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[9– 13]—long enough to span substantial environmental and ecological change and complicating the
interpretation of past conditions. Because death
assemblages form over a much longer time than can
be investigated through direct experimentation, modelling is an essential tool for understanding their genesis
and properties [14 – 16].
The primary data of community ecology consist of
the identities and abundances of species in local assemblages, so the fundamental question regarding the
fidelity of death assemblages concerns the factors determining the dead abundance of individual species. The
abundance of a single species in a death assemblage
can be conceptualized as a balance between influx and
loss [7,17,18]. The influx (F(t)) of skeletal material to
an accumulating death assemblage is governed by the
standing crop and mortality rate of the living population
as well as physical processes influencing the rate and episodicity of sedimentation and reworking (figure 1a;
physical processes can influence loss as well as influx).
Once dead skeletal elements have been added to an
accumulation, they stand some chance per unit time of
being destroyed or removed (G(t)) [6], resulting in a
decaying post-mortem age distribution (figure 1b)
[17,18]. Mathematically, the abundance of a species in
a death assemblage is the Ðintegral of the product of the
influx and loss functions: F(t)G(t)dt (figure 1c). Consecutive application of this integral to an influx time
series (a mathematical convolution or moving average
process) provides a means of modelling changes in the
dead abundance of a single species (figure 1d ).
Biol. Lett. (2012) 8, 131–134
doi:10.1098/rsbl.2011.0337
Published online 8 June 2011
Palaeontology
Remembrance of things
past: modelling the
relationship between
species’ abundances
in living communities
and death assemblages
Thomas D. Olszewski*
Department of Geology and Geophysics and Interdisciplinary Research
Program in Ecology and Evolutionary Biology, Texas A&M University,
College Station, TX 77843, USA
*[email protected]
Accumulations of dead skeletal material are a
valuable archive of past ecological conditions.
However, such assemblages are not equivalent
to living communities because they mix the
remains of multiple generations and are altered
by post-mortem processes. The abundance of a
species in a death assemblage can be quantitatively modelled by successively integrating
the product of an influx time series and a postmortem loss function (a decay function with a
constant half-life). In such a model, temporal
mixing increases expected absolute dead abundance relative to average influx as a linear
function of half-life and increases variation in
absolute dead abundance values as a square-root
function of half-life. Because typical abundance
distributions of ecological communities are
logarithmically distributed, species’ differences
in preservational half-life would have to be very
large to substantially alter species’ abundance
ranks (i.e. make rare species common or viceversa). In addition, expected dead abundances
increase at a faster rate than their range of variation with increased time averaging, predicting
greater consistency in the relative abundance
structure of death assemblages than their parent
living community.
2. METHODS
In the model, post-mortem loss is quantified as an exponential function: G(t) ¼ exp(2t ln 2/T1/2)dt, where t ¼ time and T1/2 ¼ half-life
(the duration in which 50% of initially present material is expected to
be lost). The exponential function assumes that every individual has
an equal probability of loss at every moment in time [17 –20].
Although this is a reasonable first-order approximation in the
absence of other information, it should be noted that neither the
environmental conditions controlling loss nor the preservational
properties of skeletal material need remain constant through time.
Influx is modelled here as a stationary time series derived as the
sum of a constant average value (m) plus a random term describing
deviation from the average (ks(t); the coefficient k scales the magnitude of variation): F(t) ¼ m þ ks(t). In the results presented here,
s(t) is a series of random values with no autocorrelation drawn
from a lognormal distribution and z-transformed to have a mean of
zero and a standard deviation of one; in addition, k, s(t), and m
are independent of one another. The lognormal distribution
resembles a boom– bust dynamic: most deviations from the average
are small but they are punctuated by occasional high magnitude
events, a pattern expected for opportunistic populations in harsh
or fluctuating environments [21]. A variety of other symmetrical
and skewed distributions were also explored (normal, uniform, logistic, exponential, exponentially transformed logistic and Cauchy),
and the outcomes were consistent with the results of the lognormal
series. In all simulations, influx parameters m and k were set so
that all values in the influx series were greater than zero.
Keywords: time averaging; death assemblage; fossil
assemblage; convolution; moving average process
1. INTRODUCTION
A major objective of community ecology is to understand the processes governing the composition and
structure of multi-species assemblages. The dead
remains of organisms can provide a record of past ecological conditions, and therefore have great potential to
shed light on community dynamics and to provide vital
information on ecological baselines [1 – 4]. However,
death assemblages differ from living communities
because they are typically time-averaged—i.e. they
mix the remains of multiple generations [5 – 8]. In shallow marine settings, shells of dead molluscs and
brachiopods can be tens to thousands of years old
3. RESULTS
Integrating the product of F(t) and G(t) results in an
expression for the abundance of a species in the
death assemblage at time s:
ð1
t ln 2
dt
½m þ ksðs tÞ exp T1=2
0
ð1
mT1=2
t ln 2
þk
sðs tÞ exp ¼
dt:
ln 2
T1=2
0
One contribution of 12 to a Special Feature on ‘Models in palaeontology’.
