Soil Biology & Biochemistry 38 (2006) 1–6
www.elsevier.com/locate/soilbio
Using process-based modelling to analyse earthworm life cycles
T. Jagera,*, S.A. Reineckeb, A.J. Reineckeb
a
Vrije Universiteit Amsterdam, FALW/Department of Theoretical Biology, De Boelelaan 1085, NL-1081 HV Amsterdam, The Netherlands
b
Ecotoxicology Group, Department of Botany and Zoology, University of Stellenbosch, Matieland 7602, South Africa
Received 1 September 2004; received in revised form 6 April 2005; accepted 9 April 2005
Abstract
To understand the life cycle of an organism, it is important to understand the physiological processes that govern growth and reproduction.
In this paper, we re-analyse a life-cycle data set for the earthworm Eisenia veneta, using a process-based model. The data set comprises
measurements of body size and cocoon production over 200 days, at two temperatures (15–25 8C) and two densities (five and 10 worms per
container, but with the same worm:soil weight ratio). The model consists of a set of simple equations, derived from Dynamic Energy Budget
(DEB) theory. The dynamics of growth and reproduction are simultaneously described by the model, using very few parameters (five
parameters for four curves). This supports the use of this model for efficient analysis of earthworm life-cycle data, and to interpret the effects
of stressors. However, there was considerable inter-individual variation in the response, hampering the interpretation of the temperature and
density effects. A temperature increase corresponded to an increase in the rate constants for growth and reproduction (with the same factor),
without affecting the other parameters, as expected from DEB theory. Changing the earthworm density hardly affected the growth curves, but
had an unexpected effect on reproduction: at higher densities, the worms start to produce cocoons at a larger body size and the maximum
reproduction rate was lower. This study confirms the use of DEB as a reference model for earthworms, and using this model, we can
recognise that temperature has a predictable effect on the life cycle of E. veneta. Furthermore, this analysis reveals that the effects of density
are less clear and may involve a change in energy allocation that requires further study.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Earthworm; Eisenia veneta; Life cycle; Growth; Reproduction; Temperature; Density
1. Introduction
To understand the life cycle of an organism, it is
important to understand the physiological processes that
govern growth and reproduction. However, life-cycle
studies are usually more descriptive than mechanistic. The
various endpoints (e.g. growth and reproduction) are often
treated as isolated processes, whereas they are in fact closely
interrelated. For example, reproduction generally starts at a
certain minimum body size, which means that any treatment
that slows growth automatically leads to a delay in
reproduction (e.g. Klok and De Roos, 1996, for copper in
Lumbricus rubellus, Jager, 2005, for food density and
temperature in nematodes). Furthermore, body size determines reproduction rates, because the organism converts
* Corresponding author. Tel.: C31 20 598 7134; fax: C31 20 598 7123.
E-mail address: [email protected] (T. Jager).
0038-0717/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.soilbio.2005.04.009
food into offspring, and feeding rates relate to body size. To
increase our understanding of life histories, the different
aspects should be integrated into a single model framework.
We here use the theory of Dynamic Energy Budgets (DEB,
Kooijman, 2000; 2001; Nisbet et al., 2000) to investigate
these relationships. This theory describes how individuals
acquire and utilize energy, based on a set of simple rules for
metabolic organization, treating the organism as a system
with a closed mass and energy balance.
The advantage of such a model-based approach is that it
facilitates the interpretation of observed effects of stressors
on the life cycle, and allows for the prediction of effects
under other circumstances (e.g. for the combined effects of
toxicants and food limitation, Jager et al., 2004). To
illustrate the usefulness of this process-based model, we
analyse data from life-cycle experiments with the earthworm Eisenia veneta (formerly called Dendrobaena
veneta), using a set of simplified equations based on DEB
theory. This data set was published earlier by Viljoen et al.
(1992). E. veneta, a species from the northern hemisphere, is
not a typical soil-dwelling species, but is usually confined to
2
T. Jager et al. / Soil Biology & Biochemistry 38 (2006) 1–6
accumulations of organic debris (decaying leaves, manure
heaps). For this reason, this species has been identified as a
potential candidate for vermiculture (the transformation of
organic waste to compost using earthworms) (Edwards and
Bater, 1992; Lofs-Holmin, 1986). Furthermore, this species
is very popular among anglers, and many breeders
specialize in this species. Clearly, a thorough mechanistic
understanding of the life-history aspects of this species can
have both scientific as well as economic benefits.
