Download Population dynamics of two marine polychaetes: the

ICES Journal of Marine Science, 57: 1488–1494. 2000
doi:10.1006/jmsc.2000.0912, available online at http://www.idealibrary.com on
Population dynamics of two marine polychaetes: the relative role
of density dependence, predation, and winter conditions
Jaap van der Meer, Jan J. Beukema, and Rob Dekker
Van der Meer, J., Beukema, J. J., and Dekker, R. 2000. Population dynamics of two
marine polychaetes: the relative role of density dependence, predation, and winter
conditions. – ICES Journal of Marine Science, 57: 1488–1494.
We modelled the population dynamics of two polychaete species, a prey (Scoloplos
armiger) and its predator (Nephtys hombergii) by simple linear and non-linear
time–series models using a 29 year data series of population abundance and winter
conditions. The log reproductive rate (i.e., the difference in log density between two
succeeding years) of the prey population was density dependent and negatively related
to the density of the predator. The log reproductive rate of the predator was also
density dependent, but was to a greater extent determined by winter temperature, a
density-independent factor. Log reproductive rate of the predator was not related to
the density of the prey, nor could delayed density dependence be detected. Apparently,
the feedback mechanism between predator and prey was weak, perhaps because the
predator is a generalist. Predator density can therefore be categorized as a densityindependent factor with respect to the dynamics of the prey species. In neither of the
two species was density dependence strong enough to generate complex or chaotic
behaviour, as indicated by the weakly negative slope of the reproduction curve at the
equilibrium, the apparent lack of a non-linear component in the reproduction curve,
and the negative Lyapunov exponent.
2000 International Council for the Exploration of the Sea
Key words: chaos, density dependence, Lyapunov exponent, polychaetes, population
dynamics, predation.
J. van der Meer, J. J. Beukema, and R. Dekker: Netherlands Institute for Sea Research,
PO Box 59, 1790 AB Den Burg (Texel), The Netherlands; e-mail: [email protected]
Introduction
The controversy about the role of density-dependent
regulation of population densities has led to one of the
most acrimonious and heated debates in ecology
(Wolda, 1995; Den Boer and Reddingius, 1996; Bonsall
et al., 1998). Though the dispute has not yet been settled,
it is now understood that much of it arose from a lack of
standardization of terms (Bonsall et al., 1998). Even the
terms ‘‘density dependence’’ and ‘‘density-dependent
factor’’ have been used ambiguously (Royama, 1992;
Wold and Dennis, 1993). At present, the point of
interest no longer seems to be a strict choice between
either density-dependent regulation or not. Attention
has shifted towards quantifying the relative importance
of density-dependent and density-independent factors in
determining population fluctations, and testifying the
hypothesis of regulation per se is no longer thought
profitable (Royama, 1992; Wolda, 1995). One way
of quantifying the relative importance of densitydependent and density-independent factors is through
the use of stochastic population process models that
1054–3139/00/051488+07 $30.00/0
include both types of effects (Royama, 1992). These
models are then challenged by real long-term data; the
relative role of the two sorts of effects can be determined
by choosing the best model (Leirs et al., 1997; Grenfell
et al., 1998).
A second key topic among population ecologists is the
relevance of complex and chaotic dynamics in real
populations. May (1974, 1976) showed that very simple
deterministic non-linear models of single species in discrete time can reveal chaotic behaviour. Generally,
chaotic dynamics in a non-linear process only occur
when the level of density dependence is strong (Royama,
1992). However, the earliest applications of non-linear
stochastic process models to a number of insect data sets
suggested that complex and chaotic behaviour is
unlikely to be widespread in real populations (Hassell
et al., 1976). This conclusion has been rather influential
in discussions on the importance of chaos in population
biology, but was questioned by Turchin and Taylor
(1992). They argued that the results obtained by Hassell
et al. (1976) largely arose from using a single-species
model without delayed density dependence. Complex
2000 International Council for the Exploration of the Sea
Population dynamics of two marine polychaetes
Materials and methods
Data
S. armiger and N. hombergii belong to the most common
macrozoobenthic species of the tidal flats in the Dutch
Log biomass density
0
(a)
–1
–2
–3
–4
–5
–6
1970
6
Winter temperature
dynamics are more likely in higher-dimensional systems
(e.g., multi-species system or single-species systems with
more than one age class), and analysing such systems in
fewer dimensions will hide this complexity and yield
erroneous results. Higher-dimensional systems can be
rewritten in terms of one-dimensional systems with time
delays (Royama, 1981, 1992). Studies based on singlespecies models with time delays indeed pointed to the
existence of chaotic dynamics in natural populations
(Turchin and Taylor, 1992; Turchin, 1993, 1996). However, other studies could not confirm this result (Perry
et al., 1993; Zhou et al., 1997), and the debate over
whether chaotic dynamics are relevant to ecological
reality continues.
