is sparse and more present around the edges, shallow lakes are often filled with
aquatic plants (Scheffer, 1998).
When the nutrient level of the lake is low, the plants tend to be small and
the water clear. Increases in nutrient loading, however, encourage the devel-
opment of larger plants and of phytoplankton. These plants and the surface
layer of phytoplankton create shade and turbidity, which leads to the collapse
of the vegetation that does not tolerate shade. This further favours the devel-
opment of phytoplankton, and can result in the emergence of toxic algal bloom,
cyanobacteria, which are shade tolerant (Scheffer, 1998).
This section presents the model of a shallow lake presented by Carpenter
et al. (1999), which they claim accurately depicts long-term ecological data
and the results of limnology and eutrophication studies. The lake equation
they propose provides the constraint equation to the economic analysis that
follows. In this model, the limiting factor for eutrophication is phosphorous.
Lake eutrophication dynamics are based on total available phosphorous as the
state variable, and phosphorous input as the control variable.
Although nitrogen is also used to stimulate plant growth, a model based
on phosphorous makes sense because phosphorous is thought to be the limiting
nutrient of plant growth in many cases (Ricklefs, 1979). In addition, cyanobac-
teria have the ability to fix nitrogen from the atmosphere, and therefore their
growth will be limited by the phosphorous available (Alaouze, 1995).
Carpenter et al. (1999) identify three categories of lakes by their response to
phosphorous input and reductions: fully reversible, hysteretic and irreversible.
Our focus is an economic model of which the goal is to address eutrophication
through policy aimed at mitigating phosphorous input alone.
2.1 The General Lake Model
The different functions of the lake with respect to phosphorous are used to build
the lake-phosphorous dynamics equation as follows.
Phosphorous Sinks
Phosphorous is removed from the stock P available to algae via outflow, se-
questration into biomass and sedimentation. Sedimentation is often the largest
factor contributing to phosphorous loss and thus confers the lake its phospho-
rous waste sink function. The removal of phosphorous from the stock at time t
is modelled as a linear function -sP (t) where s is the rate of loss of phoshorous
from the available stock.
Phosphorous Sources
Nutrients are retained by the lake and recycled between the physical envi-
ronment and made available to living organisms such as fish or benthic plants
(Ricklefs, 1979). Therefore external phosphorous loading from the catchment,
such as run-off from agricultural activities or sewage effluents, contribute to
the total stock of phosphorous available to consumers. The external input of
phosphorous at time t is represented by L(t).