2. Methods
2.1. Experimental
Details of the experimental set-up have been described
by Viljoen et al. (1992). Day-old hatchlings of Eisenia
veneta (average weight 23 mg, SD 5.2) were placed into a
cattle manure substrate with a moisture content of 75–80%.
The worms were maintained under constant temperatures
(at 15 or 25 8C), and in two densities (five or 10 individuals
per container). The densities only differed with respect to
the number of worms per container; in the high density,
more substrate was used to keep the number of worms per
mass of substrate constant (the data for 10 worms/container
were not used in the original analysis by Viljoen et al.). Fifty
grams of the cattle manure substrate was provided per
worm, which ensured sufficient food per worm for the
experimental duration (ad libitum conditions). Additional
feeding took place after 20 days, and every 10 days
thereafter, by adding equal amounts of manure per worm to
the containers. At each temperature, there were five
containers with five worms, and two with ten. Every five
days, the worms were weighed, and the number of mature
worms (judged from the presence of a clitellum) and
cocoons were reported.
2.2. Models
A consequence of the DEB theory is that organisms
should grow according to a Von Bertalanffy growth curve,
as long as the food density is ad libitum, or remains
constant. The rationale, underlying this specific curve, is
that food uptake is generally proportional to a surface area
(squared length for isomorphs), whereas maintenance costs
are proportional to volume (cubed length) (Kooijman,
2000). When the organism grows, volume increases more
rapidly than surface area, and at a certain body size, all of
the resources allocated to growth and somatic maintenance
are used for the latter, and growth ceases. This conceptual
view of growth also explains why growth rates and ultimate
body size decrease at lower feeding rates.
To simplify the equations, we will use scaled lengths, i.e.
body length as fraction of the ultimate length at abundant
food (Lm). Scaled lengths are denoted by a lowercase symbol
(l), absolute length by uppercase (L). As long as the organism
does not change its shape, in practice, any length measure can
be used (e.g. total body length, body width, or length of a
specific organ). In this case, body size for E. veneta were
determined as wet weight. These weights can be translated to
a length measure by a recalculation to body volume
(assuming a constant specific density of 1 g cmK3), and
taking the cubic root of the volume. The length measure thus
obtained is called a ‘volumetric body length’.
At constant food levels, the DEB theory reduces to the
Von Bertalanffy growth equation, giving scaled length as a
function of time (Kooijman and Bedaux, 1996)
d
l Z rB ðf K lÞ
dt
(1)
where f is the scaled ingestion rate (as fraction of the
maximum rate), and rB is a growth rate constant (dimension
per time). When f and rB are constant, Eq. (1) can be
integrated
l Z f ð1 K eKrB t Þ C l0 eKrB t
(2)
where l0 is the initial scaled length. The growth rate (rB) has,
within DEB theory, a special relationship with maintenance
costs and thus with the ultimate size (see Kooijman and
Bedaux, 1996). This equation produces a sigmoid curve of
body weight or volume, but a non-sigmoid curve for body
length (i.e. a curve with strictly decreasing slope, see Fig. 1).
Changing the food level changes the relative ingestion rate
(f) in Eq. (1) and (2). Less food (or food of lower quality)
leads to a smaller ultimate size, as was shown for E. fetida
by Neuhauser et al. (1980), and for E. veneta by Fayolle
et al. (1997).
DEB theory specifies that the reproduction rate (R in
offspring per time) is a function of the body size (or scaled
length l), as given by Kooijman and Bedaux (1996)
R
g Cl 2
RðlÞ Z m 3
fl K l3p
(3)
1 K lp g C f
where Rm is the maximum reproduction rate, and lp the
scaled length at the start of reproduction. This equation is a
simplification of the full DEB theory, specifically intended
for the analysis of toxicity data. The close relationship
between body size and reproduction was also shown for E.
fetida in direct experiments with varying food levels
(Reinecke and Viljoen, 1990a). The energy investment
ratio (g) is a dimensionless parameter, with the interpretation of the cost of new biomass relative to the total
available energy for growth and maintenance. For water
fleas, g is probably close to 1 (Kooijman and Bedaux, 1996).
