The name is absent

Within the lake, phosphorous is recycled and made available from living or-
ganisms and sediment. Phosphorous release from sediment is a major source of
recycled phosphorous and the cause of the hysteresis phenomenon. Lake stud-
ies have shown that long after external loading of phosphorous has ceased, the
available stock of phosphorous remains high due to the release of phosphorous
from compounds in the sediment. Scheffer (1998) gives an example of a Danish
lake studied over a period of eight years following reduction in external loading.
During this time, phosphorous continued to be released from the sediments. At
the end of the study, it was concluded that the lake could continue to release
phosphorous for another ten years.

The recycling equation is a sigmoid function, that is, it has an inflection
point at which the curve switches from convex to concave. Phosphorous recy-
cling is given by

rP^q (t)
m^q + P^q (t)

where

r is the maximum rate of recycling of P.

m is the value of P at which recycling reaches half the maximum rate r.

q is dimensionless and determines the steepness of the curve at the point
of inflection. q ≥ 2 and the larger q , the steeper the curve.

Adding the sources of phosphorous, i.e. loading and recycling, gives us the
total phosphorous made available for consumption at time t:

_{L(t) +}   r^{p q} (t)

⁽ )⁺ m« + Pq(t)

Lake-Phosphorous Dynamics

The change in phosphorous stock in the lake at time t is given by the sum of
the phosphorous sinks and source at time t and the lake-phosphorous dynamics
are therefore modelled as:

dP                     rP^q(t)

=L = L(t) - sP(t) +--'

dt                   m^q + P^q(t)

The plots of the sinks and sources of phosphorous can be overlayed to show
how controlling phosphorous loading can be used to affect the state of the lake.
Refer to Appendix A for the graph and its analysis.

2.2 The Shallow Lake Model

The focus of this paper is the case where the eutrophic state of the lake is
reversible by reductions of phosphorous alone. This case lends itself to the
application of policy that induces agents to modify the activities that result
in external phosphorous input. Because of the possibility of hysteresis in the
response of the lake to phosphorous loading, however, policy implementations
must allow for reduction in loading not having an immediate effect on restoring

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