398
through space place on ecological dynamics. Similarly, a "stoichiometrically explicit" theory of
trophic interactions and nutrient cycling would operate within the constraints that the first law of thermodynamics operating on energy and multiple elements imposes on growth rates, biomass, and
nutrient release.
Andersen's approach considers the dynamics of
a phosphorus-autotroph-grazer system under conditions in which algal CIP ratio varies with Plimited growth rate, grazer growth rate varies in
proportion to food quantity or quality as appropriate, and grazer nutrient release is governed by mass
balance. Algal growth is modeled via the familiar
"Droop equation," which expresses P-limited algal
growth rate in terms of cellular PIC ratio. Importantly, grazer growth and reproduction are modeled
via expressions that assess whether animal growth
is limited by food abundance or by food quality,
comparing the rates of C and P ingestion relative
to current requirements of the animal. Finally, the
rate of P released by the grazer is modeled as the
difference between ingested P and P incorporated
into new biomass.
Incorporating these simple and biologically realistic assumptions has profound consequences for
predicted dynamics. Those consequences are perhaps best appreciated by considering the isoclines
for grazer and autotroph populations, which are peculiarly hump-shaped (Fig. 26.7). For the algal isocline, the rising portion in the middle of the curve
corresponds the region of saturation of feeding response of individual grazers but the isocline comes
down to the x -axis (the algal axis) as a result of the
constraint of total P in the system. That is, given
that algae have a physiological minimal PIC ratio,
algal biomass in terms of C has an upper limit set
by the total amount of P in the system divided by
that minimal PIC ratio. Even more interesting is the
grazer isocline, which is also hump-shaped due to
the constraints of matter. The first intersection of
the zooplankton isocline with the algal axis is the
conventional food quantity threshold; but what
generates the second intersection? This intersection
is a direct result of the realization that grazer biomass is also constructed of P. That is, given that the
total amount of P in the system is fixed, as algal
biomass increases there is less and less P available
to build grazer biomass!
James J. Elser
The shape of these isoclines generates two primary configurations of ecological interest that are
a function of species characteristics and environmental conditions. In the first (panel A in Fig. 26.7),
algae and grazer isoclines have one positive intersection (point AI) in the positive phase plane. This
intersection is a locally stable equilibrium point but,
beyond its immediate vicinity, stable limit cycles
are predicted. In this situation, the equilibrium
point with zero grazers (point A2) is locally unstable (i.e., the system is invasible by grazers). Alternatively, conditions might generate the situation
shown in panel B in Figure 26.7, in which there are
now two positive intersections. Andersen's analysis
shows that the second positive intersection (point
B2) is locally unstable and, importantly, that the
grazer extinction equilibrium (B3) is now locally
stable. Dynamics of the algae-grazer system in this
scenario are highly unstable.
What determines whether the interaction resembles panel A or panel B? Andersen's analysis
showed that whether or not the interaction lies in
the realm of stable or unstable dynamics with the
possibility of grazer extinction depends on a few
key parameters of algal and zooplankton physiology. Specifically, his analysis shows that transitions
between panel A versus panel B in Figure 26.7 occur as a function of:
gil = g*(Q/9)
(26.1)
where g* is the growth capacity of the grazer species, Qa is the minimal PIC of the algae (i.e., algal
PIC when algal growth rate is zero under P limitation), and 9 is the elemental composition (PIC) of
the grazer. When gil is high, it is more likely that
the system will be in a situation like that in panel
A of Figure 26.7 in which the zooplanktonphytoplankton interaction is stable. Such a situation
will occur when grazer P content (9) is low (as such
animals are less sensitive to the food quality effects
that bend the grazer isocline) and Qa is high (as this
means that algal biomass at zero growth rate will
be lower for a given P input). On the other hand,
situations dominated by algal species with superior
competitive abilities for P (lower Qa) or grazers
with higher P contents (higher 9) reduce gil, favoring unstable dynamics that promote grazer extinction (panel B in Figure 26.7). Thus, predicted dynamics may differ considerably for different grazer
species, which we now know have potentially
through space place on ecological dynamics. Similarly, a "stoichiometrically explicit" theory of
trophic interactions and nutrient cycling would operate within the constraints that the first law of thermodynamics operating on energy and multiple elements imposes on growth rates, biomass, and
nutrient release.