A third point of interest is the role predators play in
determining the dynamics of a prey species (Berryman,
1992b). Predation may act either as a density-dependent
or -independent factor. In terms of the definition provided by Royama (1992, p. 21, ‘‘if the state of existence
of an ecological factor, identified in terms of its measure
or parameter, is, in turn influenced by the population
density of the animal, the factor is said to be density
dependent. Otherwise it is said to be densityindependent’’), the criterion is whether the predator’s
rate of increase (and hence its population density) is
related to prey density. Thus, in the case of a specialist
predator the relationship may be tight and may result in
(delayed) density dependence. In the case of a generalist
predator, whose growth rate is not, or only weakly,
related to the density of a single prey species (Hanski
et al., 1993; Turchin and Hanski, 1997), predation will
more or less act as a density-independent factor.
We analysed data that were collected as part of a
long-term sampling programme of the zoobenthic community of a tidal-flat area in the Dutch Wadden Sea
from 1970. Various non-linear process models were
fitted to the time series (29 years) of two polychaete
species: Scoloplos armiger (O. F. Müller, 1776) and its
main predator Nephtys hombergii (Savigny, 1818). Using
the best-fitting model(s), we examined; (i) the relative
importance of (delayed) density-dependent and densityindependent factors in determining population growth
rate, and, specifically, the role of winter temperature, as
the most likely candidate of a density-independent factor
(Beukema, 1985); (ii) the role of predation, i.e., what is
the effect of prey density on the growth rate of the
predator and vice versa; and (iii) the dynamic behaviour
of the fitted models: do they exhibit stable or complex
behaviour? Lyapunov exponents were estimated to
answer this question.
1489
1975
1980
1985
1990
1995
1975
1980
1985
1990
1995
(b)
5
4
3
2
1
1970
Figure 1. Time series of (a) log biomass density (g ash-free dry
mass m 2) of Scoloplos armiger (continuous line) and Nephtys
hombergii (broken line) at Balgzand and (b) average surface
water temperature (C) in winter (December to February) in the
Marsdiep tidal channel.
Wadden Sea (Beukema, 1976; Dankers and Beukema,
1983). The two species merely live in the upper 10–15 cm
of the sediment, where they roam through self-built
burrow systems. S. armiger is a detritus feeder, and
N. hombergii is a predator of small worms (including
S. armiger), crustaceans, and molluscs (Blegvad, 1914;
Curtis, 1977; Schubert and Reise, 1986; Bosselmann,
1991). From March to May, females of S. armiger
produce egg cocoons that are fixed to the sediment. The
larve develop within the cocoon, and upon leaving,
the post-larvae immediately burrow in the sediment
(Hartmann-Schröder, 1996). Breeding of N. hombergii
takes place between April and June, and the larvae are
planktonic.
Data were collected at Balgzand, a 50 km2 tidal-flat
area in the westernmost part of the Wadden Sea.
Starting in 1970, 15 randomly chosen permanent
plots were sampled annually in late winter (March).
At each plot 0.95 m2 was sampled. Cores were taken to
a depth of 30 cm and immediately sieved through a
1 mm mesh. More details of the sampling procedure
are given by Beukema (1974). Abundance figures
used are the annual mean biomass averaged over all 15
plots and expressed in g ash-free dry mass per m2
(Fig. 1a). Biomass data are preferred over numerical
1490
Jaap van der Meer et al.
abundance as fragmented worms can be included
unambiguously.
