Ecological Archives A013-019-A1

Donald Ludwig, Steve Carpenter, and W. Brock. Year. Optimal phosphorus loading for a potentially eutrophic lake. Ecological Applications 13:1135–1152.

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.

Literature cited

Carpenter, S. R., D. Ludwig, and W. A. Brock. 1999. Management of eutrophication for lakes subject to potentially irreversible change. Ecological Applications 9:751–771.

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