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ICES Journal of Marine Science (2012), 69(4), 602 –614. doi:10.1093/icesjms/fss031
On balanced exploitation of marine ecosystems: results
from dynamic size spectra
Richard Law1*, Michael J. Plank 2, and Jeppe Kolding 3
1
York Centre for Complex Systems Analysis, Ron Cooke Hub, University of York, York YO10 5GE, UK
Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
3
Department of Biology, University of Bergen, High Technology Center, PO Box 7800, N-5020 Bergen, Norway
2
Law, R., Plank, M. J., and Kolding, J. 2012. On balanced exploitation of marine ecosystems: results from dynamic size spectra. – ICES Journal
of Marine Science, 69: 602 – 614.
Received 24 August 2011; accepted 30 January 2012; advance access publication 4 March 2012.
Fisheries are often managed to protect small young fish and to harvest big old fish. This can be wasteful, leading to large parts of
catches being discarded. A recent suggestion is that it could be better to distribute fishing more widely across species and body
sizes, balancing it more closely to the natural productivity of different organisms. Here, we test effects of such fishing against more
traditional methods using a model of a single fish species with a dynamic size spectrum together with a fixed spectrum of plankton.
This has the feature that productivity is determined by the bookkeeping of biomass in the model, and decreases as fish grow larger.
The results show that harvesting smaller fish (which have higher productivity) allows a greater sustainable biomass yield than harvesting larger fish (which have lower productivity); the greater spawning-stock biomass that comes from protecting large fish contributes
to this. Balanced exploitation brings fishing mortality more in line with this natural variation in productivity. In addition, the resilience
of the ecosystem to perturbations can be improved, and disruption to the size distribution of organisms in the ecosystem reduced. We
argue that there are potentially real benefits to be gained by moving towards more balanced exploitation of marine ecosystems,
unconventional though this is.
Keywords: BOFFFF, discard, disruption, exploitation pattern, exploitation rate, fishery, growth rate, productivity, resilience, size spectrum,
sustainability, yield.
Introduction
This paper is motivated by some problems that stem from the way
in which our marine resources are exploited. Fisheries management aims to select species and body sizes, often trying to
protect the young and harvest the old, while maintaining a
substantial spawning-stock biomass. However, it is difficult to
achieve this when faced with the practicalities of harvesting
species in multispecies marine communities. Commercial exploitation in reality is often wasteful, the harvest containing a bycatch
of target species that has to be discarded because it is unsuitable for
landing, together with a bycatch of unwanted species (Hall, 1996).
There is a strong imperative to reduce this waste, and to develop
new approaches to exploitation that make better use of marine
resources (Diamond and Beukers-Stewart 2011; FAO, 2011).
One suggestion is that it could be helpful to move towards
more balanced harvesting of marine ecosystems (Zhou et al.,
2010), distributing “a moderate mortality from fishing across
the widest possible range of species, stocks, and sizes in an ecosystem, in proportion to their natural productivity, so that the
relative size and species composition is maintained” (Garcia
et al., 2012). “Productivity” is defined as the rate at which
biomass flows through some entity in some fixed physical
volume (dimensions: mass, volume21, time21); the entity could
be species, organisms of a chosen body size irrespective of
species, or both species and body size. Balanced harvesting
builds on an idea about linking the exploitation rate to natural
predation and productivity (Caddy and Sharp, 1986), on which
there has been rather little quantitative analysis. Bundy et al.
(2005) applied a range of harvesting patterns to an Ecopath
model of the Gulf of Thailand, with groups of species aggregated
into trophic levels. This showed that although exploitation always
changes trophic structure, balanced harvesting across trophic
levels changes the structure less than exploitation concentrated
at high trophic levels. Rochet et al. (2011) found no simple
optimal level of size selective fishing that would best maintain
biodiversity: the effect of size-selective fishing on biodiversity
depended on the selectivity curve relative to the size structure
and species composition of the exploited community.
# 2012 International Council for the Exploration of the Sea. Published by Oxford University Press. All rights reserved.
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*Corresponding Author: tel: +44 1904 325372; fax: +44 1904 500159; e-mail: [email protected].