Mean winter temperature data used are the threemonth average surface water temperatures (December to
February) obtained from a daily sampling programme
in the Marsdiep, the nearest tidal channel (Fig. 1b). This
programme has been running since 1860 (Van der
Hoeven, 1982).
2
1
Rt =b0 +b1Nt 1 +b11N2
t +b2Nt1 +
1 2
2
b22N2
t1 +b12Nt Nt1 +t.
Turchin and Taylor (1992) considered only a restricted
number of values for i (1, 0.5, 0, 0.5, . . ., 3). For
the choice i =0, they used logN instead. Hence,
the Perry-Zhou model is a special case of the TurchinTaylor model. The first-order Royama model:
Rt =Rm exp(a0 a1Xt)+t
Models
In most stochastic population process models in discrete
time, Rt, the log reproductive rate at time t, is related to
x, a set of log population densities Xt, Xt1, Xt2, . . .,
and to z, a set of exogenous variables Zt, Zt1, Zt2,
. . ., by the function f:
Rt =Xt+1 Xt =log(Nt+1/Nt)=f(x, z, ),
where Nt is the population density at time t, and is a
set of constant parameters. Various functions f have
been proposed in the past (Berryman, 1992a). Since the
results (i.e., the relative importance of x vs. z, and
the stability properties of the model) may depend
on the model choice, we used a variety of commonly
applied models. The simplest stochastic models are the
linear models. The second-order linear model is written
as:
Rt =b0 +b1Xt +b2Xt1 +t,
where t is an independent, identically normally distributed number with zero mean. These linear models
exhibit only relatively simple dynamics and cannot show
chaotic behaviour. Furthermore, there are some conceptual problems, e.g., Rt tends to infinity when Nt tends to
zero. We used three types of non-linear models. The first
group contains the extensions of the Ricker model,
such as suggested by Royama (1992). The second-order
version can be written as:
can be re-written as a first-order Turchin-Taylor model
(first-degree polynomial):
Rt =b0 +b1Nt +t,
where Rm =b0, a0 = log(b1), a1 = . Thus, as long
as b1 <0, the models are equivalent. This is not true for
the second-order Royama model, which after re-writing
becomes:
2
Rt =b0 +b1Nt 1Nt1
+t.
The effects of known exogenous variables Zt, such as
mean winter temperature or the density of a prey or
predator species, were assumed to be additive:
Rt =f(x)+Zt +t.
This implies the assumption that any predator species
included exhibits a type-I functional response. In other
words, the predator term in the prey equation resembles
the Nicholson-Bailey model. Normalized (i.e., zero
mean and unit variance) data were used for all
exogenous variables.
A global stability analysis was performed by estimating the stochastic version of the Lyapunov exponent
(Ellner and Turchin, 1995; Desharnais et al., 1997). For
the first-order models, the exponent is given by:
Rt =Rm exp(a0 a1Xt a2Xt1)+t.
Second, polynomial models using log-transformed
population densities were used, as suggested by Perry
et al. (1993) and Zhou et al. (1997). The second-order
model, using a second-degree (quadratic) polynomial, is
written as:
Rt =b0 +b1Xt +b11X2t +b2Xt1 +
b22X2t1 +b12XtXt1 +t.
Finally, polynomial models allowing a Box–Cox transformation of population densities, N, as suggested by
Turchin and Taylor (1992), were used:
where F is the (deterministic) function that relates Xt+1
to Xt (Ft =Xt +Rt), and n the number of observations.
Exogenous variables in the model (e.g., winter temperature) were kept at the observed average. For the secondorder models the Jacobian-based method was used
(Ellner and Turchin, 1995).
Statistics
Parameters were estimed by least-squares regression.
The non-linear Turchin-Taylor model would become a
linear model if the non-linear parameters i were known.
Population dynamics of two marine polychaetes
1491
Table 1. Goodness-of-fit in terms of the residual sum of squares (RSS) and Mallow’s Cp of various
models (see text) relating the log reproductive rate Rt of Scoloplos armiger and Nephtys hombergii to
population densities Nt and Nt1. The length of the data series is 29 years, resulting in 27 tri-variate
observations Rt, Nt and Nt1.
