Differential Loss Rates and Invadability of Equilibria
173
species constructed by making them identical with respect to all growth
and uptake parameters except the maximal growth rate, which is assumed
to be highest in species 1 (P"I > P"2)' Since the initial slopes of the Monod
curves [given by Eq. (3.16)] will be identical for this particular choice of
parameters, the Monod curves will be very close to each other for S close to
zero, and diverge progressively with increasing S. This explains why !
approaches 1 for S ~ 0 and decreases towards an asymptotic level::::: jI'l Ii'l
for S ~ 00. For a grazer with a given selectivity (J for species 2 (dashed horizontalline in Fig. 6.8), a shift from dominance by species 2 to species 1 will
take place at the critical inorganic P concentration corresponding to
! = (J( dashed vertical line in Fig. 6.8).
In the following we will investigate how different kinds of competitive
handicaps can affect the ability of a grazing-resistant species to invade a
simple two-species system consisting of a grazer and its preferred prey. As
in the previous Section, we can perform simulated invasion experiments by
first letting the system converge to its stationary state (a stable focus or a
limit cycle) with only species 1 resident. The invasion is then classified as
successful if species 2 is able to increase its biomass from an initial inoculum, arbitrarily set to 1% of the initial species 1 biomass.
By making repeated invasion experiments under different phosphorus
supply rates [Lp = DP L ; (Jlg P) r
l day-I], we can express the vulnerability to
invasion as function of the external forcing variable Lp instead of the internal state variable S. As in Section 6.1, we can track the two major dynamic
modes of the target system with only species 1 resident by either slowly
increasing Lp from an initial low value (tracking the stable focus) or slowly
decreasing Lp from an initial high value (tracking the limit cycle). By
repeated simulations with different values of (J we can find numerically the
critical selectivity (J- for a given P supply rate (Lp). The search for the critical selectivity (J- is efficiently implemented as a bracketing and bisection
procedure similar to those described in Appendix A9.
We will first consider the case where the invading species has no grazingindependent disadvantage in terms of differential nutrient efflux or differential sinking loss. That is, we will assume that S'I = S'2 = 0 (no threshold
for positive net uptake), and that CT) = CT 2 = 0.0008 day") (the nominal sinking loss rate used in previous Sections). Figure 6.9 shows three different
combinations of growth and uptake parameters in the two competing
phytoplankton species, corresponding to decreasing competitive ability of
the invading species (2) with respect to the resident species (1). The maximal growth rates and maximal P uptake affinities are varied according to
Table 6.1, while the phosphorus subsistence quotas and storage capacities
are in all cases assumed identical to the "typical" values proposed in Table
3.1 [Q') = Q'2 = 3.8 (Jlg P) (mg ct and Q"I = Q"2 = 28.5 (Jlg P) (mg C)-I].
173
species constructed by making them identical with respect to all growth
and uptake parameters except the maximal growth rate, which is assumed
to be highest in species 1 (P"I > P"2)' Since the initial slopes of the Monod
curves [given by Eq. (3.16)] will be identical for this particular choice of
parameters, the Monod curves will be very close to each other for S close to
zero, and diverge progressively with increasing S. This explains why !
approaches 1 for S ~ 0 and decreases towards an asymptotic level::::: jI'l Ii'l
for S ~ 00. For a grazer with a given selectivity (J for species 2 (dashed horizontalline in Fig. 6.8), a shift from dominance by species 2 to species 1 will
take place at the critical inorganic P concentration corresponding to
! = (J( dashed vertical line in Fig. 6.8).
In the following we will investigate how different kinds of competitive
handicaps can affect the ability of a grazing-resistant species to invade a
simple two-species system consisting of a grazer and its preferred prey. As
in the previous Section, we can perform simulated invasion experiments by
first letting the system converge to its stationary state (a stable focus or a
limit cycle) with only species 1 resident. The invasion is then classified as
successful if species 2 is able to increase its biomass from an initial inoculum, arbitrarily set to 1% of the initial species 1 biomass.
By making repeated invasion experiments under different phosphorus
supply rates [Lp = DP L ; (Jlg P) r
l day-I], we can express the vulnerability to
invasion as function of the external forcing variable Lp instead of the internal state variable S. As in Section 6.1, we can track the two major dynamic
modes of the target system with only species 1 resident by either slowly
increasing Lp from an initial low value (tracking the stable focus) or slowly
decreasing Lp from an initial high value (tracking the limit cycle). By
repeated simulations with different values of (J we can find numerically the
critical selectivity (J- for a given P supply rate (Lp). The search for the critical selectivity (J- is efficiently implemented as a bracketing and bisection
procedure similar to those described in Appendix A9.
We will first consider the case where the invading species has no grazingindependent disadvantage in terms of differential nutrient efflux or differential sinking loss. That is, we will assume that S'I = S'2 = 0 (no threshold
for positive net uptake), and that CT) = CT 2 = 0.0008 day") (the nominal sinking loss rate used in previous Sections). Figure 6.9 shows three different
combinations of growth and uptake parameters in the two competing
phytoplankton species, corresponding to decreasing competitive ability of
the invading species (2) with respect to the resident species (1). The maximal growth rates and maximal P uptake affinities are varied according to
Table 6.1, while the phosphorus subsistence quotas and storage capacities
are in all cases assumed identical to the "typical" values proposed in Table
3.1 [Q') = Q'2 = 3.8 (Jlg P) (mg ct and Q"I = Q"2 = 28.5 (Jlg P) (mg C)-I].