On balanced exploitation of marine ecosystems: results from dynamic size spectra
2005). However, the degree of disruption to size structure
from different harvesting schemes is not so well understood.
Collectively, we call these three properties the YRD criteria:
yield, resilience, and disruption.
Methods
Size-spectrum model
Our setting for calculations on exploitation is a size spectrum containing a single fish species living in a well-mixed, non-seasonal
environment, supported by a plankton community. This simplicity is deliberate: the purpose is to make the flow of biomass as
basic and transparent as possible. The plankton community is
taken to be fixed, as in Equation (A15), having a spectrum (i.e.
log abundance vs. log body mass) with a gradient of 21 (equivalent to a power-law relationship between abundance and body
mass with exponent 2g ¼ 22), and a density of 100 m23 at a
body size log (1 mg), consistent with observations on marine ecosystems (e.g. San Martin et al., 2006). An upper bound on the
plankton body mass is set at 20 mg, and a lower bound is set to
ensure that food is available for even the smallest fish. Using a
fixed, empirically determined plankton spectrum avoids introducing uncertain plankton dynamics; the fish population itself is
regulated through predation by conspecifics.
Technical specification of the dynamical system is given in the
Appendix, with the meanings of parameters and their values summarized in Table 1. We use a standard life history corresponding to
a species that eats both plankton and smaller fish, with key features
as follows. Fish start life with an egg mass of 1 mg. They grow by
eating smaller organisms, here with a feeding preference function
centred on prey whose body mass is 0.01 of their own. When the
fish are small, this means that they feed on plankton and, as they
get larger, they feed increasingly on smaller individuals of their
own species. Conversion of prey mass into consumer mass is inefficient; we assume a conversion efficiency of 0.2 to allow for losses
during feeding and metabolism. Before maturation, all converted
mass contributes to somatic growth. The fish have a maturity
ogive with a midpoint at 150 g. Once maturation has occurred,
a proportion of assimilated food is allocated to reproduction,
which means that reproduction depends on incoming mass
rather than on capital (Houston et al., 2007; Stephens et al.,
2009). The allocation to reproduction is set such that reproductive
mass remains an approximately constant proportion of body mass,
and is set to bring mature fish to an asymptotic mass of 1000 g.
A check on the sensitivity of our results to small changes in
parameters is provided.
Fish die primarily as a result of being eaten. We also allow a
relatively low rate of background, non-predation mortality, recognizing that predation may not be the only cause of death; background mortality scales with body mass w approximately as
w 20.25 (Brown et al., 2004), and we therefore use this exponent.
Following Hall et al. (2006), we also allow mortality to increase
as the body mass approaches its asymptotic value, corresponding
to a process of senescence; this starts at 500 g. Such mortality prevents a build-up in density of the largest fish; these lack predators
and their density would grow in the absence of this term.
Senescent mortality might not be needed in a real community
containing other species that grow to larger body masses and
continue to act as predators. Taking the intrinsic and senescent
mortality together, gives a “U”-shaped function of body mass
(Hall et al., 2006).
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Here, models of dynamic size spectra are used to evaluate the
effects of balanced and other patterns of exploitation on marine
ecosystems (for background on size-spectra dynamics, see Benoı̂t
and Rochet, 2004; Andersen and Beyer, 2006; Blanchard et al.,
2009; Datta et al., 2010). The paper develops an approach taken
by Rochet and Benoı̂t (2011) who found variation in abundance
over time in simulations of size spectra increased when fishing
was made more size selective, and/or larger fish were targetted.
The reason for turning to size-spectra models is that they take
mechanics of mass flow through ecosystems down to the level of
organisms as they eat each other and grow. It is this flow of
biomass from small to large organisms that ultimately determines
the levels of exploitation marine ecosystems can sustain. Moreover,
body size is a good first indicator of the trophic level at which fish
feed, better in fact than taxonomic identity (Jennings et al., 2001),
because fish typically change their preferred prey from microorganisms to larger organisms further up the food chain as they
grow from egg to adulthood. Models of size spectra capture this
changing diet by internalizing predation, growth, and mortality,
thereby holding in place the feedback between growth and the
availability of food, and the feedback between mortality and predation. This is in contrast to traditional dynamic-pool and
yield-per-recruit models that use an external function such as a
parameterized von Bertalanffy equation to describe growth, together with an assumed, often constant, level of intrinsic mortality.