S. armiger
Model
Constant
Linear
Royama
Perry-Zhou – 2nd degree
Turchin – 2nd degree
Linear
Royama
Perry-Zhou – 2nd degree
Turchin – 1st degree
Turchin – 2nd degree
N. hombergii
Order
df
RSS
Cp
RSS
Cp
—
1
1
1
1
2
2
2
2
2
26
25
24
24
23
24
23
21
22
19
6.00
4.99
4.99
4.99
4.99
4.74
4.73
4.29
4.44
4.19
5.34
2.24
4.24
4.24
6.24
3.00
4.93
6.69
5.47
10.21
60.74
45.24
45.22
45.22
44.70
44.72
44.66
42.34
44.49
37.13
8.56
2.00
3.99
3.99
5.70
3.71
5.68
8.39
7.59
9.52
Consequently, linear methods could be used for fitting.
For this practical reason, Turchin and Taylor (1992)
considered only a restricted set of values for i, but this
is unnecessarily simplistic (Perry et al., 1993). We used
the Golub-Pereyra algorithm for partially linear leastsquares models implemented in S-plus, which estimates
only the non-linear parameters by an iterative optimization procedure (Venables and Ripley, 1996). At each
iteration step, the linear parameters are estimated by
linear methods. We used the same procedure for fitting
the Royama models, after a re-expression as presented
above.
Mallow’s Cp statistic was used for model judgement
and selection (Wetherill, 1986):
where RSS is the residual sum of squares, ˆ 2 is an
estimate of the error variance, n is the number of
observations, and p the number of fitted parameters.
For least-squares methods, Mallow’s Cp is equivalent to
Akaike’s information criterion.
Bootstrap pseudo-replicated time-series were generated by simulation (Falck et al., 1995). The model
parameters estimated on the basis of the real data were
used in the simulations. For each simulated observation,
the error was obtained by random sampling with
replacement from the observed set of residuals. The
initial value(s) of each simulated series were the same as
for the real data, and its length was equal to the length
of the real series. For each model, we created 500
bootstrap series, and parameters (and, for the non-linear
models, the Lyapunov exponent) for each series were
estimated in the same way as for the real data. The 95%
confidence interval for the true value of each parameter
was obtained by Efron’s first percentile method; i.e.,
the interval is given by the values that exceed 2.5%
and 97.5%, respectively, of the generated bootstrap
distribution (Manly, 1997).
Results
A first comparison of the first-order and second-order
models revealed that the simplest model, that is the
first-order linear model, had the lowest Cp statistic
(Table 1). The sample auto-correlation functions and
partial auto-correlation functions showed no indication
for higher-order terms either (Fig. 2).
Inclusion of the exogenous variable winter temperature resulted in a lower Cp statistic for N. hombergii, but
not for S. armiger (Table 2, Fig. 3). Taking into account
prey density did not improve the fit for the predator N.
hombergii. Yet, the log-reproduction rate of the prey
species, S. armiger, was related to the density of the
predator (Table 2). Second-order models with winter
temperature or prey/predator density included (not presented in Table 2) did not show any substantial improvement in goodness-of-fit over the first-order models. For
example, the second-order Royama model for N. hombergii, including winter temperature revealed an RSS of
10.43, being only slightly lower than in the first-order
model (RSS=11.12).
For N. hombergii, the first-order linear model, including the effect of winter temperature, revealed an estimated parameter value b
|1 = 0.61 (95% confidence
interval: 0.77 to 0.50). This value falls in the range
minus one to zero, indicating a stable equilibrium
(Royama, 1992). A similar conclusion could be based on
the estimated first-order Royama model. The local stability behaviour is indicated by the slope of the reproduction curve at the equilibrium point, which is given by
minus the product of b
|0 and |. The estimated value of
this slope equalled 0.52 (95% confidence interval:
0.74 to 0.41), indicating that the equilibrium is
1492
Jaap van der Meer et al.