However, the model is not particularly sensitive to this
parameter, unless different food levels are compared. To
facilitate the comparison of the model to reproduction data,
Eq. (3) is integrated and compared to the cumulative
number of cocoons produced per mature worm.
Temperature has, within DEB context, a predictable effect
on the organism. A temperature increase is expected to
T. Jager et al. / Soil Biology & Biochemistry 38 (2006) 1–6
3
14
12
10
8
volumetric body length (mm)
6
4
2
0
15˚C 5 worms
15˚C 10 worms
25˚C 5 worms
25˚C 10 worms
14
12
10
8
6
4
2
0
0
20
40
60
80 100 120 140 160 180 200
time (days)
0
20
40
60
80 100 120 140 160 180 200
time (days)
Fig. 1. Growth of Eisenia veneta at two temperatures and two densities, shown as volumetric body length (cubic root of body weight). Open symbols are mean
length with SD, without outliers (see text); outliers are shown as small black symbols.
increase physiological rates like the growth and reproduction
rate in ectotherms, because all molecular reactions will
proceed faster at higher temperatures. Other parameters,
such as the length at puberty and the maximum length, should
not be affected. Maximum size depends on the ratio of two
rates (the assimilation and maintenance rates), which can be
expected to change in a similar way with temperature. It was
shown that rate constants generally follow the relation
proposed by Arrhenius (see Kooijman, 2000)
TA
TA
kðTÞ Z kðTref Þexp
K
(4)
Tref
T
where the value of a rate constant k at a certain temperature T
(in Kelvin) is related to k at an arbitrary reference
temperature Tref. In this relationship, TA is the ‘Arrhenius
temperature’, which can realistically be assumed to have the
same value for all physiological processes (Kooijman, 2000).
2.3. Fitting the model to the data
The models (Eq. (2)–(4)) are implemented in MatLabw
6.5. The growth and reproduction data are fitted simultaneously because these sets share common parameters (see
Jager et al., 2004, 2005). The initial size was fixed to
the average of the weights at the first time point, and ad
libitum food availability was assumed (i.e. fZ1) for all
treatments. Some individual worms in the experiments grew
much slower than the average worms, or even failed to grow
at all. These worms will probably never mature, and are
considered to be outliers. The same phenomenon was
observed by Viljoen et al. (1991), who reported that one in
10 worms in the experiments remained much smaller and
never reached maturity. For the purpose of this analysis,
outliers were defined as individual weights that fall below
the median by more than the inter-quartile range (the
distance between the 25th and 75th percentile). For one set
of data (five worms per container at 25 8C), variability was
so high that we decided to use half the inter-quartile range as
criterion. All data for one density are fitted simultaneously,
assuming that the rate constants follow the temperature
dependence of Eq. (4), and that the other parameters are
independent of temperature (see Table 1). All model fitting
is done on the basis of weighted least-squares estimation.
Data points receive weighing factors according to the
number of individuals on which the data point is based, as
well as to account for the observation that variation tends to
increase with the average value. Confidence intervals are
generated using profile likelihoods (see Meeker and
Escobar, 1995).
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T. Jager et al. / Soil Biology & Biochemistry 38 (2006) 1–6
Table 1
Parameter estimates for the model fits in Figs. 1 and 2. Length measures are volumetric length (cubic root of volume, assuming a specific density of 1 g/cm3),
rates are given at a reference temperature of 20 8C
Maximum body length (Lm, mm)
Von Bertalanffy growth rate constant (rB, dK1)
Length at first reproduction (Lp, mm)
Maximum reproduction rate (Rm, coc./d)
Arrhenius temperature (TA, K)
Five worms/container (at 15 and 25 8C)
10 worms/container (at 15 and 25 8C)
13.2 (12.9–13.4)
0.0156 (0.0148–0.0167)
8.57 (8.31–8.84)
0.406 (0.382–0.431)
2810 (2630–2990)
13.2 (13.1–13.4)
0.0152 (0.0147–0.0157)
9.88 (9.74–10.0)
0.306 (0.287–0.325)
3870 (3700–4050)
The initial length (L0) is fixed to 2.82 mm (the average starting weight).
