28 The Biogeochemical Theatre - Phosphorus Cycling and Phosphorus Household in Lakes
(2.8)
The differential equation (2.7) states that the net rate of change in P
contained in popUlation i is the difference between uptake and loss, where
Y; is the net, biomass-specific P uptake rate [(~g P) (mg C).I datI] and D; is
the aggregated loss rate (datI) from popUlation i. Likewise, Eq. (2.8) says
that the net rate of change in the biomass of population i is the difference
between growth and loss, with PI being the net, specific growth rate (day'l)
of the population. The aggregated loss rate D; is the sum of population
losses due to dilution, sinking, and grazing:
Dj = D+uj + LF;j Zj.
j
(2.9)
The second term in Eq. (2.9) is the sedimentation loss rate of population
i (OJ; datI), while the last term is the summation of grazing losses from
population i over all zooplankton populations. F;j is the specific clearance
rate [l (mg ct datI) of zooplankton populationj when feeding on phytoplankton population i. The clearance rate can be interpreted as the virtual
volume of water from which all food particles can be removed by the activity of a unit grazer biomass in a unit of time. The equality of the loss terms
in Eqs. (2.7) and (2.8) is equivalent to the reasonable assumption that phytoplankton cells are lost as entire units, irrespective of whether the loss is
effected by dilution, sinking, or grazing.
The P and C dynamics of an algal population [Eqs. (2.7) and (2.8)] can
also be expressed with algal biomass (C;) and P content (QJ as state variables, which is more common in algal chemostat models (e.g., Olsen et al.
1989). From the relationship P; = Q; C;' the time derivative of Q; can be
found from the rule for taking the derivative of a ratio:
Q . (p.) (P. t' )Q
= --L. = ~_--L. •
;
C
P . C ;
,
"
(2.10)
Substitution of the expressions for P; and ~ [Eqs. (2.7) and (2.8)] into Eq.
(2.10) gives
(2.11)
which is identical to the equation developed by Droop (1974), stating that
the rate of change in P content is determined by the balance between
uptake and growth.
From the discussion in Section 2.4, it is reasonable to assume that the P
content of zooplankton is a fIXed, species-specific quantity. The implied
balance between P and C allocation in zooplankton (which is equivalent to
OJ = 0 for all j) means that a single state variable is sufficient to describe
both the P and C dynamics of a zooplankton population. If we choose biomass (Z) as the state variable, the dynamics of zooplankton population j
(2.8)
The differential equation (2.7) states that the net rate of change in P
contained in popUlation i is the difference between uptake and loss, where
Y; is the net, biomass-specific P uptake rate [(~g P) (mg C).I datI] and D; is
the aggregated loss rate (datI) from popUlation i. Likewise, Eq. (2.8) says
that the net rate of change in the biomass of population i is the difference
between growth and loss, with PI being the net, specific growth rate (day'l)
of the population. The aggregated loss rate D; is the sum of population
losses due to dilution, sinking, and grazing:
Dj = D+uj + LF;j Zj.
j
(2.9)
The second term in Eq. (2.9) is the sedimentation loss rate of population
i (OJ; datI), while the last term is the summation of grazing losses from
population i over all zooplankton populations. F;j is the specific clearance
rate [l (mg ct datI) of zooplankton populationj when feeding on phytoplankton population i. The clearance rate can be interpreted as the virtual
volume of water from which all food particles can be removed by the activity of a unit grazer biomass in a unit of time. The equality of the loss terms
in Eqs. (2.7) and (2.8) is equivalent to the reasonable assumption that phytoplankton cells are lost as entire units, irrespective of whether the loss is
effected by dilution, sinking, or grazing.
The P and C dynamics of an algal population [Eqs. (2.7) and (2.8)] can
also be expressed with algal biomass (C;) and P content (QJ as state variables, which is more common in algal chemostat models (e.g., Olsen et al.
1989). From the relationship P; = Q; C;' the time derivative of Q; can be
found from the rule for taking the derivative of a ratio:
Q . (p.) (P. t' )Q
= --L. = ~_--L. •
;
C
P . C ;
,
"
(2.10)
Substitution of the expressions for P; and ~ [Eqs. (2.7) and (2.8)] into Eq.
(2.10) gives
(2.11)
which is identical to the equation developed by Droop (1974), stating that
the rate of change in P content is determined by the balance between
uptake and growth.
From the discussion in Section 2.4, it is reasonable to assume that the P
content of zooplankton is a fIXed, species-specific quantity. The implied
balance between P and C allocation in zooplankton (which is equivalent to
OJ = 0 for all j) means that a single state variable is sufficient to describe
both the P and C dynamics of a zooplankton population. If we choose biomass (Z) as the state variable, the dynamics of zooplankton population j