Stationary Points of the Nutrients, Algae, and Herbivores Model
249
within the range u + D < P < Po and calculate the steady-state values of Z,
C, and P from Eqs. (A6.12), (A6.18), (A6.21). The corresponding input P
concentration can then be found from solving Eq. (A6.1) as
~ = (1+ ~)P+(l+ ~)OZ.
(A6.22)
By substituting Eq. (A6.21) into (A6.22), it is seen that Z > 0 implies that the
input P concentration must be above a critical level (P L >P'J, with P'L given by
~= (1+ ~)( C
N
-~C;)O.
(A6.23)
By the same kind of reasoning, we can see that Z ::;; Zo implies that the input
P concentration must be below a critical level (P'L ::;; P"L)' with PIlL given by
(A6.24)
The domain of the stationary point with both C- and P-limited grazer
growth will therefore be given in terms of the P loading conditions as
P'L < P L ::;; PIlL' When the input P concentration increases from P'L to P"L'
steady-state algal growth rate will also increase from u + D to PtJ. Over the
same range of P loading conditions, steady state grazer biomass will
increase from 0 to Zo. while algal biomass will decrease to C".
C-Limited Grazer Growth. This stationary point corresponds to the classical situation of food-limited grazer growth where algal P content is sufficient for the requirements of the grazer (Q ~ B), while algal food carbon is
below the incipient limiting level (C < C'). This case also represents the
direct continuation of the previous case to the situation where Q ~ ().
The requirement Q ~ (), means that specific algal growth rate (p) will be
constrained to Po ::;; P ::;; p' over the domain of this stationary point. As
C < C, Eq. (A6.2) will also for this stationary point yield a linear relationship, identical to Eq. (A6.12), between p and Z, implying that grazer biomass will be constrained to Zo::;; Z::;; Z'. The absence of a food composition
effect on grazer growth rate when Q ~ () implies that under the assumption
that Z:;:. 0, Eq. (A6.3) can be written as
el'(C/C')- r = 8 + D.
(A6.25)
Solving Eq. (A6.25) for C gives that the steady state algal biomass is constant C = C" over the whole domain of this stationary point. From the relationship P = QC and Eq. (A6.15), the concentration of algal P at the stationary point can be expressed as
P = Z' - Z6 0 C" ,
Z'-Z
(A6.26)
249
within the range u + D < P < Po and calculate the steady-state values of Z,
C, and P from Eqs. (A6.12), (A6.18), (A6.21). The corresponding input P
concentration can then be found from solving Eq. (A6.1) as
~ = (1+ ~)P+(l+ ~)OZ.
(A6.22)
By substituting Eq. (A6.21) into (A6.22), it is seen that Z > 0 implies that the
input P concentration must be above a critical level (P L >P'J, with P'L given by
~= (1+ ~)( C
N
-~C;)O.
(A6.23)
By the same kind of reasoning, we can see that Z ::;; Zo implies that the input
P concentration must be below a critical level (P'L ::;; P"L)' with PIlL given by
(A6.24)
The domain of the stationary point with both C- and P-limited grazer
growth will therefore be given in terms of the P loading conditions as
P'L < P L ::;; PIlL' When the input P concentration increases from P'L to P"L'
steady-state algal growth rate will also increase from u + D to PtJ. Over the
same range of P loading conditions, steady state grazer biomass will
increase from 0 to Zo. while algal biomass will decrease to C".
C-Limited Grazer Growth. This stationary point corresponds to the classical situation of food-limited grazer growth where algal P content is sufficient for the requirements of the grazer (Q ~ B), while algal food carbon is
below the incipient limiting level (C < C'). This case also represents the
direct continuation of the previous case to the situation where Q ~ ().
The requirement Q ~ (), means that specific algal growth rate (p) will be
constrained to Po ::;; P ::;; p' over the domain of this stationary point. As
C < C, Eq. (A6.2) will also for this stationary point yield a linear relationship, identical to Eq. (A6.12), between p and Z, implying that grazer biomass will be constrained to Zo::;; Z::;; Z'. The absence of a food composition
effect on grazer growth rate when Q ~ () implies that under the assumption
that Z:;:. 0, Eq. (A6.3) can be written as
el'(C/C')- r = 8 + D.
(A6.25)
Solving Eq. (A6.25) for C gives that the steady state algal biomass is constant C = C" over the whole domain of this stationary point. From the relationship P = QC and Eq. (A6.15), the concentration of algal P at the stationary point can be expressed as
P = Z' - Z6 0 C" ,
Z'-Z
(A6.26)