Donald Ludwig, Steve Carpenter,
and W. Brock. Year. Optimal phosphorus loading for a potentially eutrophic lake.
Ecological Applications 13:11351152.
Appendix A. A derivation of
the difference equations.
If the rates of sedimention (s)
and flushing (h) are not small, then there may be competition between
the corresponding processes: the simple model that is linear in these parameters
is not appropriate. For instance, if
and s + h > 1, then P may become negative, which is impossible.
Appendix A of Carpenter et al. (1999) addressed this
issue by deriving the difference equation from a differential equation. The analogous
equation for P would be
If we ignore changes in L and the recycling term (this is by no means justified
except by expediency), the differential equation may be integrated after multiplying
it by exp[(s + h)t]. This yields
Notice that if s + h is small, then Eq. A.4 reduces to Eq. A.1,
provided that the loading is adjusted to correct for loading reduction through
sedimentation and flushing. The loss in P due to sedimentation and flushing
is (1 esh)Pt. The fraction of this due
to sedimention (namely s/(s + h)) adds to the M equation.
A similar argument applies to the terms involving loading and recycling. Hence
Note that if s and h are small, then g(s, h)
s/2. Hence Eq. A.5
reduces to Eq. A.2 if s and h are small, except for a term Ls/2,
which represents the load that is sedimented during the year.
Carpenter, S. R., D. Ludwig, and
W. A. Brock. 1999. Management of eutrophication for lakes subject to potentially
irreversible change. Ecological Applications 9:751771.
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