|
| |
|
|
|
|
||||
| Home Help Feedback Subscriptions Archive Search Table of Contents | ||||
First published online April 26, 2005
Journal of Experimental Biology 208, 1749-1769 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01588
Review article: Lifespan, reproduction and ecology |
Scaling and power-laws in ecological systems
1 Center for Advanced Studies in Ecology and Biodiversity (CASEB) and
Departamento de Ecología, Facultad de Ciencias Biológicas,
Pontificia Universidad Católica de Chile, casilla 114-D, Santiago,
Chile
2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501,
USA
3 Centro de Investigación Oceanográfica en el Pacífico
Sur-Oriental (COPAS) and departamento de Ocenografía, Universidad de
Concepción, Casilla 160-C, Concepción, Chile
* Author for correspondence (e-mail: pmarquet{at}bio.puc.cl)
Accepted 14 March 2005
Summary
Scaling relationships (where body size features as the independent variable) and power-law distributions are commonly reported in ecological systems. In this review we analyze scaling relationships related to energy acquisition and transformation and power-laws related to fluctuations in numbers. Our aim is to show how individual level attributes can help to explain and predict patterns at the level of populations that can propagate at upper levels of organization. We review similar relationships also appearing in the analysis of aquatic ecosystems (i.e. the biomass spectra) in the context of ecological invariant relationships (i.e. independent of size) such as the `energetic equivalence rule' and the `linear biomass hypothesis'. We also discuss some power-law distributions emerging in the analysis of numbers and fluctuations in ecological attributes as they point to regularities that are yet to be integrated with traditional scaling relationships and which we foresee as an exciting area of future research.
Key words: scaling, power-law, metabolism, complexity
Introduction
Living entities are embedded in and constituted by, networks at any level
of organization, from cells to ecosystems (e.g.
Ulanowicz, 1986
;
Pahl-Wostl, 1995;
Strogatz, 2001;
Barabási and Oltvai,
2004). The structure and dynamics of these networks emerge as a
result of the processes whereby energy, materials and information are
acquired, stored, distributed and transformed. Biological networks typically
consist of a large number of non-identical elements whose interaction are
usually localized, although their effects are not and whose emergence,
maintenance and dynamics represent a challenge to understanding let alone
prediction (e.g. Weng et al.,
1999; Levin,
1998,
1999,
2002). Biological networks
represent the most complex physical system in the universe and yet, as most
complex systems they can be described by simple relationships
(West, 1999;
Brown et al., 2000). These
relationships are of the form
 | (1) |
where Y is some response or dependent variable, x
represents an independent or explanatory variable, ß is a normalization
constant and 
is the scaling exponent. Depending on the value of the
exponent these relationships are called allometric (a11) or
isometric (=1). The functional form of the relationship in Equation 1
is also called a power-law relationship, where some quantity can be expressed
as some power of another. Power-laws are ubiquitous in physical and social
systems where they most commonly arise as probability or frequency
distributions, of the form
f(x)=ßx, different from the
usual exponential or Gaussian distributions. For example power-law
distributions describe phenomena such as the frequency of earthquakes of
different magnitudes (the Gutenberg-Richter law), the distribution of income
among individuals (Pareto's law) and the rank-frequency distribution of words
in natural languages and city sizes (Zipf's law). Power-laws are well-known to
biologists in the form of bivariate relationships of power-law type, called
scaling relationships (e.g. Peters,
1983; Niklas,
1994; Wiesenfeld,
2001; Brown and West,
2000; Brown et al.,
2002; Chave and Levin,
2003) by which molecular, physiological, ecological and life
history attributes relate to some attribute of organisms raised to a power as
in Equation 1. Although the history of the term scaling in biology probably
has deep roots in time, its use has been associated with relationship where
the independent variable is the size of an organism (Calder,
1983,
1984;
Peters, 1983;
Schmidt-Nielsen, 1984). For
the sake of consistency we will retain the use of scaling as related to
relationships involving body size and will differentiate them from power-law
distributions as defined above.
This special issue is devoted to explore the consequences of organismal
size as affecting biological processes. Most of the papers in this special
issue have addressed scaling relationships where the response variable is an
individual level attribute, such as metabolic rate, life span and running
speed, and where the independent variable is body size. However, as pointed
out above, scaling relationships are common at higher levels of organizations
as well, such as at the level of populations, communities and ecosystems, and
they are usually an allometric function of body size
(Peters, 1983;
Calder, 1984;
Schmidt-Nielsen, 1984;
Bonner, 1988;
Brown, 1995). This fact
underscores the importance of body size at all levels of organization, and
opens the way for synthesis and integration across levels. In fact, it has
been a pressing challenge for ecologists and evolutionary biologists to
develop a conceptual and quantitative framework bringing together disciplines
traditionally viewed as distinct, such as physiology, ecology, biogeography
and macroevolution (e.g. Brown and Maurer,
1987,
1989;
Ricklefs, 1987; Brown,
1995,
1999;
Marquet and Taper, 1998) and
much of this quest for a synthetic framework has been based on empirical
statistical patterns relating body size with physiological, ecological and
evolutionary traits (e.g. Lawton,
1990; Blackburn et al.,
1993a; Brown et al.,
1993; Brown,
1995). In what follows we review some of this relationships as
they emerge at the population, community and ecosystem levels, and emphasize
their connections as well as future developments.
Our main focus in this review paper will be scaling relationships were body size features as the independent variable, however we will restrict ourselves to scaling relationships related to energy acquisition and transformation primarily at the level of populations. Our aim will be to show how individual level attributes can help to explain and predict patterns at the level of populations, communities and ecosystems. In addition, we will also discuss power-law distributions emerging in the analysis of numbers and fluctuations in ecological attributes as they point to regularities that are yet to be integrated with traditional scaling relationships, and that we foresee as an exciting area of future research. However, before delving into the main theme of our paper we will make a brief detour to introduce some general concepts associated with scaling and power-law relationships that will be used throughout the paper.