Andersen's approach considers the dynamics of
a phosphorus-autotroph-grazer system under conditions in which algal CIP ratio varies with Plimited growth rate, grazer growth rate varies in
proportion to food quantity or quality as appropriate, and grazer nutrient release is governed by mass
balance. Algal growth is modeled via the familiar
"Droop equation," which expresses P-limited algal
growth rate in terms of cellular PIC ratio. Importantly, grazer growth and reproduction are modeled
via expressions that assess whether animal growth
is limited by food abundance or by food quality,
comparing the rates of C and P ingestion relative
to current requirements of the animal. Finally, the
rate of P released by the grazer is modeled as the
difference between ingested P and P incorporated
into new biomass.
Incorporating these simple and biologically realistic assumptions has profound consequences for
predicted dynamics. Those consequences are perhaps best appreciated by considering the isoclines
for grazer and autotroph populations, which are peculiarly hump-shaped (Fig. 26.7). For the algal isocline, the rising portion in the middle of the curve
corresponds the region of saturation of feeding response of individual grazers but the isocline comes
down to the x -axis (the algal axis) as a result of the
constraint of total P in the system. That is, given
that algae have a physiological minimal PIC ratio,
algal biomass in terms of C has an upper limit set
by the total amount of P in the system divided by
that minimal PIC ratio. Even more interesting is the
grazer isocline, which is also hump-shaped due to
the constraints of matter. The first intersection of
the zooplankton isocline with the algal axis is the
conventional food quantity threshold; but what
generates the second intersection? This intersection
is a direct result of the realization that grazer biomass is also constructed of P. That is, given that the
total amount of P in the system is fixed, as algal
biomass increases there is less and less P available
to build grazer biomass!
James J. Elser
The shape of these isoclines generates two primary configurations of ecological interest that are
a function of species characteristics and environmental conditions. In the first (panel A in Fig. 26.7),
algae and grazer isoclines have one positive intersection (point AI) in the positive phase plane. This
intersection is a locally stable equilibrium point but,
beyond its immediate vicinity, stable limit cycles
are predicted. In this situation, the equilibrium
point with zero grazers (point A2) is locally unstable (i.e., the system is invasible by grazers). Alternatively, conditions might generate the situation
shown in panel B in Figure 26.7, in which there are
now two positive intersections. Andersen's analysis
shows that the second positive intersection (point
B2) is locally unstable and, importantly, that the
grazer extinction equilibrium (B3) is now locally
stable. Dynamics of the algae-grazer system in this
scenario are highly unstable.
What determines whether the interaction resembles panel A or panel B? Andersen's analysis
showed that whether or not the interaction lies in
the realm of stable or unstable dynamics with the
possibility of grazer extinction depends on a few
key parameters of algal and zooplankton physiology. Specifically, his analysis shows that transitions
between panel A versus panel B in Figure 26.7 occur as a function of:
gil = g*(Q/9)
(26.1)
where g* is the growth capacity of the grazer species, Qa is the minimal PIC of the algae (i.e., algal
PIC when algal growth rate is zero under P limitation), and 9 is the elemental composition (PIC) of
the grazer. When gil is high, it is more likely that
the system will be in a situation like that in panel
A of Figure 26.7 in which the zooplanktonphytoplankton interaction is stable. Such a situation
will occur when grazer P content (9) is low (as such
animals are less sensitive to the food quality effects
that bend the grazer isocline) and Qa is high (as this
means that algal biomass at zero growth rate will
be lower for a given P input). On the other hand,
situations dominated by algal species with superior
competitive abilities for P (lower Qa) or grazers
with higher P contents (higher 9) reduce gil, favoring unstable dynamics that promote grazer extinction (panel B in Figure 26.7). Thus, predicted dynamics may differ considerably for different grazer
species, which we now know have potentially