Received 24 March 2011
Accepted 16 May 2011
131
This journal is q 2011 The Royal Society
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(a)
8
influx abundance
132 T. D. Olszewski
6
Modelling time averaging
4
2
0
50
100
150
time (before present)
200
proportion surviving
(b) 1.0
0.8
0.6
0.4
0.2
0
50
100
post-mortem age
150
200
second term on the right-hand side,
Ð The
1
k 0 sðs tÞ expðt ln 2=T1=2 Þdt; describes the effect
of influx variation. The magnitude, k, has a linear
effect on variation of dead abundance—i.e. doubling
influx variation doubles dead abundance variation. As
a moving average process, the standard
deviation
of
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
dead abundances is expected to be T1=2 =2 ln 2 times
the standard deviation of influx [22] (figure 2b,c). This
relationship indicates that although dead abundance
variation increases with time averaging, it does so at a
slower rate than the linear increase in the expected
value of dead abundance. In contrast to standard deviation, both skewness and kurtosis of dead abundances
decrease as half-life increases (figure 2c)—i.e. temporal
mixing leads to dead abundances that are more symmetrical and less prone to extreme values than the
influx series [23]. Lastly, temporal mixing induces a
degree of autocorrelation in dead abundances equal to
exp(2s ln 2/T1/2) at a lag of s [22] (figure 2d). The
degree of autocorrelation is a measure of a dead abundance series’ memory of its own past: autocorrelation is
reduced to approximately 0.5 after one half-life and is
negligible after five half-lives.
dead abundance
(c) 2.0
1.5
1.0
0.5
0
50
100
post-mortem age
150
200
100
150
time (before present)
200
(d) 30
abundance
25
20
15
10
5
0
50
Figure 1. Dead abundance as a convolution of influx and
post-mortem loss. (a) Influx function, F(t; m ¼ 1, k ¼ 1).
(b) Post-mortem loss function, G(t; T1/2 ¼ 16). (c) Product
F(t) . G(t) at time 0—i.e. the distribution of post-mortem
shell ages in the present. (d) Time series of dead abundance
obtained by integrating the product F(t) . G(t) at each timestep.
The first term on the right-hand side, mT1/2/ln 2,
indicates that when influx is constant (s(t) ¼ 0),
dead abundance is the product of average influx
and post-mortem half-life. Differences in half-lives
among species [20] have a linear effect on their dead
abundances—i.e. doubling the half-life doubles the
expected dead abundance. This implies that in
the case of an assemblage of species with the same rate
of post-mortem loss, average dead relative abundances
will be the same as average living relative abundances.
Biol. Lett. (2012)
4. DISCUSSION
The model presented here indicates that rather than converging on an expected value [5–7,24,25], variation of
absolute dead abundance values increases with increasing time averaging. The reason is that ‘taphonomic
inertia’ [26] makes dead abundances hard to push
away from their value at any moment. Once they have
deviated from their expected value owing to a few
above-average or below-average influx events, they have
a tendency to stay deviated, and the magnitude and duration of this effect increases with time averaging.
Despite the increase in variance with time averaging,
meta-analysis of 85 live– dead comparisons of shallow,
marine mollusc communities [27] indicated that an
average of 20 per cent of the variation in dead abundance
ranks could be explained by living abundance ranks
(mean Spearman r 2 ¼ 0.203). This degree of rank fidelity appears to be owing to the fact that ecological
abundances are typically distributed on a logarithmic
scale [16,28], whereas the effects of temporal mixing
on expected dead abundances are linear—i.e. substantial shifts in abundance rank from live to dead would
require species to have differences in influx or loss
much larger than their rank differences. In addition,
the model predicts that the range of dead abundance
variation relative to expected absolute abundance
should decrease with increased time averaging as a
function of (T1/2)21/2, consistent with recent multispecies models which have found that time-averaged
death assemblages are less variable than sympatric live
assemblage [15,16,25].
5. CONCLUSION
The model of time averaging presented here clearly
glosses over the full complexities of population dynamics,
post-mortem alteration and sedimentary burial. Nevertheless, by stripping a wide array of interacting
processes down to simple fundamentals, it refines our
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Modelling time averaging
emphasizing that death assemblages are no more static
than living communities, it highlights the need to acquire
dead abundance series for comparison with corresponding living series over durations relevant to time averaging
[29]. Even as data improve, modelling will provide a
powerful tool both for understanding deep-time fossil
assemblages as well as for using death assemblages as
baselines of ecological change in the present day.
(a) 120
abundance
100
80
60
40
20
0
1000
deviation from mean influx
(b)
2000
3000
4000
time (before present)
5000
This work was supported by NSF grant EAR-0617355.
Insightful and constructive reviews were provided by
Michał Kowalewski and Adam Tomašových.
20
10
0
–10
–20
0.5
1
2
4
16
8
half-life
32
64
128
0.5
1
2
4
16
8
half-life
32
64
128
(c) 100
10
1
(d) 1.0
correlation
T. D. Olszewski 133
0.8
0.6
0.4
0.5
0.2
1 2
4
8
16
64
32
128
0
1
2
4
8
16 32 64 128 256 512 1024
lag
Figure 2. Effect of increasing half-life on the distribution
of dead abundances. (a) Dead abundance time series for
T1/2 ¼ 1, 4, 16, and 64 timesteps from bottom to top
(respective mean dead abundance values are 1.4, 5.8, 23.1
and 92.3; based on an influx with m ¼ 1 and k ¼ 1).
(b) Solid black line is median deviation from expected average dead abundance; dashed lines denote 95% quantile
interval. (c) Effect of time averaging on the properties of
the statistical distribution of dead abundances. Influx distribution had standard deviation: (squares) ¼ 1.00, skewness
(circles) ¼ 8.56 and kurtosis (triangles) ¼ 241.86. (d) Autocorrelation of dead abundance time series based on different
post-mortem half-lives (indicated for each curve). Autocorrelation of influx series and cross-correlation of dead and influx
series were not significantly different from zero for all halflives and lags (dashed line).
understanding of the effects of time averaging on variation in death assemblages and connects processes of
influx, burial and loss to measurable patterns like
post-mortem age frequency distributions [20]. By
Biol. Lett. (2012)
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