Why bother with scaling and power-law relationships?
There are two notions or characteristics associated with power-law
relationships that stand out because of their theoretical and empirical
importance. (1) Power-laws, as well as scaling relationships as used here, are
scale-invariant (e.g. Sornette,
2000; Stanley et al.,
2000; Gisiger,
2001), that is, they display invariance under scale change. This
can be seen if we consider a scale transformation in x such that
x 
x then
f(x)=ßx
ß
x=
f(x), thus a change in the scale of the independent variable
preserves the functional form of the original relationship. Scale invariance
describes phenomena that are not associated with a particular or
characteristic scale and are also known as scale-free or true on all scales,
that is they posses the same statistical properties at any scale. In practical
terms, this means that the same principles or processes are at work no matter
what the scale of analysis (Milne,
1998). This property makes scaling and power-law relationships
very well suited for the study of ecological systems, which show variability
at different temporal, spatial and organizational scales such that there is no
single `correct scale' for their analysis
(Levin, 1992). (2) The notion
of universality. This concept was introduced into physics in association with
critical phenomena (e.g. Biney et al., 1992) to describe the state and
dynamics of systems as they approach a phase transition (such as water turning
into ice or the onset of magnetization when temperature is changed or the
transition between dynamical regimes through bifucartions in deterministic
dynamical systems). Near phase transitions, systems are said to become
critical and relevant quantities to describe their state (e.g. magnitude of
fluctuations, correlation length) behave as power-law relationships with
critical exponents (e.g. Maris and
Kadanoff, 1978; Solé
et al., 1996; Milne,
1998; Stanley et al.,
2000; Gisiger,
2001). Interestingly, it has been shown that systems that are
completely different away from a critical point, show similar critical
exponents near a phase transition (e.g. Biney et al., 1992). These
non-arbitrary exponents are said to be universal and define disjoint classes
(universality classes) into which different physical systems can be
classified. A system can arrive to a critical state through changes in a
variable external to it (e.g. temperature), but also as a result of its own
internal dynamics, in which case we speak of self-organized criticality a
concept introduced by Bak et al.
(1987,
1988). During the past decade
or so, several empirical and theoretical investigations have suggested that
biological systems in general, and ecological systems in particular, seem to
operate near a critical state, which results in the ubiquity of power-law
behavior in several descriptors of their dynamics (e.g.
Miramontes 1995;
Bak, 1996;
Keitt and Marquet, 1996;
Rhodes et al., 1997;
Ferrier and Cazelles, 1999;
Solé et al., 1999,
2002;
Gisiger, 2001;
Roy et al., 2003;
Pascual and Guichard, 2005)
and might even belong to the same universality class as other complex systems
such as economic systems (Stanley et al.,
2000). Thus the analysis of power-law and scaling relationships
can help us to identify general principles that apply across a wide range of
scales and levels of organizations, revealing the existence of universal
principles within the seemingly idiosyncratic nature of ecological systems.
However, it should be borne in mind that power-laws might emerge as a
consequence of several processes not necessarily related to critical points
and phase transitions (Brock,
1999; Sornette,
2000; Mitzenmacher,
2001; Allen et al.,
2001) such that the claim that ecological systems are maintained
near a critical state is still an open question.
Scaling in ecology
Life on earth has diversified in form and function to occupy virtually all
kinds of environments forming spatially and temporally diffuse associations of
organisms, or ecological systems, wherein energy acquisition, allocation and
transformation is carried out through complex webs of interacting species. To
understand the structure and dynamics of these complex ecological systems, two
major approaches have predominated among ecologists. On the one hand,
experimental microscopic approaches have emphasized the highly variable and
idiosyncratic nature of communities with regard to the relative importance of
specific biotic interactions (e.g. competition, predation, mutualism) and
their effect on local coexisting populations
(Diamond and Case, 1986;
May, 1986;
Lawton, 1999). Two
representative quotations from major figures in the field of ecology can help
as to clarify this point further. May
(1986, p. 1116) in his
MacArthur Award address wrote: `Ecology is a science of contingent
generalizations, where future trends depend (much more than in the physical
sciences) on past history and on the environmental and biological
setting.' A view that is also sponsored by Diamond and Case
(1986) in an edited volume
that is appreciated as representing the last synthesis in community ecology,
in their own words (Diamond and Case,
1986; p.x): `The answers to general ecological questions are
rarely universal laws, like those of physics. Instead, the answers are
conditional statements such as: for a community of species with properties
A1 and A2 in habitat B and latitude C, limiting factors
X2 and X5 are likely to predominate.'
On the other hand, macroscopic non-experimental approaches have emphasized
the existence of statistical patterns in the structure of communities that
seemingly reflect the operation of general principles or natural laws and
emerge as scaling relationships with similar or related exponents
(West et al., 1997). These
regularities underlie two recent research programs in ecology, the first is
macroecology (Brown and Maurer,
1989; Brown, 1995;
Gaston and Blackburn, 2000;
Marquet, 2002a;
Storch and Gaston, 2004) and
the second is the `metabolic theory of ecology'
(Brown et al., 2004). The
change in the conceptualization of ecological systems entailed by this latter
approach, as opposed to the idiosyncratic view expressed by Diamond and Case
(1986) and May
(1986), is apparent in the
following excerpt (Brown et al.,
2003; p. 411): '...Our own recent research is based on the
premise that the general statistical patterns of macroecology... are emergent
phenomena of complex ecological systems that do indeed reflect the operation
of universal law-like mechanisms.'