Models of size-spectrum dynamics currently used for fish communities can be thought of as two types: (i) aggregated community
models that do not distinguish between species and do not include
explicit assumptions about renewal, and (ii) models disaggregated
to species that use body-size information on maturation and reproduction to make the renewal explicit. Because of the potentially
important effects of fishing on stock renewal, we adopt approach
(ii) here. However, it is enough to consider just a single fish
species to see some of the major consequences of harvesting
small or large fish. We take this simplified approach here, the
fish being supported by a plankton spectrum when they are
small in body size, and shifting to feeding on smaller conspecific
fish as they grow larger. Within this framework, we compare
balanced harvesting with some alternative patterns. These alternatives include: (i) a traditional fishery designed to protect small fish
and harvest large fish; (ii) the reverse, in which the fishery protects
large fish and harvests small fish; and (iii) a slot fishery, in which
fish in some intermediate size range are harvested.
We judge the success of fishing patterns based on three criteria.
The first is the size of the sustainable yield, taking “sustainable” to
mean that yields are evaluated at steady state. Harvesting reduces
the steady-state abundance and, if too heavy, may cause extinction
of the species. Second is the effect of exploitation on the resilience
of the system to disturbances. We adopt a dynamical-systems interpretation of resilience, as the tendency to return to a reference
state (here the steady state). Exploitation is unlikely to be
neutral in its effects on resilience (Hsieh et al., 2010); for instance,
truncating adult lifespan concentrates reproduction in young
adults, reduces mixing across cohorts, and is likely to be destabilizing. Third is the disruption to the natural size structure of the
system caused by exploitation, which we measure as the deviation
of the exploited from the unexploited steady-state system. That
exploitation alters the size structure of marine ecosystems is not
in doubt (Hsieh et al., 2010); indeed, slopes of size spectra have
been investigated as potential indicators of exploitation (Rice
and Gislason, 1996; Blanchard et al., 2005; Piet and Jennings,
603
604
R. Law et al.
Table 1. Model parameters and values.
Parameter
Value
Unit
Comments
Fish life history
Mass of fish egg
Mass at 50% maturity [Equation (A16)]
Asymptotic mass [Equation (A16)]
Exponent for reproduction function [Equation (A16)]
Controls the width of transition from immaturity to maturity
[Equation (A16)]
0.001
150
1 000
0.2
10
g
g
g
–
–
K
a
A
b
s
mi,0
ji
w0 exs
js
0.2
0.8
600
5
2
1
0.25
500
5
–
–
m3 year−1 g−a
–
–
year−1
–
g
–
Dynamic size spectrum
Food conversion efficiency [Equation (A1)]
Search rate scaling exponent [Equation (A8)]
Feeding rate constant [Equation (A8)]
Natural log of mean PPMR [Equation (A9)]
Diet breadth (Equation 9)
Intrinsic mortality rate at mass x0 [Equation (A18)]
Exponent for intrinsic mortality rate [Equation (A18)]
Mass at onset of senescent mortality [Equation (A19)]
Exponent for senescent mortality rate [Equation (A19)]
w0 ex p,min
w0 ex p,max
u0,p
2g
4.54 × 10−8
0.02
100
22
g
g
m−3
–
Fixed plankton size spectrum [Equation (A15)]:
Lowest body mass of plankton
Greatest body mass of plankton
Plankton density at x0
Exponent of plankton spectrum
The equations referenced are those in which the symbols are used, and are given in the Appendix.
A consequence of the bookkeeping of biomass above is that the
rate of growth of fish depends on the availability of food to eat.
The average growth trajectory of the fish is therefore an output
of the model obtained by solving Equation (A21), and yields are
calculated without the need for an external model of growth,
such as a parameterized von Bertalanffy growth equation. The
same applies to death, insofar as this is caused by predation.
Also, the system is internally regulated by larger fish eating
smaller fish: as the density of adults increases, adults consume
more small fish, reducing the recruitment of further adults. This
means that renewal of the fish population is also an output of
the model. In this way, the growth, death, and the renewal processes are all internalized, and determined by the feedbacks from
predation within the system.
