Nutrients, Algae, and Grazers - a Minimal Model
121
If we assume that algal growth is a process controlled only by phosphorus
availability, this can conveniently be described by the Droop model [Eq.
(3.2)], which with the present choice of state variables can be expressed as
.u=.u'(l-Q'~}
(5.4)
The medians from the frequency distributions in Figs. 3.2 and 3.3 are
probably the best estimates we can make for typical parameter values in
Eq. (5.4). From this, we can assume a phosphorus subsistence quota
Q' = 3.8 (J.Lg P) (mg Cr l and a maximal growth rate J.I" = 1.2 day"l. As a
consequence of the approximations made in the model equations (5.3),
(5.4), algal P quota is not constrained below an upper limit Q", as in
Eq. (3.3). Allowing the algae to have infinite P storage capacity means that
the asymptote of the Droop model will be equal to the maximal growth rate
(p' = p"). As neither algae nor grazers are likely to be phosphorus-limited
when the P quota exceeds Q", the chosen representation should have only
minor consequences for the performance of the model.
While large, nonmotile plankton algae are likely to have sinking loss
rates that are at least an order of magnitude higher than observed net loss
rates of phosphorus in lakes, flagellates with active vertical movement
should probably suffer only negligible losses from sinking. As a compromise between these two extremes, one can assume the algae to have a
sinking loss rate equal to average net loss rate of phosphorus, as estimated
for the retention model [Eq. (2.5); that is, (j = 0.0008 d"I]. This choice of
parameter value will make the P retention in an ungrazed system equal to
the observed average for lakes with the same dilution rate, and at the same
time ensure that phytoplankton sinking will constitute only a minor P flux
under most circumstances.
If we want the unstructured grazer model to conform as closely as possible to the age-structured Daphnia model of Chapter 4, we can require that
the grazer population must have a net growth rate less than or equal to the
maximal intrinsic rate of increase (A., determined in Section 4.7, that is,
g - (j $ A.'. Assuming that A. '= 0.38 day"1 and that 8 = 0.02 day"1 gives a
maximal zooplankton growth rate g'= 0.4 day"l. If & is the assimilated
fraction of ingested food carbon, I' is the saturated ingestion rate (day"I),
and r is the specific rate of respiration (day"I), we can express the
zooplankton growth rate under food-sufficient conditions as the difference
between assimilation and respiration, or as g = &1' - r. If we assume that &
and r are equal to the size-independent parameter values determined in
Section 4.3 (that is, & = 0.8 and r = 0.25 day"I), the maximal ingestion rate
will be given by I' = (g' + r)I&= 0.81 day"l.
Ifwe use the piecewise linear functional response [Eq. (4.9)], the specific
ingestion rate I (day"l) as function of food concentration can be written as
1= I'Min(I, cl C'),
(5.5)
121
If we assume that algal growth is a process controlled only by phosphorus
availability, this can conveniently be described by the Droop model [Eq.
(3.2)], which with the present choice of state variables can be expressed as
.u=.u'(l-Q'~}
(5.4)
The medians from the frequency distributions in Figs. 3.2 and 3.3 are
probably the best estimates we can make for typical parameter values in
Eq. (5.4). From this, we can assume a phosphorus subsistence quota
Q' = 3.8 (J.Lg P) (mg Cr l and a maximal growth rate J.I" = 1.2 day"l. As a
consequence of the approximations made in the model equations (5.3),
(5.4), algal P quota is not constrained below an upper limit Q", as in
Eq. (3.3). Allowing the algae to have infinite P storage capacity means that
the asymptote of the Droop model will be equal to the maximal growth rate
(p' = p"). As neither algae nor grazers are likely to be phosphorus-limited
when the P quota exceeds Q", the chosen representation should have only
minor consequences for the performance of the model.
While large, nonmotile plankton algae are likely to have sinking loss
rates that are at least an order of magnitude higher than observed net loss
rates of phosphorus in lakes, flagellates with active vertical movement
should probably suffer only negligible losses from sinking. As a compromise between these two extremes, one can assume the algae to have a
sinking loss rate equal to average net loss rate of phosphorus, as estimated
for the retention model [Eq. (2.5); that is, (j = 0.0008 d"I]. This choice of
parameter value will make the P retention in an ungrazed system equal to
the observed average for lakes with the same dilution rate, and at the same
time ensure that phytoplankton sinking will constitute only a minor P flux
under most circumstances.
If we want the unstructured grazer model to conform as closely as possible to the age-structured Daphnia model of Chapter 4, we can require that
the grazer population must have a net growth rate less than or equal to the
maximal intrinsic rate of increase (A., determined in Section 4.7, that is,
g - (j $ A.'. Assuming that A. '= 0.38 day"1 and that 8 = 0.02 day"1 gives a
maximal zooplankton growth rate g'= 0.4 day"l. If & is the assimilated
fraction of ingested food carbon, I' is the saturated ingestion rate (day"I),
and r is the specific rate of respiration (day"I), we can express the
zooplankton growth rate under food-sufficient conditions as the difference
between assimilation and respiration, or as g = &1' - r. If we assume that &
and r are equal to the size-independent parameter values determined in
Section 4.3 (that is, & = 0.8 and r = 0.25 day"I), the maximal ingestion rate
will be given by I' = (g' + r)I&= 0.81 day"l.
Ifwe use the piecewise linear functional response [Eq. (4.9)], the specific
ingestion rate I (day"l) as function of food concentration can be written as
1= I'Min(I, cl C'),
(5.5)