Much of the connections between individual, population, community and
ecosystem level scaling relationships has been exceptionally synthesized in
Brown et al. (2004) in the
context of the metabolic theory of ecology, which attempts to explain material
and energetic fluxes, in ecological systems, from first principles of
thermodynamics, chemical reaction kinetics and fractal-like biological
structures and which is expressed in a `master equation' relating metabolism
to body size and temperature (Gillooly et
al., 2001; Brown et al.,
2004). In the following we will revisit some of the relationships
that are at the core of Brown et al.
(2004), but with an emphasis
in the connections between individual and population level scaling
relationships and predominantly on the scaling of population number and
fluctuations, to show how these can help us to explain and predict
relationships emerging at other levels of organization and at different scales
in time and space.
Individual and population level scaling
The most basic property of a population is the number of individuals it
contains. Furthermore, since both the turnover as well as the maintenance of
each individual requires resources available in the environment, everything
else being equal, the maximum number of individuals that a species can achieve
in a given area (or maximum density N) will be proportional to the
ratio between rate of resource supply per unit area of the environment
(
) and the average per individual rate
of resource use (
). This can be
written as:
 | (2) |
Since environmental resources are used by individuals to sustain their
metabolism (or the complex set of chemical reactions that allow the organisms
to sustain its living) the rate of resources used by an individual can be
assimilated to its metabolic rate, which is well known to scale with body size
(M) as:
 | (3) |
where the scaling exponent b has been shown to be 3/4 both on
empirical and theoretical grounds (West et al.,
1997,
1999;
Savage et al., 2004a)
although the issue is still contentious
(Dodds et al., 2001). Assuming
b=3/4 leads to:
 | (4) |
where C1 contains both the effects of variability in
resource supply rates as well as other sources of variability
(C0) affecting body size and density (i.e.
C1=C0/).
Compilation studies based on the analysis of published data for closely
related species worldwide (e.g. Damuth,
1981,
1987,
1991) typically report that
the slope of the relationship between density and body mass approximate -3/4.
Although this relationship seems to be stronger in mammals than in other taxa,
such as birds (Bini et al.,
2001; Dobson et al.,
2003), and might be affected by the scale of analysis, level of
data aggregation, type of environment, latitude, taxa, trophic position,
census area and method of statistical analysis (see reviews in Cotgrave, 1993;
Cyr, 2000;
Gaston and Blackburn, 2000;
Silva et al., 2001), recent
analysis of this relationship underscores the empirical generality of the -3/4
scaling exponent as well as its strong theoretical support
(Li, 2002;
Belgrano et al., 2002,
Brown et al., 2004).
Furthermore, as shown by Marquet et al.
(1990), this relationship
holds in local communities when a wide spectrum of taxonomic groups are
included (see also Cyr et al.,
1997; Schmid et al.,
2000; Cohen et al.,
2003; Mulder et al.,
2005; but see Dugan et al.,
1995; Navarrete and Menge,
1997), although the exponent is closer to -1 (which is expected
when analyzing species in more than one trophic level, see discussion below)
and is maintained in the face of perturbations affecting changes in the
abundance and identity of species (Fig.
1, see also de Boer and Prins,
2002; Cohen et al.,
2003). The existence of temporal invariance in this relationship
further testifies to its importance in understanding ecological dynamics
(Marquet, 2000).
|
Although most studies do not usually try to disentangle the effect of both
and C0 (but see
discussion below), Equation 4 is widely accepted as an accurate description of
the relationship between maximum density and body size, although most of the
time it is not explicitly realized that energetic limitation through average
per individual rate of resource use (ß) should be stronger in
the boundary of maximum density at carrying capacity (i.e. it is a boundary
condition) and when resource supply is constant and bounded within similar
levels among species (Enquist et al.,
1998; Brown et al.,
2004; Savage et al.,
2004b). As discussed below, when this is not the case deviations
are expected. A case in point is the scaling of secondary consumers. Since
energy available to secondary consumers (i.e. those feeding on other animals)
is less than that available to primary consumers
(Lindeman, 1942), it is
expected that they will reach lower densities than similar sized herbivores
(Marquet, 2002b;
Ernest et al., 2003;
Brown et al., 2004) as was
first described by Mohr
(1940). However, what has
puzzled ecologists for a long time is that its allometric exponent is
considerably smaller (i.e. steeper slopes in the range -1.0 to -0.8, see
Fig. 2) than -3/4. Explanations
for this discrepancy have been elusive and usually based on presumed
systematic (allometric) variation in prey biomass and productivity with
predator body mass (Peters and Raelson,
1984). However, Carbone and Gittleman
(2002) solved this problem by
showing that the relationship between population density and size in mammalian
carnivores is constrained by metabolic rate and also by variability in their
resource base (prey species) such that the -3/4 power law only emerges if the
local productivity of prey species, experienced by a carnivore population, is
taken into account. Thus, the answer to the anomalous scaling of mammalian
secondary consumers is found in local resource availability.
|
The fact that resources () are
distributed in space allow us to calculate how much space or area would an
individual require or its home range (H). This can be calculated as
the inverse of Equation 2, assuming that individuals use just the sufficient
area to sustain their energy demands (ß)
(McNab, 1963) and that
resources are homogeneously distributed in space. Furthermore, if we assume
that Equations 3 and 4 hold then it is expected that
 | (5) |
However, there are two important considerations to make regarding this
relationship: (1) empirical analysis of home range scaling in mammals shows
that the exponent is larger than 3/4 and (2) it is non-monotonic, showing a
change in slope at a threshold body size MT (around 100 g
in mammals) (Marquet and Taper,
1998; Kelt and Van Vuren,
1999,
2001). The first anomaly was
recently explained by Jetz et al.