Exploitation
Four contrasting schemes for harvesting the ecosystem are considered. These are intended to provide general understanding of how
to exploit size-structured fish stocks through their effects on the
YRD criteria, rather than to examine the practical issues of implementing them.
The first scheme corresponds to the familiar convention that
small fish should be protected, and large ones harvested. In this
case, fish enter the fishery at a chosen body mass and experience
a fixed fishing mortality rate at larger sizes (a size-at-entry
fishery). Second is the reverse scheme in which an upper limit to
harvested body sizes is chosen (in practice, there must also be a
lower limit of vulnerability, for convenience set here at 1 g). This
scheme is motivated by recent suggestions that there might be a
benefit to fisheries of protecting large fecund fish, sometimes
referred to as BOFFFFs (big old fat fecund female fish)
(Longhurst, 2002; Scott et al., 2006; Wright and Trippel, 2009;
Brunel, 2010; Hsieh et al., 2010). Third is a slot fishery in which
both lower and upper limits on body mass are chosen in
keeping with trap and gillnet fisheries, the upper limit here
being a constant multiple of the lower one. These three schemes
require a fixed fishing mortality rate F, together with a lower
limit wmin, or an upper limit wmax, or both, on the body sizes
harvested.
The fourth scheme makes exploitation proportional to sizedependent productivity, i.e. the production per unit time at each
body mass. Productivity at some body mass w is measured as
the total rate at which biomass flows through organisms of size
w in some volume of sea, here with units g m23 year21. It is measured as the product of: (i) the per capita rate at which prey mass is
converted into somatic predator mass, (ii) the density of organisms of that mass, and (iii) the body mass itself; see equation
(A20). To make fishing mortality proportional to productivity, a
fishing mortality rate F1 is chosen for the smallest harvested
body mass (1 g), and fishing mortality rates at larger body
masses multiply F1 by the productivities at the larger body
masses, relative to that at 1 g. The pattern of exploitation is therefore decided simply by the value of F1, together with the calculated
productivity. Note that productivity has to be measured in
context: how fast fish grow depends on the availability of food,
which in turn depends on how the ecosystem is exploited. We
use the unexploited ecosystem at steady state as our context; see
the “Discussion” section for more about this assumption.
YRD criteria
Yield Y is measured for an ecosystem at steady state under exploitation; this means that the yield is potentially sustainable in the long
term. (This is with the caveat that the steady state is only of interest
if the ecosystem comes to rest at the steady state, i.e. if it is an attractor.) There is always a steady state at which fish are absent, but
an interior steady state at which fish are present may or may not
exist, depending on how intense exploitation is. The interior
steady state is found by applying Newton’s method to the discretized version of the dynamical system (see Appendix).
Multiplying the steady-state densities by the corresponding body
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w0 ex0
w0 exm
w0 ex1
r
rm
On balanced exploitation of marine ecosystems: results from dynamic size spectra
21. This means that the conservation of biomass density in logarithmic mass intervals is not exactly fulfilled. Indeed, it should not
be expected to do so, because the conservation is thought to be a
property of a multispecies community, not of a single species
(Andersen and Beyer, 2006).
Productivity at steady state, the total rate at which mass passes
through the ecosystem at each body size, falls substantially as body
size increases (Figure 1c). We use this function below to set fishing
mortality proportional to productivity. The “bumps” in the productivity function come from nonlinearities in the size spectrum in
Figure 1b. This happens at very small body masses, and at body
masses around the size at maturation where somatic growth is
arrested causing a build-up in density of fish. If the conservation
of biomass was perfect, there would be an exact correspondence
between productivity and the relative growth rate (see the
“Discusssion” section). Despite the less than perfect conservation
in the model here, the resemblance between productivity and relative growth rate remains strong (Figure 1c). The steeper drop in
growth rate around 150 g comes from the slowing of somatic
growth that accompanies maturation.
