Differential Loss Rates and Invadability of Equilibria
177
Case E in Fig. 6.10 illustrates the effects of letting the invading species
suffer from an additional competitive disadvantage in terms of increased
sinking loss rate (0'1 < 0'2)' While nutrients lost by the efflux process considered in case D will be immediately available for the competitor, nutrients
lost by sinking will be equally unavailable to all members of the community. In order to compare the two types of differential loss processes, we
can, for example, assume that the invading species in both cases D and E
have equal loss rates at zero inorganic P concentration. By setting S = 0 in
Eq. (3.10), we find that this will be the case if we assume species 2 to have a
sinking loss rate 0'2 = 0.066 dati, which will be in the lower half of the range
considered as typical for large, nonmotile algae in Section 2.3. If we keep all
other parameters identical to those in case C; the Monod curves will also be
identical in cases C and E, while the ,,·-curve as function of S will be similar
to the one from case D, but with a slightly lower asymptote due to the
contribution from 0'2 in the nominator of Eq. (6.14). Up to the bifurcation
point, the ,,·-curves as functions of P loading are nearly identical for cases
D and E. After the onset of limit cycling behavior, the two cases differ in
that the ,,·-curves in case E starts to decrease while it remains at a rather
constant level in case D. This difference between the two loss processes
might be caused by an increasing sinking loss of phosphorus from the
species 2 compartment in the undergrazed phase of the limit cycle.
The cases illustrated in Figs. 6.9 and 6.10 indicate that for a grazer-controlled plankton community to be resistant to invasion by phytoplankton
species with a grazing refuge, the resident species must have higher capacities for both growth and nutrient uptake. Superiority in only one of these
aspects will leave the system open to invasion by species with only a modest
level of predator defence at either the low or the high end of the nutrient
loading gradient. Resistance to invasion by completely inedible phytoplankton species seems possible only if this level of predator defence can
only be realized at the expense of increased loss rates due to grazing-independent processes. If the completely inedible species is capable of attaining
positive net growth at all [that is, if !i'2 - (D + 0'2)]' then it will nevertheless
be able to invade at a sufficiently high P loading rate.
The takeover by species 2 was always absolute for selectivities below,,',
with no intermittent interval allowing coexistence between the two species.
It thus appears that the activity of an invariant selective grazer does not
promote the coexistence of its prey species, like the non-selective grazing
considered in Section 6.1. This also means that the present model is unable
to represent a situation with stable coexistence of large, inedible "canopy"
species and small, edible "undergrowth" species identified as step 8 in the
PEG model of plankton succession (Sommer et aI. 1986).
177
Case E in Fig. 6.10 illustrates the effects of letting the invading species
suffer from an additional competitive disadvantage in terms of increased
sinking loss rate (0'1 < 0'2)' While nutrients lost by the efflux process considered in case D will be immediately available for the competitor, nutrients
lost by sinking will be equally unavailable to all members of the community. In order to compare the two types of differential loss processes, we
can, for example, assume that the invading species in both cases D and E
have equal loss rates at zero inorganic P concentration. By setting S = 0 in
Eq. (3.10), we find that this will be the case if we assume species 2 to have a
sinking loss rate 0'2 = 0.066 dati, which will be in the lower half of the range
considered as typical for large, nonmotile algae in Section 2.3. If we keep all
other parameters identical to those in case C; the Monod curves will also be
identical in cases C and E, while the ,,·-curve as function of S will be similar
to the one from case D, but with a slightly lower asymptote due to the
contribution from 0'2 in the nominator of Eq. (6.14). Up to the bifurcation
point, the ,,·-curves as functions of P loading are nearly identical for cases
D and E. After the onset of limit cycling behavior, the two cases differ in
that the ,,·-curves in case E starts to decrease while it remains at a rather
constant level in case D. This difference between the two loss processes
might be caused by an increasing sinking loss of phosphorus from the
species 2 compartment in the undergrazed phase of the limit cycle.
The cases illustrated in Figs. 6.9 and 6.10 indicate that for a grazer-controlled plankton community to be resistant to invasion by phytoplankton
species with a grazing refuge, the resident species must have higher capacities for both growth and nutrient uptake. Superiority in only one of these
aspects will leave the system open to invasion by species with only a modest
level of predator defence at either the low or the high end of the nutrient
loading gradient. Resistance to invasion by completely inedible phytoplankton species seems possible only if this level of predator defence can
only be realized at the expense of increased loss rates due to grazing-independent processes. If the completely inedible species is capable of attaining
positive net growth at all [that is, if !i'2 - (D + 0'2)]' then it will nevertheless
be able to invade at a sufficiently high P loading rate.
The takeover by species 2 was always absolute for selectivities below,,',
with no intermittent interval allowing coexistence between the two species.
It thus appears that the activity of an invariant selective grazer does not
promote the coexistence of its prey species, like the non-selective grazing
considered in Section 6.1. This also means that the present model is unable
to represent a situation with stable coexistence of large, inedible "canopy"
species and small, edible "undergrowth" species identified as step 8 in the
PEG model of plankton succession (Sommer et aI. 1986).