(2004) by noticing that the
realized home range of an individual might be smaller than that assumed by
Equation 5 due to intraspecific overlap in space use as a consequence of
intrusions from foraging conspecific neighbors, which would reduce the supply
of resources and hence the area exclusively used by an individual (but see
Haskell et al., 2002). With
regard to the non-monotonicity issue, it has been hypothesized that it
reflects energetic constraints upon small-sized organisms that lead to trophic
specialization on energetically rich but widely dispersed resources
(Brown and Maurer, 1987;
Brown et al., 1993;
Marquet and Taper, 1998;
Kelt and Van Vuren, 1999) and
this could also explain the observed departures from Equation 4 for small
species (see Silva and Downing,
1995; Marquet et al.,
1995; Siemann et al.,
1996; Armbruster et al.,
2002; McClain,
2004). Available estimates for the scaling exponent of H
below MT range between -1.81 to -2.4
(Marquet and Taper, 1998;
Kelt and van Vuren, 2001).
These considerations imply that for mammals:
 | (6) |
 | (7) |
On extreme body sizes, extinction and minimum viable populations
Marquet and Taper (1998)
first realized that Equations 6 and 7 allow us to predict the maximum and
minimum body size able to persist in a landmass of a given area. Their
argument starts by estimating the minimum area of a landmass required for
persistence (Am) as:
 | (8) |
where Nm is the minimum number of individuals required
to avoid extinction in the absence of immigration (see also
McNab, 1994). Thus if
Nm is equal to 500 (individuals) and H is equal
to 0.1 (km2 per individual), then the minimum area required for the
persistence of this species (Am) would be 50
km2. Substituting Equations 6 and 7 into Equation 8 we arrive at:
 | (9) |
 | (10) |
Equations 9 and 10 set the boundary for persistence and apply to the
largest and smallest species able to persist in a given landmass in the face
of extinction. Although the scaling of Nm is not known it
can be estimated by estimating the exponents associated with
Am vs M. Marquet and Taper
(1998) tested for these
relationships using data on mammals found in land bridge islands, mountaintops
and continents, whose actual species compositions are mainly the result of a
selective extinction process associated with relaxation phenomena (e.g.
Diamond,
1984a,b).
As seen in Fig. 3A, as the
size of the largest mammal species within an insular fauna increases so does
the landmass area required for persistence. Furthermore, as the size of the
smallest species decreases the area of the landmass where it is found also
increases. These patterns were found to be highly significant within
archipelagoes, across continental landmasses and when all cases are analyzed
jointly in one general regression. A similar pattern has been reported for
snakes by Boback and Guyer
(2003, see
Fig. 3B). Since the estimated
exponents for mammals reported by Marquet and Taper
(1998) did not differ from
the expected ones it is possible to conclude that
 | (11) |
|
or, in other words, the minimum number of individuals required to avoid stochastic extinction is invariant i.e. independent of body size.
A similar analysis carried out by Burness et al.
(2001) using mammals, birds
and reptile species found in oceanic islands and continents during the last
65,000 years, confirmed our predictions for the maximum size of species and
show that part of the variability in these scaling relationships can be
explained by diet and thermoregulatory physiology. Furthermore, the patterns
shown in Fig. 3, suggest the
existence of an evolutionary advantage for medium sized species linked to
reduced extinction probability, and is consistent with macroevolutionary and
microevolutionary changes in mammalian body size (see
Brown et al., 1993;
Alroy, 1998;
Schmidt and Jensen, 2003),
although the mechanistic basis of these changes are not yet fully
comprehended.
If Equation 11 is correct, the further away a species is from
Nm the better its chances to persist. In macroevolutionary
time scales, this implies that small-sized species will tend to survive longer
and likely accumulate by resisting extinction, thus implying that the number
of species should decrease with body size. If this is correct in a given area,
such as the South American continent, the number of species (S) of a
given size (M) should be characterized by a power-law with a scaling
exponent close to the one characterizing density (N) and body size
i.e.:
 | (12) |
A preliminary test of this idea is presented in
Fig. 4 where we plot the size
frequency distribution of the number of species of mammals in South America
(Marquet and Cofre, 1999)
using exponentially increasing size classes. As predicted the exponent is not
different from the expected -3/4 (P.A.M. and S.A., unpublished).
|
To derive the patterns in extreme body size, minimum population size and
size frequency distributions just discussed, we assumed that habitat and food
resources are either homogeneously distributed in space or clumped for species
below a threshold size, but did not specify any particular spatial pattern.
While this could be a reasonable assumption at large spatial scales, at finer
scales (i.e. at the level of landscapes), the spatial distribution of
resources (which is usually fractal and can be described by a power-law) can
affect the distribution, abundance and interaction of species (e.g.
Milne et al., 1992) and
affect community patterns (Ritchie and
Olff, 1999; Schmid,
2000).
Ecological invariants
As noted by West (1999),
one of the most intriguing consequences of biological scaling laws is the
emergence of invariant quantities. These are seen, for example, in association
with longevity (Peters, 1983;
Calder, 1984;
Schmidt-Nielsen, 1984).
Because lifespan increases like M1/4, it follows that most
rates (such as heart-rate and specific metabolic rate), which decrease as
M-1/4, give rise to relationships that are size invariant
(i.e. they scale as M0) at the scale of a lifetime. So,
for example, the number of heartbeats in a lifetime is the same for all
mammals and so is the total energy needed to support a given mass of an
organism during its lifetime. And they are also common in life history theory
(Charnov, 1993) but associated
with the timing of life history events. However, at present it is not known if
these fundamental symmetries in living entities are just a by-product of
fundamental scaling laws or have a deeper ecological and evolutionary meaning.
