‘sources’ curve;
• By decreasing recycling r to push the sigmoid down such that the only
point of intersection with the ‘sinks’ curve lies in the oligotrophic basin
of attraction;
• By lowering external phosphorous input L; or
• via a combination of the above.
Increasing the sinks and decreasing recycling requires measures that are
independent of phosphorous input. These measures include sediment treatment,
such as the addition of aluminum sulfate to precipitate the sediments, injections
of oxygen into the hypolimnion, and biomanipulation, such as the introduction
of consumers of phosphorous, e.g. fish, or large aquatic plants. These methods
tend to be costly and therefore reductions in phosphorous input will tend to be
preferable as a measure to restore lakes to an oligotrophic state.
Lowering external loading to an effective level may not always be possible
however, because the minimum phosphorous input may not be controllable by
human intervention when it is due to factors such as soil chemistry and airborne
phosphorous deposition. This is likely to be the case for lakes in phosphorous-
rich regions, for lakes that have been subject to a high level of external loading
for extended periods of time, and for shallow lakes (Carpenter et al., 1999).
B Parameters that Determine the Type of Lake
As done by Maler et al. (2003), analysis of equation (3) give rises to the following
results.
- For high values of a, the equation has one stable equilibrium.
- For low values of a, that is a < 0.3, three situations occur depending on
the value of b.
By solving the equation for values of x between 0 and 2, the curve for
equation (3) can be used to plot a(t) against x(t) and to recreate the graphs
provided in Maler et al. for different values of b. The value of b is what affects
the lake’s reversibility from a state of eutrophication.
• b ≥ 8 √3 ≡ b ≈ 0.6495
For these values of b, all values of a lead to one stable equilibrium. This
implies that the sedimentation and recycling rates are such that the lake
can continue to be used as a waste-sink without regard to the amount
of phosphorous loading a. This further implies that the lake requires no
management as far as controlling phosphorous loading is concerned. (See
Figure 6.)
• b ≤ 2
For b ≤ 2, values of a below the local maximum (where da/dx = 0) yield
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