Differential Loss Rates and Invadability of Equilibria
181
are above the diagonal. The isolines are seen to diverge when food elemental composition becomes non-limiting (QN > ON and Q,. > 0,.); that is,
the effect of differential nutrient utilization by grazers on N:P ratios of
recycled nutrients diminishes when the food contains a surplus of both
elements.
Competition in an N:P Loading Gradient. We have now reached the point
where we can assemble the submodels of phytoplankton competition and
zooplankton recycling into a system driven by the external supply rates of
Nand P. In order to do so, we must first rewrite the phytoplankton massbalance equations (3.18), (3.19) to include the loss rate due to grazing. The
addition of a non-selective grazer, as discussed in Section 6.1, to the system,
gives the modified mass-balance equations as
~ =vR.i~ -(D+FZ)~
(6.19)
C/ = (Pi - (D + FZ»)~
(6.20)
for i = 1,2 and R = N, P (that is, a total of six differential equations). As in
Eqs. (6.2), (6.3), Z is grazer biomass [(mg C) rl] and F is the clearance rate
of the grazers [1 (mg C)"I day"I]; otherwise the notation is unchanged from
Eqs. (3.18), (3.19). Ifwe, as in Section 6.1, denote the concentrations offood
C, N, and P that are available to the grazers by C = C I + C 2 , N = NI + N 2 ,
and P = PI + P 2 , the P:C and N:C ratios of available food become Q,. = PIC
and Q N = NIC. With this notation, we can directly use the expressions
(6.15), (6.16) for zooplankton growth and nutrient release as function of
food composition. Thus, the modified mass-balance equations for the
dissolved inorganic nutrient pools can be written as:
SR = P R Z + D(R L - SR)- L v RJ C i
(6.21)
for R = N, P. The first term in Eq. (6.21) is the zooplankton release rate of
element R, given by Eq. (6.16), while the remaining terms describing
loading, dilution loss, and uptake are the same as in Eq. (6.29). Finally, the
zooplankton biomass mass-balance equation will be the same as Eq. (6.4),
but with the growth rate given by the threshold model [Eq. (6.15»):
Z=(g-(D+o»)z
(6.22)
All phytoplankton parameters will be assumed to be the same as those in
Table 3.4, which were deliberately chosen to conform with the conditions for
coexistence from Appendix Al (as illustrated in Fig. 3.9). With the exception
of the grazer N:C ratio (ON), all zooplankton parameters will be identical to
those in Table 5.1. Using a typical N:C ratio for Daphnia, as reported by
Andersen and Hessen (1991), ONcan be set to 190 (Ilg N) (mg Crl.
181
are above the diagonal. The isolines are seen to diverge when food elemental composition becomes non-limiting (QN > ON and Q,. > 0,.); that is,
the effect of differential nutrient utilization by grazers on N:P ratios of
recycled nutrients diminishes when the food contains a surplus of both
elements.
Competition in an N:P Loading Gradient. We have now reached the point
where we can assemble the submodels of phytoplankton competition and
zooplankton recycling into a system driven by the external supply rates of
Nand P. In order to do so, we must first rewrite the phytoplankton massbalance equations (3.18), (3.19) to include the loss rate due to grazing. The
addition of a non-selective grazer, as discussed in Section 6.1, to the system,
gives the modified mass-balance equations as
~ =vR.i~ -(D+FZ)~
(6.19)
C/ = (Pi - (D + FZ»)~
(6.20)
for i = 1,2 and R = N, P (that is, a total of six differential equations). As in
Eqs. (6.2), (6.3), Z is grazer biomass [(mg C) rl] and F is the clearance rate
of the grazers [1 (mg C)"I day"I]; otherwise the notation is unchanged from
Eqs. (3.18), (3.19). Ifwe, as in Section 6.1, denote the concentrations offood
C, N, and P that are available to the grazers by C = C I + C 2 , N = NI + N 2 ,
and P = PI + P 2 , the P:C and N:C ratios of available food become Q,. = PIC
and Q N = NIC. With this notation, we can directly use the expressions
(6.15), (6.16) for zooplankton growth and nutrient release as function of
food composition. Thus, the modified mass-balance equations for the
dissolved inorganic nutrient pools can be written as:
SR = P R Z + D(R L - SR)- L v RJ C i
(6.21)
for R = N, P. The first term in Eq. (6.21) is the zooplankton release rate of
element R, given by Eq. (6.16), while the remaining terms describing
loading, dilution loss, and uptake are the same as in Eq. (6.29). Finally, the
zooplankton biomass mass-balance equation will be the same as Eq. (6.4),
but with the growth rate given by the threshold model [Eq. (6.15»):
Z=(g-(D+o»)z
(6.22)
All phytoplankton parameters will be assumed to be the same as those in
Table 3.4, which were deliberately chosen to conform with the conditions for
coexistence from Appendix Al (as illustrated in Fig. 3.9). With the exception
of the grazer N:C ratio (ON), all zooplankton parameters will be identical to
those in Table 5.1. Using a typical N:C ratio for Daphnia, as reported by
Andersen and Hessen (1991), ONcan be set to 190 (Ilg N) (mg Crl.