Several invariant relationships associated with the density scaling
relationship shown in Equation 4 have been postulated for ecological systems.
These are seen in: (1) the total energy used by a population or population
energy use scaling; (2) the distribution of biomass in ecosystems; and (3) the
minimum size of populations. Since we have already elaborated on (3), in this
section we will devote our attention to (1) and (2).
The first invariant relationship has been dubbed the `energetic equivalence
rule' (Nee et al., 1991). Its
derivation follows. Because metabolic rate (B) scales with body mass
raised to the 3/4 power, the existence of the same scaling exponent for
N has been taken as evidence that the abundance of species is limited
by energetic requirements (Damuth,
1981,
1991). Similarly, the total
energy used by a species' population per unit area (EU) can be assessed by
multiplying the average energy used by an individual times the density of
individuals. Thus, by multiplying Equations 3 and 4 one gets:
 | (13) |
such that the energy used by different species should be roughly equal and
independent of body mass. This pattern was first pointed out by Damuth
(1981), although not exempt
from criticism (e.g. Marquet et al.,
1995; Taper and Marquet,
1996) as it depends on the exact value of the scaling exponents
associated with B and N and has yet to be mechanistically
understood (Damuth, 1998, but
see Charnov et al., 2001)
especially for mobile organisms that utilize a broad spectrum of resources and
inhabit different ecosystems around the world
(Marquet et al., 1995;
Brown et al., 2004). For
tree-dominated communities, this relationship has been shown to hold at local,
regional and worldwide scales (Enquist and
Niklas, 2001; see also
Enquist et al., 1998 for
plant species in general, see Fig.
5) and have been hypothesized to emerge from the allometric rules
that influence the behavior of individual plant species
(Niklas and Enquist, 2001)
competing for space and limiting resources. Similarly it has been recently
documented in local parasite communities of fishes
(George-Nascimento et al.,
2004).
|
The second ecological invariant is that related to biomass distribution in
ecosystems. Unlike previous invariants, this one is associated with work
conducted mostly in aquatic ecosystems. In brief, this invariant was proposed
by Sheldon et al. (1972) who,
by doing what they called a size spectra (see below), concluded that there is
a `tendency for roughly similar amounts of particulate materials to occur
in logarithmically equal size ranges...'
(Sheldon et al., 1972: p.
336). This relationship can be expressed as a power-law of the form:
 | (14) |
where Z(s) stands for the number or biomass of particles
of size or volume (s). It is usual practice to work with the normalized
biomass spectra (see below), thus after normalizing we arrive at:
 | (15) |
Because the equal biomass invariant, when normalized, implies a linear
proportional decrease in biomass across size classes it has been dubbed `the
linear biomass hypothesis' (Sheldon et
al., 1986). This relationship has been shown to be a well-known
pattern in pelagic ecosystems (Sheldon et al.,
1972,
1977;
Rodriguez, 1994;
Rinaldo et al., 2002;
Quiñones et al.,
2003). It should be noted that although this relationship is
different from the one depicted in Equation 4, for this does not rely on
distinguishing species (i.e. it is ataxonomic), they are related
(Rinaldo et al., 2002). As
shown by these authors, the linear biomass hypothesis implies that the scaling
exponent of the relationship between number of individuals and average size
should be -1 instead of the observed -3/4. Brown and Gillooly
(2003) hypothesized that an
exponent of -1 is expected when analyzing species in more than one trophic
level, as is the case in marine ecosystems and in size-structured food webs in
general. The -3/4 exponent and, hence, the energetic equivalence rule, is
expected in situations where all species use the same source of energy (i.e.
within trophic levels). However, the same relationships for size-structured
food webs need to account for energy transfer efficiency as well as body mass
differences between trophic levels. In this case the prediction is an exponent
of -1. A formal test of this hypothesis using a marine food web has shown that
it accurately predicts observed patterns
(Jennings and Mackinson,
2003) thus narrowing the gap between two research traditions
(marine vs terrestrial), both of which have appreciated the value of
scaling approaches (e.g. Platt,
1985; Cyr and Pace,
1993; Brown et al.,
2004). Notice that this result implies that both invariant
relationships are not independent, such that the existence of invariance in
energy use within trophic levels entails the existence of biomass invariance
across them. To our knowledge the only terrestrial biomass spectra so far
reported is that carried out by Enquist and Niklas
(2001) for tree-dominated
communities using a worldwide data set of 227 plots of 0.1 ha assembled by the
late Alwyn Gentry. These authors show, as expected, that the number of
individuals in logarithmic size classes decreases as M-3/4
implying that population energy use is invariant.
In a later analysis of the same data set Enquist et al.
(2002) introduces a related
invariant by showing that total biomass [i.e. total standing above ground dry
biomass (Mtot) per 0.1 ha plot] is invariant with respect
to number of species (S) (i.e.
Mtot
S0) implying that an
increase in species richness within communities results in a finer division of
biomass instead of an increase in total biomass. Notice that this invariant,
as well as the `energetic equivalence rule', entail the existence of an
ecological zero-sum dynamic (Van Valen,
1980) consistent with recent symmetric models of community
assembly (Hubbell, 2001).
However, it remains to be seen if the biomass invariant described by Enquist
et al. (2002) applies to taxa
other than trees and how it changes when more than one trophic level is
analyzed.
Ecological scaling and biomass size spectra
The study of the distribution of biomass by size in the pelagic systems has
been a significant step in the search for generalizations in aquatic ecology.
