246
Appendices
c=Jl'-(a+D) D p,.
Jl'Q'
a+D L
(A6.5)
In other words; algal biomass will be directly proportional to the P supply rate and non-negative for all dilution rates D :s; '" - a. At a critical dilutioQ
rate D = '" - 0; the grazer extinction point will be coincident with the washout
point; the algae are unable to compensate for the losses through flushing an~
sedimentation, and positive net growth thus becomes impossible.
Apart from the special case where a grazer population growing at its innate
capacity is exactly balanced by the losses through the outflow (D = g' - b), the
grazer growth rate must be controlled by the properties of the prey
population (so that g = 0 + D) in order to have a stationary point with
nonzero grazer biomass (Z,* 0). We can distinguish between three cases as
grazer growth rate can be limited by either food composition, food concentration, or both. The use of piecewise linear relationships descnbing ingestion and utilization of food implies that the stationary points will be
determined by a different set of equations for each of the three cases.
P-Limited Grazer Growth. This stationary point corresponds to the situation
where algal P content is below the requirements of the grazer (Q < 0), whi Ie
algal food carbon is above the incipient limiting level and thus non-limitin!S
to grazer growth (C ~ C~. This means that food ingestion is at its maximal
value (J = n and Eq. (A6.3) can be written as
g = (£1' - r~ = t5 + D·
(A6.6)
o
By solving Eq. (A6.6) for algal P content (Q), it is seen that Q will be a
constant which depends on the dilution rate, but not on the input P
concentration (P L ). Thus, algal growth rate will also be a constant which
will be denoted by
jl = /J' (l -...L. ~) ,
(A6.7)
o+D ()
where g' = &/' - r is the maximal grazer growth rate. By eliminating Z from
Eqs. (A6.1) and (A6.2), utilizing the relationship between P and C implied
by Eq. (A6.6), the state variables at the stationary point can be expressed
recursively as
p=
DI'
p',
(0' + D)I' + (ii -(0' + D »)R' L
c= 8+D!:, and
g' 0
(A6.8)
(A6.9)
Appendices
c=Jl'-(a+D) D p,.
Jl'Q'
a+D L
(A6.5)
In other words; algal biomass will be directly proportional to the P supply rate and non-negative for all dilution rates D :s; '" - a. At a critical dilutioQ
rate D = '" - 0; the grazer extinction point will be coincident with the washout
point; the algae are unable to compensate for the losses through flushing an~
sedimentation, and positive net growth thus becomes impossible.
Apart from the special case where a grazer population growing at its innate
capacity is exactly balanced by the losses through the outflow (D = g' - b), the
grazer growth rate must be controlled by the properties of the prey
population (so that g = 0 + D) in order to have a stationary point with
nonzero grazer biomass (Z,* 0). We can distinguish between three cases as
grazer growth rate can be limited by either food composition, food concentration, or both. The use of piecewise linear relationships descnbing ingestion and utilization of food implies that the stationary points will be
determined by a different set of equations for each of the three cases.
P-Limited Grazer Growth. This stationary point corresponds to the situation
where algal P content is below the requirements of the grazer (Q < 0), whi Ie
algal food carbon is above the incipient limiting level and thus non-limitin!S
to grazer growth (C ~ C~. This means that food ingestion is at its maximal
value (J = n and Eq. (A6.3) can be written as
g = (£1' - r~ = t5 + D·
(A6.6)
o
By solving Eq. (A6.6) for algal P content (Q), it is seen that Q will be a
constant which depends on the dilution rate, but not on the input P
concentration (P L ). Thus, algal growth rate will also be a constant which
will be denoted by
jl = /J' (l -...L. ~) ,
(A6.7)
o+D ()
where g' = &/' - r is the maximal grazer growth rate. By eliminating Z from
Eqs. (A6.1) and (A6.2), utilizing the relationship between P and C implied
by Eq. (A6.6), the state variables at the stationary point can be expressed
recursively as
p=
DI'
p',
(0' + D)I' + (ii -(0' + D »)R' L
c= 8+D!:, and
g' 0
(A6.8)
(A6.9)