1.0
(a)
95 89
9083
75 92
9387
73 71
8474
81
72
77
1
Predicted
0.5
r
98
2
0.0
–0.5
0
–1
–3
0
4
2
6
8
97
91
82
–2
–1.0
86 9476
78
85
88
80
79
96
–2
–4
10
0
2
Observed
1.0
Figure 3. Observed log reproduction rate Rt of Nephtys hombergii versus the one-step-ahead predictions (rs =0.90). Predictions of the first-order model, Rt =b0 +b1Nt +b2Zt +t, where Zt
is winter temperature.
(b)
r
0.5
0.0
The results for S. armiger were similar. The first-order
linear model (including predator density P) revealed the
estimate b
|1 = 0.59 (95% confidence interval: 0.89 to
0.42). The first-order Royama model yielded a slope
at the equilibrium point of 0.60 (95% confidence
interval: 1.05 to 0.41). The estimated Lyapunov
exponent was negative, i.e. |n = 0.90 (95%
confidence interval: 2.38 to 0.50).
We estimated the local and global stability in the same
way for all other first-order and second-order models
(although for these models the decrease in the residual
sum of squares did not keep pace with the increase in the
number of model parameters), and the conclusion was
always the same: the equilibrium is globally stable.
Apparently, the fluctuations of the two polychaete
populations cannot be categorized as chaotic.
–0.5
–1.0
2
4
6
Lag
10
8
Figure 2. Sample (a) auto-correlation functions and (b) partial
auto-correlation functions of the time series of log biomass
density of Scoloplos armiger (continuous lines) and Nephtys
hombergii (broken lines) at Balgzand. Lag in years.
locally stable (Royama, 1992). The estimated Lyapunov
exponent was negative, i.e. |n = 0.76 (95% confidence
interval: 1.44 to 0.53), pointing also to global
stability (Ellner and Turchin, 1995).
Table 2. Goodness-of-fit in terms of the residual sum of squares (RSS) and Mallow’s Cp of various
first-order models (see text) relating the log reproductive rate Rt of Scoloplos armiger and Nephtys
hombergii to population densities Nt, winter temperatures Zt, and the density of the ‘‘other’’ species Pt.
The length of the data series is 29 years, resulting in 28 multi-variance observations Rt, Nt, Zt and Pt.
S. armiger
Model
Constant
Linear
Royama
Perry-Zhou – 2nd degree
Turchin – 2nd degree
Linear
Royama
Perry-Zhou – 2nd degree
Turchin – 2nd degree
Linear
Royama
Perry-Zhou – 2nd degree
Turchin – 2nd degree
N. hombergii
Covariate
df
RSS
Cp
RSS
Cp
—
—
—
—
—
Z
Z
Z
Z
P
P
P
P
27
26
25
25
24
25
24
24
23
25
24
24
23
6.03
4.99
4.99
4.99
4.99
4.99
4.99
4.99
4.99
3.35
3.35
3.35
3.35
19.02
13.23
15.23
15.23
17.23
15.23
17.22
17.23
19.22
3.00
4.98
4.98
6.98
61.20
45.65
45.64
45.64
45.20
11.81
11.12
11.22
11.11
43.54
43.20
43.02
42.77
106.10
74.52
76.52
76.52
77.56
3.49
4.00
4.21
5.98
71.99
73.24
72.85
74.32
Population dynamics of two marine polychaetes
Discussion
One problem with using simple non-linear time-series
models to describe and characterize the dynamics of
populations is that different models may fit equally well,
but result in qualitatively different characterizations
of the dynamics of the population (Morris, 1990).
