52
Algae and Nutrients: Uptake and Utilization of Limiting ...
Droop 1983). The threshold model is equivalent to the Liebig law of the
minimum in that growth is limited by the single element least in supply.
Whether an essential resource becomes growth-limiting to a given organism
will then depend on the relative supply rate of this resource compared to other
essential resources.
Tilman (1980) introduced a graphical method for analyzing equilibrium
situations with two limiting resources based on the zero net growth isoclines,
which are the set of equilibrium resource levels where growth and losses are
exactly balanced, and thus net growth is zero. For the case of two essential
resources, the zero net growth isocline will consist of two straight line segments,
parallel to the resource axes, and intersecting at right angles at a point where
both resources are equally limiting (Fig. 3.8). For a pair of species to be able to
coexist at fixed densities on two essential resources, the isoclines must intersect;
that is, none of the species can be competitively superior for both resources. At
the intersection point, each species will be limited by the resource on which it is
competitively inferior.
Tilman (1980) has shown that the intersection of the zero net growth isoclines is a necessary, but not sufficient, condition for the stable coexistence of
two species on two limiting resources. In order to investigate the stability
properties of the equilibrium point, we will also have to consider the relative
resource consumption rates of the two species. For each species, we can
associate a resource consumption vector, which will be a directed line with
origin at the equilibrium point, and slope equal to the ratio of the resource
consumption rates.
Figure 3.8 shows the isoclines with respect to two essential resources (I and
II) for two species (1 and 2), such that neither species is competitively superior
for both resources. In the situation where the consumption vector u belongs to
species 1 and vector v to species 2, a slight increase in species 1 biomass would
result in a proportionally larger decrease in the equilibrium level of resource 1
than resource ll. The increase in species 1 biomass would therefore decrease the
growth rate of species 2 more than it would for species 1, thus giving species 1 a
growth advantage that would lead to further increase in species 1 biomass. The
resulting positive feedback loop would eventually lead to the exclusion of
species 2. A similar positive feedback cascade leading to the exclusion of species
1 could be initiated by a slight increase in species 2 biomass.
In the opposite situation, where the consumption vector u belongs to species
2 and vector v to species 1, a slight increase in species 1 biomass would result in
a proportionally larger decrease in the equilibrium level of resource II than
resource I. The increase in species 1 biomass would therefore decrease the
growth rate of species 1 more than it would for species 2, thus giving species 2 a
growth advantage that would prevent species 1 from increasing its biomass any
further. The resulting negative feedback loop would eventually restore the
equilibrium biomasses of both species after a perturbation.
Algae and Nutrients: Uptake and Utilization of Limiting ...
Droop 1983). The threshold model is equivalent to the Liebig law of the
minimum in that growth is limited by the single element least in supply.
Whether an essential resource becomes growth-limiting to a given organism
will then depend on the relative supply rate of this resource compared to other
essential resources.
Tilman (1980) introduced a graphical method for analyzing equilibrium
situations with two limiting resources based on the zero net growth isoclines,
which are the set of equilibrium resource levels where growth and losses are
exactly balanced, and thus net growth is zero. For the case of two essential
resources, the zero net growth isocline will consist of two straight line segments,
parallel to the resource axes, and intersecting at right angles at a point where
both resources are equally limiting (Fig. 3.8). For a pair of species to be able to
coexist at fixed densities on two essential resources, the isoclines must intersect;
that is, none of the species can be competitively superior for both resources. At
the intersection point, each species will be limited by the resource on which it is
competitively inferior.
Tilman (1980) has shown that the intersection of the zero net growth isoclines is a necessary, but not sufficient, condition for the stable coexistence of
two species on two limiting resources. In order to investigate the stability
properties of the equilibrium point, we will also have to consider the relative
resource consumption rates of the two species. For each species, we can
associate a resource consumption vector, which will be a directed line with
origin at the equilibrium point, and slope equal to the ratio of the resource
consumption rates.
Figure 3.8 shows the isoclines with respect to two essential resources (I and
II) for two species (1 and 2), such that neither species is competitively superior
for both resources. In the situation where the consumption vector u belongs to
species 1 and vector v to species 2, a slight increase in species 1 biomass would
result in a proportionally larger decrease in the equilibrium level of resource 1
than resource ll. The increase in species 1 biomass would therefore decrease the
growth rate of species 2 more than it would for species 1, thus giving species 1 a
growth advantage that would lead to further increase in species 1 biomass. The
resulting positive feedback loop would eventually lead to the exclusion of
species 2. A similar positive feedback cascade leading to the exclusion of species
1 could be initiated by a slight increase in species 2 biomass.
In the opposite situation, where the consumption vector u belongs to species
2 and vector v to species 1, a slight increase in species 1 biomass would result in
a proportionally larger decrease in the equilibrium level of resource II than
resource I. The increase in species 1 biomass would therefore decrease the
growth rate of species 1 more than it would for species 2, thus giving species 2 a
growth advantage that would prevent species 1 from increasing its biomass any
further. The resulting negative feedback loop would eventually restore the
equilibrium biomasses of both species after a perturbation.