Regularities in the size structure of pelagic communities have been observed
in offshore systems (e.g. Sheldon et al.,
1972; Beers et al.,
1982; Platt et al.,
1984; Rodriguez and Mullin,
1986a,b;
Witek and Krajewska-Soltys,
1989; Quiñones et al.,
2003) and lakes (e.g. Sprules et al.,
1983,
1991;
Sprules and Knoechel, 1984;
Sprules and Munawar, 1986;
Echevarría et al.,
1990; Ahrens and Peters,
1991; Gaedke,
1993). In coastal pelagic ecosystems the biomass size distribution
does not present patterns as regular as those observed in oligotrophic systems
but biomass is not randomly distributed across body size (e.g. Jimenez et al.,
1987,
1989;
Rodriguez et al., 1987). A
regular pattern in the biomass size distribution has also been found in salt
marshes (Quintana et al.,
2002) and benthic communities (e.g.
Schwinghamer, 1981;
Warwick 1984;
Schwinghamer, 1985;
Saiz-Salinas and Ramos, 1999;
Quiroga et al., 2005).
On the other hand, aquatic food webs are strongly size-structured with
larger predators eating smaller prey
(Sheldon et al., 1972;
Dickie et al., 1987). Many
species grow in mass by five orders of magnitude; cannibalism, cross-predation
and transient predator-prey relationships are common
(Cushing, 1975;
Kerr and Dickie, 2001).
However, mean body mass of species is only weakly correlated with body mass in
the whole food web (Fry and
Quiñones, 1994; France
et al., 1998; Jennings et al.,
2001,
2002). These observations
provide compelling reasons to adopt size-based rather than species-based
analyses of food web structure in pelagic ecosystems
(Jennings and Mackinson,
2003).
In the study of biomass size distribution of pelagic communities, the most
common representation used has been the construction of biomass size spectra.
In this formulation every individual in the system is assigned to one of a
series of size classes represented on a logarithmic scale conforming to an
un-normalized spectrum (Fig.
6). The high degree of aggregation of such an ataxonomic approach
reduces the complexity of the system to a manageable level. Platt and Denman
(1977,
1978) indicated that a
normalization procedure was required to represent and cross-compare biomass
size distributions adequately, because the width of the size classes varies
significantly through the size spectra. In brief, the normalization procedure
consists of taking the variable of interest Z(s) (i.e.
usually biomass or numerical abundance) in the size class characterized by the
weight or volume (s) and dividing it by the width of the size class,
s. Thus the normalized version of the variable z
(i.e. Z(s); see Fig.
7) is equal to:
 | (16) |
|
|
A detailed analysis about constructing normalized (NBS) and un-normalized
size spectra can be found in Blanco et al.
(1994;
1998).
On the other hand, Vidondo et al.
(1997) have argued in favor
of using the Pareto type II distribution, which is widely used in many
disciplines to describe size distributions, for representing and modeling size
spectra. To apply such an approach adequately, each particle should contribute
one point to the Pareto plot and, therefore, all the information contained in
the observations is used. The Pareto approximation is ideal for automatic
sizing instruments, such as flow cytometers and electronic or laser particle
counters. Although in theory it is possible to estimate the parameters of the
underlying Pareto distribution from the NBS-spectra, this procedure is not
recommended from a statistical standpoint
(Vidondo et al., 1997). It is
important to note that, in systems that are far from equilibrium, there may be
size distributions that cannot be appropriately described by the Pareto nor
the normalized biomass-size-spectrum model, such as multimodal distributions
(Gasol et al., 1991;
Havlicek and Carpenter,
2001).
The size-spectrum approach is rooted in the well-accepted concepts of the
pyramids of biomass and numbers (Cousins,
1980,
1985;
Platt, 1985) and research in
this field can be traced back to the first half of the century (e.g.
Elton, 1927;
Ghilarov, 1944). However, it
is the work of Sheldon et al.
(1972,
1973) that provided new
impetus to the field by publishing a set of particle-size spectra from oceanic
areas (for a historical perspective see
Platt, 1985). Sheldon et al.
(1972), based on his field
observations, proposed the `linear biomass hypothesis', which states that in
the pelagic system there is roughly the same biomass at all size classes. The
regularities in pelagic size structure observed by Sheldon et al.
(1972,
1973) and the fact that most
aspects of energy and material flow of an organism are size dependent
(Peters, 1983;
Calder, 1984;
Schmidt-Nielsen, 1984) led to
the development of theoretical models to explain and quantify the regularities
(e.g. Kerr, 1974;
Sheldon et al., 1977; Platt
and Denman, 1977,
1978; Silvert and Platt,
1978,
1980; Borgmann,
1982,
1983,
1987;
Dickie et al., 1987;
Boudreau and Dickie, 1989;
Boudreau et al., 1991). The
first theoretical models about the size structure of the pelagic ecosystem
were proposed by Kerr (1974),
Sheldon et al. (1977) and
Platt and Denman (1977,
1978). Whereas the two first
models were based on the trophic-level concept, the last stands on the
consideration of a continuous flow of energy from small to large organisms.
Kerr and Sheldon's models propose that biomass is constant when organisms are
organized in logarithmic size classes. On the other side, Platt and Denman's
model predicts a slight decrease of biomass with organism size with a slope of
-0.22 and proposes an allometric structure for the pelagic ecosystem (Platt
and Denman, 1977,
1978). Until now the most
comprehensive biomass size spectra constructed in close to steady state
systems (i.e. North pacific Central Gyre,
Rodriguez and Mullin, 1986b;
oligotrophic areas of the Northwest Atlantic,
Quiñones et al., 2003)
support the Platt and Denman's model. It is important to note that in Platt
and Denman's model the exponent (-0.22) represents a balance between
catabolism and anabolism and, consequently, from a scaling standpoint it is
coherent with the recently proposed `metabolic theory of ecology'
(Brown et al., 2004).