The average growth trajectory (Figure 1d) is computed just from
feeding in the steady-state size spectrum, together with an assumption about asymptotic mass; see Equations (A16) and (A21). There
are no grounds, a priori, to expect this modelled growth to resemble
that observed in the sea. Figure 1d thus compares the growth trajectory with those of two species that grow over a similar size
range, a planktivore the herring (Clupea harengus), and an omnivore the Atlantic mackerel (Scomber scombrus). We obtained the
growth trajectories of these species from the von Bertalanffy
growth equation, with parameters taken from Fishbase for
herring (Froese and Pauly, 2000), and from Santos et al. (2002)
and Villamor et al. (2004) for mackerel. The growth trajectory in
the size-spectrum model is quite similar to the growth of these
species (Figure 1d), suggesting that the modelled feeding is
similar to that observed in the sea. The growth of the fish is accompanied by a major decline in predation, the death rate from predation falling by over an order of magnitude (Figure 1d).
In sum, the unexploited ecosystem goes from a highproductivity high-mortality regime for fish that are young and
small (body mass ≈1 g), to a low-productivity low-mortality
regime for fish that are old and mature.
Yield
Results
Here, we consider four schemes of exploitation applied to the
system. In each case, the results are calculated for the system at
steady state under the prevailing pattern of fishing mortality.
Before embarking on this, it helps to understand some general
properties of the steady state in the absence of exploitation.
Unexploited steady-state ecosystem
Taking body mass as a measure of trophic level (Jennings et al.,
2001), the fish species feeds at several trophic levels, with
biomass density concentrated at small body size (and low
trophic level) (Figure 1a), giving a pyramid of biomass like that
of an ecosystem with discrete trophic levels. Thus, Figure 1a is
analogous to a conventional biomass pyramid for trophic levels,
laid on its side. The pyramid of numbers hugs the axes still
more closely, and is better depicted as the standard size spectrum
using the logarithm of body mass (Figure 1b). The relationship is
approximately (but not exactly) linear, with a slope greater than
Biomass yields at steady state are shown in Figure 2, covering a
range from no exploitation up to levels that cause collapse and extinction of the exploited species. In keeping with the high productivity of small young fish, the greatest yields at steady state are
obtained when these small fish are heavily exploited and large
fish are protected (Figure 2b).
There is however an obvious danger in high rates of exploitation of small fish (Figure 2b). Yields get larger as the upper
limit on body mass wmax increases, up to a threshold near the
body size at maturation, at which point the stock collapses.
There is no warning of this impending catastrophe as the upper
limit on body mass is increased. From this perspective, a fishery
with a minimum body size wmin is less risky (Figure 2a). A slot
fishery that limits the size range exploited is also less risky, as
long as this range is sufficiently small, although this is at the cost
of smaller yields (Figure 2c).
Exploitation proportional to productivity (Figure 1c) is also a
safer option than heavy fishing at a fixed rate on smaller fish.
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mass and fishing mortality rate and summing over body mass give
the yield (g m23 year21) as in Equation (A22).
Resilience R describes the tendency to (and speed of) return to
the interior (non-zero) steady state when the ecosystem is perturbed from the steady state by a small amount. Importantly,
the existence of a steady state does not necessarily mean that the
ecosystem will come to rest there: this requires in addition that
the steady state is an attractor (Datta et al., 2011; Plank and Law,
2011). Moreover, even if it is an attractor, small perturbations
from the steady state may decay quickly or slowly; larger perturbations may decay or may grow. In technical terms, the local asymptotic stability of the steady state is determined by the largest real
part l* of the eigenvalues (with units year21) of the operator linearized for small perturbations around the steady state. If l* . 0,
the steady state is not an attractor; if l* , 0, the steady state is an
attractor and the more negative l* is, the faster the return to the
steady state is. Unfortunately, l* does not give any indiation of
how far the system can be perturbed and still return to the
steady state in the long term. This is more difficult to quantify formally and we do not attempt to do so in this paper. Instead, we use
the quantity 2l* as the measure of resilience: the larger this is, the
more resilient the system is to (small) external perturbations.
Disruption D of size spectra is a natural consequence of their
exploitation. We measured disruption as a dimensionless deviation of the logarithm of the harvested steady state from logarithm
of the unharvested steady state at each log body mass. Summing
the absolute value of the deviation over all body masses gives a
scalar measure of the degree of disruption, as described in
Equation (A23). This is one of a number of ways in which disruption could be measured. We also considered a measure in which
the exploited spectrum was first rescaled so that its density at
the smallest body mass (egg mass) would be the same as in the unexploited spectrum; the surfaces had a shape similar to the ones
given here.
