, the expected pollutant level in the following period will be less
- by an amount equal to r > 0 - than when the current pollutant level is above Xc. A model
specification similar to ours was used by Peterson et al. [25] to study the dynamics of a fresh-
water lake ecosystem. In that setting, the pollutant X represented phosphorus loading to the lake,
and r represented additional phosphorus recycling that occurred when the lake switched between
oligotrophic (desirable) and eutrophic (undesirable) states at the threshold Xc. Peterson et al.’s
model assumes that current management actions and pollutant loading have no effect on recycling
in the current period, but only in future time periods. In this paper, we make the more realistic
assumption that threshold crossings (such as caused by phosphorus recyling) depend not only on
the carryover from the previous period but also on current loading and an error component vt . An
intuitive interpretation of the error component vt is that it represents uncertainty in the natural
system. This may be because the threshold itself may be subject to some movement, ecosystem
processes operate at differing rates, or real ecological thresholds may involve multiple interacting
slow and fast variables [4]. The advantage of including vt is that the resulting value function is
concave, continuous, and differentiable, even at Xc . As a result, we are able to obtain an exact
analytical solution to the optimization problem, rather than requiring numerical approximations
such as those used in previous studies [5, 18]. Our approach allows us to analyze the range and
characteristics of optimal behavior in much greater detail than existing studies that use numerical
solution methods.
We assume that society derives economic benefits from the ability to increase the pollutant
loading of the environmental system. These benefits are given in each period by the utility function
U (lt, Xt) = klt - Xt2. Examples of such benefits might include the capacity of ecosystems to
assimilate waste by-products from industry or agriculture, or the value of ecosystem functionality
in maintaining habitat. Note that from society’s point of view, the utility function shows a tradeoff
between the benefits of allowing increased pollutant loading and the negative consequences of the
increased pollutant stock.