Linearity, smoothness and continuity in biomass size spectra
Evidence to date shows that oligotrophic ecosystems close to a steady state
present more or less linear normalized biomass size spectra (log-log scale).
The slope of the NBS-spectrum of oceanic pelagic systems seems to be close to
-1 or -1.2 depending on whether biomass is expressed as volume or carbon,
respectively (Rodriguez and Mullin,
1986b; Quiñones et
al., 2003). By contrast, ecosystems far from the steady state may
present non-linear normalized biomass spectra and under extreme conditions the
biomass-size spectra can present discontinuities
(Quiñones, 1994).
Havlicek and Carpenter (2001)
show that size distributions in lake communities have multiple lump and gap
regions within each functional group of phytoplankton, zooplankton and fish.
Simulations showed the gaps could not be explained by incomplete censuses of
species or by systematic underestimation of intraspecific size variation.
Nevertheless lakes that differed widely in nutrient status, trophic structure,
species diversity and area had similar size distributions. A detailed analysis
of the discontinuities in the marine biomass spectra of close to steady state
systems has not been conducted to date.
Dickie et al. (1987)
analyzed the distribution of specific production by size in ecosystems. They
identified two kinds of slopes in the relationship between log-specific
production and log body size (Fig.
8). First, a unique primary slope reflecting the size dependence
of metabolism. This primary slope is uniform, low and negative (approximately
-0.18). Second, a collection of secondary slopes, which represent an
ecological scaling of production related to rapid changes of log
annual-specific-production with log body size within groups of organisms with
similar production efficiencies. These secondary slopes are steeper than the
primary slope. Boudreau et al.
(1991) have pointed out that
such ecological scaling would produce dome-like patterns in the biomass size
spectra. In fact, dome-like patterns have been observed in several ecosystems
(e.g. Sprules and Munawar,
1986; Sprules et al.,
1988; Rodriguez et al.,
1990; Sprules et al.,
1991). However, oligotrophic oceanic systems present NBS-spectra
that can be properly described by a straight line (e.g.
Rodriguez and Mullin, 1986a;
Witek and Krajewska-Soltys,
1989; Quiñones et al.,
2003). In fact, in addition to linearity, perhaps the second
most-characteristic feature of the size structure of plankton in the
oligotrophic ocean seems to be the similarity between primary and secondary
scales. The linearity of the NBS-spectra in oligotrophic oceanic waters
suggests the dominance of the metabolic scaling over the ecological scaling in
these areas. It is interesting to note that the primary slope of the
normalized biomass spectra seems to be strongly related to the slope of the
normalized metabolic spectra as shown empirically by Quiñones
(1992) and Quiñones et
al. (1994) in planktonic
communities from the North Atlantic and Mediterranean Sea, respectively.
|
The complete absence or scarcity of conspicuous dome-like patterns in the
biomass size distribution in some pelagic ecosystems can also be explained in
trophodynamic terms by several hypotheses that are not mutually exclusive.
First, if the food web in a particular system is unstructured (sensu
Isaacs, 1972,
1973) the domes, if any, will
tend to be minor. Second, the dome-like patterns will also be less conspicuous
in systems with a more-structured food web but where there is a large range of
prey/predator body-size ratios (Thiebaux
and Dickie, 1993). Indeed, the assumption of a constant
prey/predator ratio for the pelagic ecosystem is erroneous as shown by
Longhurst (1989,
1991). Third, if the trophic
positions (i.e. groups of organisms having a common production efficiency,
Boudreau and Dickie, 1992) are
not sufficiently characterized by different size ranges, the domes will not be
conspicuous in the biomass size spectra. Evidently, not all observed dome-like
patterns are produced by the secondary scaling described by Dickie et al.
(1987). In fact, dome-like
patterns may result from mere methodological artifacts
(García et al., 1994).
In addition, some observed dome-like patterns in pelagic systems could be the
by-product of the propagation of a peak of biomass or energy (Silvert and
Platt, 1978,
1980;
Han and Straskraba, 2001)
through the size spectrum. Waves of energy changing the shape of the biomass
spectrum have been observed both in coastal (Rodríguez et al., 1987;
Jiménez et al., 1989) and oceanic waters
(Rodríguez and Mullin,
1986a).
Environmental variables determining and/or affecting biomass size spectra
Sprules and Munawar (1986)
proposed a relationship between the numerical value of the slope of the
NBS-spectra and the trophic state of a pelagic ecosystem. Eutrophic ecosystems
would present more positive slopes than oligotrophic ecosystems. However, due
to both methodological difficulties and to the lack of sufficient data this
hypothesis is still far from being validated.
It is known that several size-dependent processes can alter community size
structure. Size-selective predation can be a primary organizing force in some
communities (Brooks and Dodson,
1965; Hall et al.,
1976; Vanni,
1986) and the size structure of the grazers can influence the size
structure of the phytoplankton community
(Carpenter and Kitchell, 1984;
Bergquist et al., 1985). In
fact, Rassouldagan and Sheldon
(1986) and Sheldon et al.
(1986) have experimentally
shown that predation can play a major role in structuring size spectra.
Abiotic forcing has also the potential to modify biomass size distribution.
For instance, Havens (1992)
demonstrated that acidification could change the parameters of freshwater
plankton size spectra and Samuelsson et al.
(2002) show that nutrient
enrichment in mesocosms resulted in higher biomass and changed plankton size
structure.