We caution that the model used here is not guaranteed to have
a unique, globally stable steady state. For instance, there may be
multiple interior steady states (Hartvig, 2011) or attracting
cycles (Law et al., 2009). A rigorous study of the existence and
type of attractors that are present in the system is beyond the
scope of this paper. However, it is worth noting that the YRD criteria could be calculated for alternative stable states or multiple
cycles if they were found to be present.
605
606
R. Law et al.
This is because the calibration of fishing mortality to productivity
puts the fishing mortality on an overall downward path with increasing body mass. Yield continues to increase with F1 until
very high levels of exploitation are reached, for the life history
investigated here (Figure 2d).
Resilience
Figure 3 shows the effect of fishing on resilience of the system,
measured as the tendency to return to the steady state l* following
small perturbations. In the absence of fishing, the steady state has
the property of local asymptotic stability, which means that small
perturbations from the steady state die out. Exploitation of small
fish has little effect on this (Figure 3b and c). Fishing large
mature individuals makes l* more negative, as does fishing at
moderate levels when smaller body sizes are included as well
(Figure 3a), thereby increasing resilience in the sense that the
system returns faster to the steady state.
However, there are some striking non-linearities in resilience
that stem from a collision between the size range of exploitation
and the size range over which fish become mature (Figure 3a and
c). Heavy exploitation that starts around the body mass at maturation is strongly destabilizing, giving rise to a “ridge” of high values
of l* in Figure 3a and c. For sufficiently high levels of F, this can
turn the steady state from an attractor into a repellor, generating
a qualitatively different class of dynamics with periodic behaviour
and the possibility of stock collapse (Datta et al., 2011; Plank and
Law, 2011). We think the reason for the apparent regime shift is
that the large increase in mortality at body sizes close to maturation
makes renewal disproportionately reliant on a narrow range of
body sizes. This has the effect of moving the stock away from its
natural iteroparous life history towards one of effective semelparity.
Semelparity decreases mixing across cohorts as long as individuals
mature at a similar size, and can lead to strong fluctuations (in a
seasonal environment, experienced as large variations in density
at size from one cohort to the next).
In contrast, when fishing is proportional to productivity, resilience continues to increase up to very high values of F1 (Figure 3d).
The productivity function (Figure 1c), and hence the fishing mortality rate, is always lower for large fish than for small fish,
although there is a small increase around the size of maturation.
There is no major increase in fishing mortality near the body
mass at maturation that could drive the stock towards semelparity.
Disruption
Figure 4 shows how four size-specific fishing patterns affect the
size spectrum at steady state. These Fs were chosen to give
similar biomass yields under entirely different patterns of
fishing: (Figure 4a) 0.10, (Figure 4c) 0.11, (Figure 4e) 0.11,
(Figure 4g) 0.10 g m23 year21. The key message is that heavy exploitation of adult fish, especially around the size of maturation,
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Figure 1. Properties of an unexploited system at steady state. (a) Biomass pyramid. (b) Numbers pyramid, the size spectrum.
(c) Productivity (continuous line); relative growth rate of soma (dashed line). (d) Body mass (continuous line) and predation rate
(dotted line) as functions of age. The dashed line in (d) is the growth trajectory of herring with parameter values for the von Bertalanffy
b
growth equation: w(t) = a L1 1 − exp(k(t − t0 )) , with parameter values: k = 0.4 year−1 , L1 = 45 cm, and allometric parameters:
a = 0.0049 g cm−b , b = 3.089. Similarly, the dash-dot line in (d) is the growth trajectory of mackerel with parameter values for the von
Bertalanffy growth equation: k = 0.27y−1 , L1 = 42.7 cm, and allometric parameters: a=0.0064 g cm2b, b=3.079. In both cases, t0 sets the
egg mass at 1 mg at t ¼ 0.
On balanced exploitation of marine ecosystems: results from dynamic size spectra
607
Figure 3. Resilience of steady state under contrasting schemes of exploitation. Resilience is measured as -l* as defined in the text.
