106
Herbivores and Algae: Food Utilization, Growth and Reproduction ...
3. The threshold food level for positive net population growth (C"-t): a
measure of competitive ability toward other zooplankton populations
exploiting the same food resource.
These population properties are derived from coupled equations
involving a total of 11 parameters describing both energetic and demographic aspects of Daphnia biology, making it is not immediately obvious
how a perturbation in a given parameter might influence population
performance. Some insight into the direction and magnitude of the effect of
a small change in a parameter can be gained from sensitivity analysis.
Partial derivatives ozlop of a population property z with respect to a
parameter p were computed as central, fmite difference approximations
ozlop ::::IIh ..1p'! [z(P + Ap) - z(p - ~p)], where ~p is a small perturbation to
the nominal parameter value. As parameters differ in both units and
magnitude, the relative importance of a parameter cannot be inferred from
the partial derivatives alone. Results of the sensitivity analysis are therefore
reported as relative parameter sensitivities, which can be interpreted as the
exponent of a power function fitted to a small interval containing the
nominal parameter value. The relative sensitivity, which by economists is
usually termed the elasticity of a function z with respect to a parameter p, is
defined as (Plz)ozlop = o(1n z)lo(1n p)]. A relative sensitivity of 1 implies
direct proportionality, while -1 implies inverse proportionality. Relative
sensitivities different from unity imply suh- or superlinear proportionality
(for example, a relative sensitivity of 2 means that a 1 % change in the input
parameter gives 2% change in the output value).
By inspection of Fig. 4.21, it is apparent that the two parameters of greatest
overall importance to the the performance of a Daphnia population are the
maximal ingestion rate in neonates (I'.) and the assimilation efficiency (&). The
equality of relative sensitivities for these two parameters probably reflects that
they influence all population properties as the product & l'., i.e., the gross assimilation rate. Although not shown in Fig. 4.21, we can expect that food elemental composition, as discussed in Section 4.5, will also have a strong influence on population performance through the constraints placed on food assimilation efficiency by the maintenance of animal elemental composition.
The size dependency of food ingestion, determined by the parameter B'«l,
seems to be of comparatively minor importance. The relative sensitivities to the
maintenance costs of respiration and molting (r and h) are closely correlated,
with sensitivities to both parameters roughly proportional to their relative
magnitudes (5:1). As expected, the incipient limiting food concentration (C1
influences only the threshold level for population growth with relative sensitivity of 1; that is, C"-t is directly proportional to C.
Herbivores and Algae: Food Utilization, Growth and Reproduction ...
3. The threshold food level for positive net population growth (C"-t): a
measure of competitive ability toward other zooplankton populations
exploiting the same food resource.
These population properties are derived from coupled equations
involving a total of 11 parameters describing both energetic and demographic aspects of Daphnia biology, making it is not immediately obvious
how a perturbation in a given parameter might influence population
performance. Some insight into the direction and magnitude of the effect of
a small change in a parameter can be gained from sensitivity analysis.
Partial derivatives ozlop of a population property z with respect to a
parameter p were computed as central, fmite difference approximations
ozlop ::::IIh ..1p'! [z(P + Ap) - z(p - ~p)], where ~p is a small perturbation to
the nominal parameter value. As parameters differ in both units and
magnitude, the relative importance of a parameter cannot be inferred from
the partial derivatives alone. Results of the sensitivity analysis are therefore
reported as relative parameter sensitivities, which can be interpreted as the
exponent of a power function fitted to a small interval containing the
nominal parameter value. The relative sensitivity, which by economists is
usually termed the elasticity of a function z with respect to a parameter p, is
defined as (Plz)ozlop = o(1n z)lo(1n p)]. A relative sensitivity of 1 implies
direct proportionality, while -1 implies inverse proportionality. Relative
sensitivities different from unity imply suh- or superlinear proportionality
(for example, a relative sensitivity of 2 means that a 1 % change in the input
parameter gives 2% change in the output value).
By inspection of Fig. 4.21, it is apparent that the two parameters of greatest
overall importance to the the performance of a Daphnia population are the
maximal ingestion rate in neonates (I'.) and the assimilation efficiency (&). The
equality of relative sensitivities for these two parameters probably reflects that
they influence all population properties as the product & l'., i.e., the gross assimilation rate. Although not shown in Fig. 4.21, we can expect that food elemental composition, as discussed in Section 4.5, will also have a strong influence on population performance through the constraints placed on food assimilation efficiency by the maintenance of animal elemental composition.
The size dependency of food ingestion, determined by the parameter B'«l,
seems to be of comparatively minor importance. The relative sensitivities to the
maintenance costs of respiration and molting (r and h) are closely correlated,
with sensitivities to both parameters roughly proportional to their relative
magnitudes (5:1). As expected, the incipient limiting food concentration (C1
influences only the threshold level for population growth with relative sensitivity of 1; that is, C"-t is directly proportional to C.