3. Results
The fit of the Von Bertalanffy growth curve (Eq. (2)) is
good, although there are quite some data points that are
considered outliers, especially at five worms/container (Fig.
1). Apparently, some worms failed to grow under the
experimental conditions. The variation in the length
measures is quite high, and generally increases with the
average value. The maximum length does not depend on the
treatment, although the worms of the low density group at
15 8C appear to remain somewhat smaller than the model
predicts. The growth rate constant (rB, referenced at 20 8C)
is also very similar at both densities. However, rB cannot be
directly compared between densities as the temperature
dependence (TA) differs (the confidence intervals do not
overlap). Although significant, the absolute effect of the
difference in TA is very small; the calculated growth rate
constants at 15 and 25 8C (using Eq. (4)) differ only some
10% between both densities. Overall, the growth curves
reveal a predictable effect of temperature, and are very
similar for both densities.
The reproduction data are also well described by the
model of Eq. (3) (Fig. 2). A decrease in temperature has the
predicted effect that the onset of reproduction is delayed (as
a result of slower growth), and the maximum rate of
reproduction is decreased (see Eq. (4)). The body size for
the start of reproduction is unaffected by temperature.
Density has, in contrast to growth, a profound effect on
the reproduction pattern; at higher densities, reproduction
starts at a larger body size and the maximum reproduction
rates are lower (Table 1).
4. Discussion
The process-based model provides a good fit on growth
and reproduction data together, with few parameters (the
10 parameters of Table 1 for the eight curves of Fig. 1
and 2), which all have a direct physiological interpretation, in contrast to purely descriptive models. These
parameters are themselves build up from parameters with
a more direct interpretation in terms of energy (Kooijman,
2000). Even though the compound parameters have a
simple interpretation, the underlying theory ensures
consistency with regard to energy allocation and mass
flows. Furthermore, growth and reproduction are linked in
a consistent manner.
Overall, drawing conclusions from the E. veneta data set
is hampered by the large inter-individual variation. Part of
this variation may be related to the considerable variation in
starting weights (although hatchlings are difficult to weigh
accurately). We tried to minimise the effects by removing
the poorly growing worms from the growth curve (see
Fig. 1), and by expressing reproduction as the total number
of cocoons per clitellate worm (assuming that the slow
growing worms hardly contribute to the total reproduction).
60
cumulative cocoons per worm
5 worms
10 worms
50
25˚C
40
30
25˚C
20
15˚C
10
15˚C
0
0
20
40
60
80
100 120 140 160 180 200
time (days)
0
20
40
60
80
100 120 140 160 180 200
time (days)
Fig. 2. Cumulative reproduction (cocoons per adult worm) of Eisenia veneta at two densities and two temperatures.
T. Jager et al. / Soil Biology & Biochemistry 38 (2006) 1–6
Nevertheless, this large variation may have biased our
results. Owing to this large amount of variation, the current
data set is not optimal for application of the DEB model.
When setting up a dedicated experiment for analysis with
this model, care must be taken to ensure a homogenous test
population, or alternatively, the organisms can be followed
individually. However, the large number of observations on
both body size and reproduction in time, and the two
treatments (temperature and density), in this data set are
quite valuable for testing the model.
Temperature has a predictable effect on the life cycle;
an increase in temperature increase the values for the rate
constants for growth (rB) and reproduction (Rm) to the
same extent (see Eq. (4)), without affecting the maximum
length or the length at the start of reproduction. The net
effect is that lower temperatures will slow down growth,
delay reproduction, and slow down cocoon production.
This is consistent with the observations by Fayolle et al.