Fishing scenarios are set by the fishing mortality rate F and lower and upper limits on body masses in the harvest, as described in the legend of
Figure 2.
causes major disruption to the structure of size spectra (compare
Figure 4b and f with d and h). The structure is changed in two
ways: directly through fishing mortality on large fish, and indirectly
through the loss in renewal when these large fish are removed. In
contrast, fishing entirely (Figure 4d) or mostly (Figure 4h) on
small fish leaves the structure quite close to that of the unexpoited
ecosystem. Thus, in keeping with heuristic arguments (Garcia
et al., 2012), exploitation schemes that match productivity (here
the small fish are most productive) have a relatively small effect
on the shape of the size spectrum.
Figure 5 expands the results to describe deviations from the unexploited size spectrum over the full range of fishing schemes. As
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Figure 2. Yields at steady state under contrasting schemes of exploitation. Fishing scenarios are set by the fishing mortality rate F and lower
and upper limits (wmin, wmax) on body masses in the harvest. (a) All body masses from wmin onwards harvested. (b) All body masses from 1 g
up to wmax harvested. (c) Body masses over a range wmin to wmax harvested, where wmax = wmine 2. (d) Harvesting in proportion to
productivity; F1 sets the fishing mortality at the smallest harvested body mass 1 g; fishing mortality at larger body masses is scaled by the
productivity at these masses relative to that at 1 g, i.e. F = F1 P(log(w/w0 ))/P(log(1)), where w0 ¼ 1 g and P is given by Equation (20).
608
R. Law et al.
suggested by the examples in Figure 4, large deviations to the size
structure are caused by heavy fishing of adults. It is fishing on the
smallest adults that has the greatest effect (Figure 5c); we think
this is because, among the adults, these are the most abundant,
and also because they have the greatest expected future production
of eggs. In contrast, fishing concentrated on small individuals has
little effect on the size structure (Figure 5b and c). Fishing in proportion to productivity causes more disruption than fishing just
on small individuals, but F1 would have to be made very large to
achieve the disruption caused by removal of large individuals.
Sensitivity analysis
Table 2 shows the gradients of the YRD criteria with respect to the
parameters used, computed in the neighbourhood of the plankton
spectrum and fish life history for a size-at-entry fishery. Most
parameter values generate slopes for yield and resilience of an
order 0.1 or less. This provides evidence that the results on yield
and resilience are robust to changes in most model parameters.
Exceptions to this are the slope of the plankton spectrum g and
the conversion efficiency K: these play an important role in
moving mass from plankton to fish. Also, the parameters b and
s of the feeding kernel have somewhat greater effects on resilience,
in keepng with previous analyses of stability of size spectra (Plank
and Law, 2011). The search-rate scaling exponent a is notable for
causing larger changes; a value for a of 0.8 is widely used in
modelling the dynamics of size spectra (Benoı̂t and Rochet,
2004; Andersen and Beyer, 2006; Blanchard et al., 2009) based
on calculations on how cruising speed of fish scales with body
mass (Ware, 1978). Disruption to the size spectrum caused by
fishing is more sensitive to parameters than yield and resilience.
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Figure 4. Examples of steady-state size spectra arising from contrasting patterns of exploitation. Fishing mortality rates F in first column.
Corresponding size spectra in second column (continuous lines) compared with the size spectrum of the unexploited ecosystem (dashed line).
(a and b) Fishing over a body mass range 100 g to the maximum fish mass. (c and d) Fishing over a range 1 to 18 g. (e and f) Fishing over a
range 40– 250 g. (g and h) Fishery with F proportional to productivity in the unexploited ecosystem. Deviations from unexploited steady state
are: (b) 3.8, (d) 0.29, (f) 2.0, and (h) 0.57.
On balanced exploitation of marine ecosystems: results from dynamic size spectra
609
Table 2. Sensitivity analysis: rate of change in YRD measures
corresponding to small changes in the value of each parameter,
for a size-at-entry fishery with wmin = 60 g and F ¼ 1 year21.