Berryman (1992a) replied that model choice should not
only be based on a goodness-of-fit criterion (or on
subjective personal preference), but also on the basis of
ecological credibility. A similar view was expressed by
Royama (1992), who criticized the generalization of the
Ricker model by Turchin and Taylor (1992) to cases
with more than one time lag. Royama did not consider
this a good model because it is hard to find a possible
ecological mechanism for its form. A lack of ecological
credibility also holds for the so-called non-linear selfexciting threshold autoregressive (SETAR) models,
which have been advanced recently (Grenfell et al., 1998;
Stenseth and Chan, 1998). Future population size is
modelled as a discontinuous piecewise linear function of
the previous population size. This approach allows for a
sudden drop from a strongly positive population growth
rate to a negative rate when the population size increases
at only an infinitely small step. Such response is ecologically rather unlikely, but it fully characterizes the
dynamical properties of the (deterministic skeleton of
the) model. Berryman (1992a) goes one step further, and
takes the radical view that conclusions drawn from
incorrect models must themselves be incorrect. Although
we sympathize with the plea for ecological credibility,
this must be an overstatement. It is widely accepted that
all models in ecology are in a way incorrect (Levin,
1981), but the chief question is the extent to which
conclusions are robust given imperfect models. However, we were not really confronted with this problem,
since none of the fitted models indicated complex
dynamics, and therefore we may safely conclude that the
occurrence of chaos is rather unlikely in this system.
Another point of concern with the use of non-linear
time series to characterize the type of dynamics is the
risk of over fitting. The response–surface–methodology
models as applied by Turchin (1993), for example, had
as many as 13 parameters, which is high if there are only
as few as 16 to 28 data points. Therefore, Falck et al.
(1995) suggest that a better approach for short time
series would be to test the hypothesis of linearity first. If
that hypothesis cannot be rejected, it does not make
sense to search for chaos. We only found marginal
differences in goodness-of-fit (in terms of RSS) between
linear and non-linear models. The product of the two
parameters of the non-linear Royama model that indicates local stability could be estimated with reasonable
accuracy, but the parameter estimates separately had
much wider confidence intervals. For both species the
95% confidence interval of | (0.58 to +0.19 for N.
1493
hombergii; 1.67 to +2.06 for S. armiger) contained
zero, implying that the non-linear component of the
model could not reliably be estimated and that the
hypothesis of linearity could not be rejected.
Generally speaking, predation may lead to a delayed
density-dependent response of prey. The feedback mechanism works through high abundance of prey resulting
in large numbers of predators, followed by high levels of
prey mortality. We found a negative effect of predator
abundance on the log reproductive rate of the prey, but
no effect of prey abundance on the predator. Delayed
density dependence could not be detected either. Apparently, the feedback mechanism was interrupted. This
may be because N. hombergii is a generalist predator
(Blegvad, 1914; Curtis, 1977; Schubert and Reise, 1986;
Bosselmann, 1991), implying that the density of merely
one of its prey species could be of minor importance for
its total food supply and population growth rate. Note
that prey and predator biomass densities were of the
same order of magnitude, and energetic considerations
make a tight predator–prey coupling also rather
unlikely. Thus, the density of N. hombergii can be
categorized as a density-independent factor, with respect
to the dynamics of S. armiger, one of its prey species.
For both species, evidence was obtained for densitydependent regulation, but the impact of winter temperature, a density-independent factor, on the population
growth rate of N. hombergii was remarkably strong. One
might state that winter temperature also had a considerable impact, albeit indirect and delayed, on the population growth rate of S. armiger. Including winter
temperature of the year before had indeed almost the
same effect on the population growth rate of S. armiger
as the density of N. hombergii.
In fitting simple time-series models for a two-species
situation, we were able to categorize each effect included
as either density-dependent or not. For example, prey
density did not affect predator dynamics and, following
the definition by Royama (1992), predator density was
therefore classified as a density-independent factor with
respect to prey dynamics. But suppose that the predator’s rate of change had been influenced both by
weather, and prey density. According to Royama’s
definition, predator density would then have to be
regarded as a density-dependent factor with respect to
prey dynamics, despite the fact that predator dynamics
were to a large extent determined by weather. This
would imply that the (indirect) effects of weather on prey
dynamics should be classified as density dependent,
which does not seem to make sense. Thus, although it
was possible to construct a simple model that indicates
the way various factors generate the dynamics of the
two polychaete populations, in multidimensional
system it is not easy to quantify in a single measure
the relative importance of density-dependent versus
density-independent factors.
1494
Jaap van der Meer et al.
Acknowledgements
W. de Bruin and J. Zuidewind assisted with the field
work. We thank H. van Aken for providing recent
weather data. Since 1994, part of the benthos sampling
programme has been financed by Rijkswaterstaat/
RIKZ.
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