122
Nutrients. Algae and Herbivores - the Paradox of Bnrichment Revisited
where e'is the incipient limiting food level [(mg C) }"I]. If we require that the
unstructured grazer model should have the same threshold food level for
positive net population growth as determined in Section 4.7 [e"'), = 0.0725
(mg C) }"I], we must have g = 8when C = C'~, or e' = el'C''A /(r + 8) = 0.17
(mgen l •
From the discussion of the phosphorus economy in Daphnia in Section
4.5, it was concluded that phosphorus-limited growth could be described
by a piecewise linear function of food P content. This can be expressed in
terms of the present state variables as
g= (el-r)Min(1,P/8C),
(5.6)
where {lis the grazer P content [which is found to be close to 30 (J18 P) (mg C)"I
in Daphnia species, and generally lower in other zooplankton taxa].
The last two model parameters, the dilution rate (D; day"l) and the input
P concentration [P L ; (JIg P) }"I], are considered external forcing variables
much in the same way as in total phosphorus loading models. The dilution
rate enter the mass-balance equations both as the product DP L , which is the
volumetric P load to the pelagic zone, and as the loss rate due to flushing.
The model parameter P L will be less than the measured, flow-weighted
input concentration to a given lake if some of the load is lost before entering the pelagic zone. If we assume an average 22% load decay, as suggested
by Fig. 2.2, P L will be 78% of the total external P load.
5.2 Equilibrium Points and Local Stability
Every combination of the three state variables P. e, and Z will correspond
to a point in the state space of the system [Eqs. (5.1)-(5.3)]. As all state
variables represent physical entities, they must necessarily all be nonnegative; thus, the state space is confined to the positive cone (P ~ 0, C ~ 0,
Z ~ 0). At certain points in state space, which are called the stationary
points of the system, all the right-hand sides in Eqs. (5.1)-(5.3) will evaluate
to zero (that is, P= C= Z= 0 ). If the system is initially located exactly at
such a stationary point, it will remain there indefinitely, unless it is driven
away from it by some external disturbance. If the system returns to the
stationary point after some small perturbation, the stationary point is said
to be locally stable.
In Appendices A6 and A7 it is shown that the differential equation
system (5.1)-(5.3) possesses several stationary points, whose existence and
stability properties depend on the phosphorus loading and water renewal
conditions in terms of the input P concentration P L and the dilution rate D.
Two of the stationary points, which in Appendix A6 are called the washout
point and the grazer extinction point, are located on the boundary of the
positive cone. The washout point, corresponding to the absence of both algae
Nutrients. Algae and Herbivores - the Paradox of Bnrichment Revisited
where e'is the incipient limiting food level [(mg C) }"I]. If we require that the
unstructured grazer model should have the same threshold food level for
positive net population growth as determined in Section 4.7 [e"'), = 0.0725
(mg C) }"I], we must have g = 8when C = C'~, or e' = el'C''A /(r + 8) = 0.17
(mgen l •
From the discussion of the phosphorus economy in Daphnia in Section
4.5, it was concluded that phosphorus-limited growth could be described
by a piecewise linear function of food P content. This can be expressed in
terms of the present state variables as
g= (el-r)Min(1,P/8C),
(5.6)
where {lis the grazer P content [which is found to be close to 30 (J18 P) (mg C)"I
in Daphnia species, and generally lower in other zooplankton taxa].
The last two model parameters, the dilution rate (D; day"l) and the input
P concentration [P L ; (JIg P) }"I], are considered external forcing variables
much in the same way as in total phosphorus loading models. The dilution
rate enter the mass-balance equations both as the product DP L , which is the
volumetric P load to the pelagic zone, and as the loss rate due to flushing.
The model parameter P L will be less than the measured, flow-weighted
input concentration to a given lake if some of the load is lost before entering the pelagic zone. If we assume an average 22% load decay, as suggested
by Fig. 2.2, P L will be 78% of the total external P load.
5.2 Equilibrium Points and Local Stability
Every combination of the three state variables P. e, and Z will correspond
to a point in the state space of the system [Eqs. (5.1)-(5.3)]. As all state
variables represent physical entities, they must necessarily all be nonnegative; thus, the state space is confined to the positive cone (P ~ 0, C ~ 0,
Z ~ 0). At certain points in state space, which are called the stationary
points of the system, all the right-hand sides in Eqs. (5.1)-(5.3) will evaluate
to zero (that is, P= C= Z= 0 ). If the system is initially located exactly at
such a stationary point, it will remain there indefinitely, unless it is driven
away from it by some external disturbance. If the system returns to the
stationary point after some small perturbation, the stationary point is said
to be locally stable.
In Appendices A6 and A7 it is shown that the differential equation
system (5.1)-(5.3) possesses several stationary points, whose existence and
stability properties depend on the phosphorus loading and water renewal
conditions in terms of the input P concentration P L and the dilution rate D.
Two of the stationary points, which in Appendix A6 are called the washout
point and the grazer extinction point, are located on the boundary of the
positive cone. The washout point, corresponding to the absence of both algae