102
Herbivores and Algae: Food Utilization, Growth and Reproduction ...
the maternity function will be identically zero everywhere. This threshold
food level for reproduction, C"R' can be found by setting iJ = 0 and B = B'in
Eqs. (4.14)-(4.17) and solving for e:
C" = r + h B' .. - ~ C',
R
£1' B' -B'
1
..
(4.32)
or C"R = 0.063 (mg e) }"' with the parameter values in Table 4.2.
Asymptotic Population Properties - the Intrinsic Rate of Increase. In
Appendix AS it is shown that, from a given pair of maternity and survival
functions [m(x) and l(x)], the corresponding asymptotic net population
growth rate (.It), can be found by solving the characteristic equation .
..
J ,,(x)l(x)e-kdx = l'
(4.33)
o
Repeating this procedure for maternity functions generated by a range of
food concentrations gives the relationship between net population growth
rate and food level (Fig. 4.18). Due to the piecewise linear functional response
[Eq. (4.11)], all population statistics will be constant for food levels above the
incipient limiting concentration (C ~ e'), giving a maximal intrinsic rate of
increase .It' = 0.381 day-'. The maximal intrinsic rate of increase predicted by
the model compares well with an average .It' = 0.37 ± 0.06 day-' estimated from
a number of life-table experiments on different Daphnia species, as
summarized in Table AI0.7.
Population growth rate falls off quite linearly from the incipient limiting
level, and then drops abruptly when food concentration approaches the
singularity at the threshold level for reproduction (C"R)' At a food concentration somewhat above C"R' births and deaths are exactly balanced in the
stable age distribution, so that net population growth will be zero (.It = 0).
This threshold food level for positive net population growth (C"..l.) can be
found more precisely by setting .It = 0 in the characteristic equation (AS.S),
and locating the value of C corresponding to a net maternity rate of unity
[Eq. (A5.6)]. Using the same root-finding procedure as above, the threshold
food level for net population growth was found to be C"..l. = 0.0725 (mg e) }"'.
For two or more zooplankton populations exploiting a common food
resource, the population with the lowest threshold food level will be competitively superior, because it will eventually depress the food resource to a
level where the other populations are unable to maintain positive net
population growth. As conjectured by Lampert (1977), the food level (C"..l.)
needed for positive population growth will be higher (16% in this case),
than is necessary to reproduce (C"R)'
Herbivores and Algae: Food Utilization, Growth and Reproduction ...
the maternity function will be identically zero everywhere. This threshold
food level for reproduction, C"R' can be found by setting iJ = 0 and B = B'in
Eqs. (4.14)-(4.17) and solving for e:
C" = r + h B' .. - ~ C',
R
£1' B' -B'
1
..
(4.32)
or C"R = 0.063 (mg e) }"' with the parameter values in Table 4.2.
Asymptotic Population Properties - the Intrinsic Rate of Increase. In
Appendix AS it is shown that, from a given pair of maternity and survival
functions [m(x) and l(x)], the corresponding asymptotic net population
growth rate (.It), can be found by solving the characteristic equation .
..
J ,,(x)l(x)e-kdx = l'
(4.33)
o
Repeating this procedure for maternity functions generated by a range of
food concentrations gives the relationship between net population growth
rate and food level (Fig. 4.18). Due to the piecewise linear functional response
[Eq. (4.11)], all population statistics will be constant for food levels above the
incipient limiting concentration (C ~ e'), giving a maximal intrinsic rate of
increase .It' = 0.381 day-'. The maximal intrinsic rate of increase predicted by
the model compares well with an average .It' = 0.37 ± 0.06 day-' estimated from
a number of life-table experiments on different Daphnia species, as
summarized in Table AI0.7.
Population growth rate falls off quite linearly from the incipient limiting
level, and then drops abruptly when food concentration approaches the
singularity at the threshold level for reproduction (C"R)' At a food concentration somewhat above C"R' births and deaths are exactly balanced in the
stable age distribution, so that net population growth will be zero (.It = 0).
This threshold food level for positive net population growth (C"..l.) can be
found more precisely by setting .It = 0 in the characteristic equation (AS.S),
and locating the value of C corresponding to a net maternity rate of unity
[Eq. (A5.6)]. Using the same root-finding procedure as above, the threshold
food level for net population growth was found to be C"..l. = 0.0725 (mg e) }"'.
For two or more zooplankton populations exploiting a common food
resource, the population with the lowest threshold food level will be competitively superior, because it will eventually depress the food resource to a
level where the other populations are unable to maintain positive net
population growth. As conjectured by Lampert (1977), the food level (C"..l.)
needed for positive population growth will be higher (16% in this case),
than is necessary to reproduce (C"R)'