Differential Loss Rates and Invadability of Equilibria
171
respect to species 2. If we introduce the effective food concentration C,
defmed by Eq. (6.9), we will have that F =IIC with the ingestion rate I given
by the piecewise linear function [Eq. (6.10)]. As cells are ingested as whole
entities, we can define the effective concentration of algal P in analogy with
Eq. (6.9) as
(6.13)
The remaining mass-balance equations for zooplankton carbon [Eq.
(6.4)] and dissolved inorganic phosphorus [Eq. (6.5)] will be unaffected by
redefining the symbols C and P [by Eqs. (6.9) and (6.13)] from total concentrations to effective concentrations of algal carbon and phosphorus. In
fact, the present model will be identical to the one in the previous section
when ¢= 1.
Edibility and Invadability in a Eutrophication Gradient. As in the previous
section, the condition for coexistence of two phytoplankton species with
different edibilities at a fIXed equilibrium point is that all derivatives in the
mass-balance equations vanish. Setting ~ = 0 and ~ = 0 in Eqs. (6.11) and
(6.12), and eliminating the loss terms, implies that Vi = Jl..Q i for i = 1, 2. In
other words, uptake and growth must balance in both species, such that
both species must satisfy their respective Monod relationships (cf. Section
3.3). Equation (6.11) implies that PI = D + CT I + FZ and Il2 = D + CT 2 + ¢ FZ
at the equilibrium point. Eliminating the grazing term (FZ) gives the
following relationship between the equilibrium growth rates for a given
value of the selectivity index ¢
f/J = J.l2 - (D+ CT 2),
(6.14)
J1t -(D+CTt)
If we substitute the Monod equations (3.10) for PI and JJ,. into Eq. (6.14),
we find that the equilibrium condition [Eq. (6.14)] is equivalent to a quadratic equation in terms of the dissolved inorganic P concentration [S; (Jig P)
rl]. In other words, coexistence of the two species at fixed densities is only
possible for (at most) two specific values of the equilibrium dissolved inorganic P concentration, corresponding to the roots of this quadratic equation. For all other combinations of equilibrium growth rates (PI and 1l2),
species 2 will be able to invade a species 1 equilibrium and competitively
exclude species 1 whenever i. e.,
~ -(D+CT2»f/J(P. -(D+CT.)).
Equation (6.14) implies that if species 2 is competitively superior in the
absence of grazing [that is, if Il2 - (D + CT 2 ) > PI - (D + OJ) for all S > 0], then
species 2 will also be competitively superior in the presence of grazing (that
is, for all 0 ~ 4> ~ I). This means that for the resident species to have any
chance at all of resisting invasion by less edible species, we must assume a
171
respect to species 2. If we introduce the effective food concentration C,
defmed by Eq. (6.9), we will have that F =IIC with the ingestion rate I given
by the piecewise linear function [Eq. (6.10)]. As cells are ingested as whole
entities, we can define the effective concentration of algal P in analogy with
Eq. (6.9) as
(6.13)
The remaining mass-balance equations for zooplankton carbon [Eq.
(6.4)] and dissolved inorganic phosphorus [Eq. (6.5)] will be unaffected by
redefining the symbols C and P [by Eqs. (6.9) and (6.13)] from total concentrations to effective concentrations of algal carbon and phosphorus. In
fact, the present model will be identical to the one in the previous section
when ¢= 1.
Edibility and Invadability in a Eutrophication Gradient. As in the previous
section, the condition for coexistence of two phytoplankton species with
different edibilities at a fIXed equilibrium point is that all derivatives in the
mass-balance equations vanish. Setting ~ = 0 and ~ = 0 in Eqs. (6.11) and
(6.12), and eliminating the loss terms, implies that Vi = Jl..Q i for i = 1, 2. In
other words, uptake and growth must balance in both species, such that
both species must satisfy their respective Monod relationships (cf. Section
3.3). Equation (6.11) implies that PI = D + CT I + FZ and Il2 = D + CT 2 + ¢ FZ
at the equilibrium point. Eliminating the grazing term (FZ) gives the
following relationship between the equilibrium growth rates for a given
value of the selectivity index ¢
f/J = J.l2 - (D+ CT 2),
(6.14)
J1t -(D+CTt)
If we substitute the Monod equations (3.10) for PI and JJ,. into Eq. (6.14),
we find that the equilibrium condition [Eq. (6.14)] is equivalent to a quadratic equation in terms of the dissolved inorganic P concentration [S; (Jig P)
rl]. In other words, coexistence of the two species at fixed densities is only
possible for (at most) two specific values of the equilibrium dissolved inorganic P concentration, corresponding to the roots of this quadratic equation. For all other combinations of equilibrium growth rates (PI and 1l2),
species 2 will be able to invade a species 1 equilibrium and competitively
exclude species 1 whenever i. e.,
~ -(D+CT2»f/J(P. -(D+CT.)).
Equation (6.14) implies that if species 2 is competitively superior in the
absence of grazing [that is, if Il2 - (D + CT 2 ) > PI - (D + OJ) for all S > 0], then
species 2 will also be competitively superior in the presence of grazing (that
is, for all 0 ~ 4> ~ I). This means that for the resident species to have any
chance at all of resisting invasion by less edible species, we must assume a