In relation to benthic size spectra, the physical characteristics of the
sediment (Schwinghamer, 1981;
Drgas et al., 1998;
Duplisea, 2000), the gradient
of organic matter (Schwinghamer,
1985), the life-history strategies of dominant taxa
(Warwick, 1984) and oxygen
levels (Quiroga et al., 2005)
are thought to constrain the size spectrum of faunal species. However, the
spectra seem to be quite conservative. For instance, Rafaelli et al. (2000)
imposed size-specific perturbations (enrichment and predation) on marine
sediment assemblages. Perturbations significantly affected the densities and
relative abundance of the main invertebrate taxa and these effects were
consistent with the known effects of enrichment and predation. However, there
was little evidence of significant treatment effects on the overall benthic
biomass or abundance size spectrum, supporting the contention that the
spectrum is conservative and is probably constrained by habitat
architecture.
The applications of biomass size spectra
Since the late 1970s, the NBS approach has found application in several
fields, such as fisheries research and pollution studies. In fisheries, the
NBS approach has been applied to predict fish production from phytoplankton
standing stock (Moloney and Field,
1985) and from primary and zooplankton production
(Sheldon et al., 1977;
Borgmann, 1982,
1983;
Borgmann et al., 1984). The
NBS approach has formed the basis of models to estimate fish mortality rates
(Peterson and Wrobleski,
1984) and to analyze multispecies fisheries (e.g.
Pope et al., 1988;
Murawski and Idoine, 1989;
Duplisea and Kerr, 1995,
2000). Also, models to
estimate production of multispecific fisheries based on size structure and the
allometric relation of the production to biomass ratio have been developed
(Dickie et al., 1987; Boudreau
and Dickie, 1989,
1992). Recently, there is a
growing interest in generating sound ecological indicators to support an
ecosystem approach to fisheries as stated in the International Symposium on
Quantitative Ecosystem Indicators for Fisheries Management (March-April 2004,
UNESCO, France). Thus, size-based indicators have become one of the main
avenues of research (e.g. Bianchi et al.,
2000; Rice, 2000;
Zwanenburg, 2000). In
pollution studies, the NBS approach has been used to model the flow of
contaminants up the food web (Thomann,
1979,
1981;
Griesbach et al., 1982;
Borgmann and Whittle, 1983;
Vezina, 1986).
The metabolic spectra
It has become evident that to understand the dynamic of the pelagic
ecosystem it is necessary to allocate more efforts in the empirical study of
community processes, such as respiration and production, from an allometric
point of view (Quiñones,
1994). The distribution of biomass by size, despite its linkage to
the energetics of the system, is essentially a measure of ecosystem structure.
The simultaneous study of size structure and processes, such as respiration,
should permit a better understanding of the relationship between structure and
function in the ecosystem. The only metabolic spectra to date are those
described for the Northwest Atlantic
(Quiñones et al.,
1991; Quiñones,
1992) and for the Alboran Sea
(Quiñones et al.,
1994). These spectra covering from bacteria to zooplankton show
that respiration in the pelagic system diminishes as a power function of body
size at the community level of organization, with a slope close to -1.2
(normalized metabolic spectra). Further research in this field is crucial to
connect the metabolic theory of ecology
(Brown et al., 2004)
adequately with size-spectrum theory in the pelagic ecosystem.
Some ecological power-laws related to population abundance and fluctuation
As with scaling relationships associated with body size, power-laws are
ubiquitous in ecological systems, for example in the size and duration of
epidemic events (Rhodes and Anderson,
1996; Rhodes et al.,
1997), in patterns of abundance, distribution and richness (e.g.
Frontier, 1985;
Banavar et al., 1999; Harte et
al., 1999,
2001) and in food web
attributes (e.g. Brose et al.,
2004; Garlaschelli et al.,
2003). In the following paragraphs we will present some power-laws
associated with population dynamics, which highlight phenomena also seen in
the context of scaling relationships (such as zero-sum dynamics) and
relationships that can be categorized in terms of body size.
Power-laws in population growth rates
Standard ecological wisdom asserts that population size is expected to
follow a lognormal distribution, given that it is the product of a
multiplicative renewal process (e.g.
Lawton, 1989;
Blackburn et al., 1993b;
Halley and Inchausti, 2002).
Furthermore, several single species population models give rise to normal or
lognormal population abundance distributions (e.g.
Keeling, 2000). If population
abundance follows a lognormal distribution, it is expected that the ratio of
successive abundances N(t+1)/N(t) also has a lognormal
distribution and, hence, the logarithm of such a ratio
=ln[N(t+1)/N(t)],
should show a normal or Gaussian distribution. In other words, under an
expectation of lognormal population abundances, population growth rates should
exhibit a Gaussian probability distribution. Interestingly, as shown by Keitt
and Stanley (1998), the
growth rates in an avian ensemble over a large geographical scale in North
America are not distributed following a Gaussian distribution, but rather
follow a power-law with a characteristic tent shape
(Fig. 9A), which is well
described by an exponential or log-Laplace distribution
(Keitt and Stanley, 1998;
Keitt et al., 2002).
Furthermore, the same tent-shaped power-law form is also observed when
examining the conditional probability density distributions of growth rates
s given an initial
abundance class
p(s|N),
defined by grouping observations into bins or categories of initial total
abundance (Fig. 9B). The width
of the distribution, as measured by the standard deviation of the growth
rates, widens as the initial population abundance decreases
(Fig. 9B). It is remarkable
that when the scaled growth rate
scal=[s-<s>]/
and the scaled probability density
pscal=p(s|N)
are calculated for these conditional probability distributions
(Fig. 9C), all the data from
the different bins collapse onto the same universal power-law curve
pscal
(-|r scal|). This non-trivial
rescaling suggests that in spite of differences in body size, life history and
ecology, all the species under study fall along a single power-law
relationship, which suggests that they share a common universal probability
density distribution of growth rates. This powerful statement is further
strengthened by the fact that this universal distribution is a power-law.