(1997), who studied the life cycle of E. veneta on paper
sludge and horse manure over a range of temperatures
(15–25 8C). However, using horse manure instead of paper
sludge as food source clearly decreased the maximum size
and the reproduction rate. Within the DEB context, such a
change is most likely related to changes in food availability
or food quality (changing f in Eq. (2) and (3)). Even though
temperature has a predictable effect, the Arrhenius
temperature (TA) differs between both densities; at high
density, a change in temperature seems to have more effect
on the rate constants than at lower density. We have no
explanation for this phenomenon, but the absolute effect of
this difference on the rate constants is very small
(approximately 10%). Compared to other organisms
(Kooijman, 2000; Jager, 2005), the TA in this study is
quite low, indicating that E. veneta is not heavily affected
by a change in temperature.
The number of worms per container had an unexpected
influence on the life cycle: the growth curves are hardly
affected, but reproduction starts at a larger body size and
commences at a lower rate. The reason for this particular
behaviour is unclear, but may relate to a change in their
strategy for energy allocation. Perhaps, worms experiencing
higher densities devote less of the available energy to
maturation and reproduction, resulting in a larger body size
at first cocoon production and a lower reproduction rate.
However, such a change in allocation would also result in
larger animals with a higher growth rate, which was not
clearly observed in this data set (although the worms at
15 8C and low density are indeed somewhat smaller than the
model predicts). It is possible that the worms, when
encountering many others in their surroundings, follow a
different strategy in using the allocated energy for
reproduction. For example, it may be that the reproduction
rate is lower at high densities, but that the investment per
cocoon is higher (e.g. resulting in larger cocoons, providing
a better start for the hatchlings). Generally, and in
accordance with previous findings however, worms at
5
a higher density performed less well than those at lower
densities: they reproduced at a larger body size and at a
lower rate.
It should be noted that density per se (in terms of
number of worms per volume of substrate) was not the
governing factor, because the amount of substrate per
worm at both densities was very similar. It is therefore
unclear what would trigger this change in energy allocation
or use. Higher numbers per container may perhaps have
subtle effects on food quality (e.g. through effects of faeces
on microbial activity), interaction, feeding and mating
behaviour, that were not explicitly included in the model.
Other experiments with epigeic species (E. fetida and E.
andrei) demonstrated that worm density in terms of
number of worms per volume of substrate could influence
reproduction as well as growth rate and maximum size
considerably (Neuhauser et al., 1980; Reinecke and
Viljoen, 1990b; Domı́nguez and Edwards, 1997). Because
these investigators used different densities in the strict
sense, the negative effects of density are most likely caused
by a decrease in the quantity or quality of food, possibly
through detrimental effects of excreta (see Neuhauser et al.,
1980).
The current analysis shows that growth and reproduction of E. veneta largely agree with the DEB model, and
can be analysed together. Simultaneous analysis of the
endpoints is necessary, as effects on reproduction cannot be
properly interpreted without the growth data. For example,
temperature leads to a delay in reproductive output, which
is fully explained by the reduced growth rate (reproduction
starts at the same body size). However, at the same time,
the model analysis helps to reveal that density has a rather
unexpected effect on the reproductive pattern (a larger size
at first reproduction, and a lower rate of cocoon
production), that may involve changes in the earthworm’s
strategy for energy allocation. An additional benefit of the
simultaneous model fit is that the parameters and their
confidence intervals are based on the entire data set. For
example, the size at first reproduction at five worms per
container (and its confidence interval) results from the
model fits on growth and reproduction at two temperatures
(thus using the information present in four curves to
estimate this parameter).
In conclusion, the DEB model can be used a platform for
the interpretation of effects of stressors on the life cycle.
Especially the deviation from the model predictions (such as
density in our example) is valuable, as this provides
directions for further investigation. The simultaneous
analysis of the parameters means that the information
available in the data set is used with optimal efficiency,
leading to well-defined parameter estimates that are based
on the entire data set. Apart from density and temperature,
other stressors may be analysed in a similar fashion, such as
toxicants (Kooijman and Bedaux, 1996) or food limitation
(Jager, 2005).
6
T. Jager et al. / Soil Biology & Biochemistry 38 (2006) 1–6
Acknowledgements
This research was supported by the Technology Foundation STW, applied science division of NWO and the
technology programme of the Ministry of Economic Affairs.
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