Parameter
x p,max
xm
x1
rm
g
K
a
A∗
b
s
mi,0
xs
js
Yield
0.0625
20.0104
0.00964
20.000602
22.06
1.66
0.958
0.0000468
0.110
20.00925
20.0118
0.000224
20.0000045
Resilience
20.0164
0.000561
20.149
0.00966
2.60
20.589
22.98
20.000122
0.301
20.651
20.0180
20.00145
0.0000306
Disruption
21.92
3.76
21.37
0.0810
63.3
266.1
2137
20.00415
0.559
23.59
0.534
20.331
20.0143
A* is an aggregated parameter, A w0 a u0,p , used in the numerical analysis.
Discussion
These results from analysis of dynamical size spectra call into question the basic principle of protecting the small and killing the large
fish. Targetting large body sizes takes the catch from a part of the
ecosystem in which productivity is relatively low, and gives a correspondingly smaller sustainable biomass yield. Targetting large
body sizes can also reduce resilience, and cause a basic shift in
the dynamical regime away from a steady state, if fishing mortality
rates are high enough. Targetting large body sizes can also cause
major disruption to the size structure of the ecosystem.
Yet, in practice, fisheries are almost always regulated to protect
small fish, with a minimum body size for exploitation. In keeping
with this, there is a widespread trend towards truncation of age
and size structure in exploited fish stocks over time (Hsieh et al.,
2010), and a trend towards more negative slopes of size spectra
has also been observed (Rice and Gislason, 1996; Blanchard
et al., 2005). The results here are consistent with these
observations. The effects of truncation on resilience are harder
to document, but (Hsieh et al. (2006) reported greater temporal
variability in exploited than in unexploited fish species in the
California Current System which they attributed to truncation of
the age distribution. Also Rochet and Benoı̂t (2011), running
simulations of a biomass size-spectrum model which often
showed unstable behaviour, noted that oscillations in the size spectrum increased in amplitude when fishing was made more size selective or when larger fish were targetted (or both). Results from
dynamic size spectra given here support the argument that exploitation of adults is destabilizing, if such exploitation is heavy and
starts close to the size at maturation.
In recent years, there has been increasing interest in regulations
designed to protect large fish (BOFFFFs), as well as to protect small
ones. Slot fisheries with maximum as well as minimum harvested
sizes include, for example, the Maine lobster fishery, sturgeon fisheries on the west coast of North America, and Nile perch in Lake
Victoria (Kolding et al., 2008). There are several reasons for the
current interest in protecting BOFFFFs. First, maternal age and
size are often positively correlated with egg quality and larval survival in marine fish species (Hsieh et al., 2010). Second, a pool of
large fish helps to buffer stocks against years of poor recruitment
and should lead to less variability in year-class strength (Hsieh
et al., 2006). Third, the selection pressures generated by such
fishing may have less deleterious evolutionary effects on body
growth in the long term (Law, 2007). The results here show that
the yields are relatively small, although this is a relatively safe
pattern of exploitation if focused on small individuals, in that
the system remains resilient with little disruption to the size
structure.
A strong reason for protecting the small and immature, and
killing the large and mature fish is that it is risk averse: it
reduces the risk of inadvertently breaking the path from egg to
adulthood by overexploitation of small fish. The benefits of the
“spawn-at-least-once principle” are supported by recent empirical
results showing that stocks are deleteriously affected by a large
Downloaded from http://icesjms.oxfordjournals.org/ at University of Canterbury on June 10, 2012
Figure 5. Effects of harvesting on size structure of size spectra at steady state. Deviations describe the difference between exploited size
spectra and the unexploited one as described in the text. Fishing scenarios are set by the fishing mortality rate F and lower and upper limits on
body masses in the harvest, as described in the legend of Figure 2.
610
instance, feeding behaviour in which prey is made smaller relative
to the predator (larger b) and/or in which predators feed on a narrower range of prey sizes (smaller s), would shift the dynamics
from an attracting steady state to wave-like solutions moving
along the size spectrum from small to large body sizes (Plank
and Law, 2011; Rochet et al., 2011). Also the results are contingent
on there being an appropriate rate of survival from egg to maturation. This means that they are sensitive to how the upper boundary of the plankton body mass and the body mass at maturation
are related. We believe that the basic advantages of balanced harvesting found here will be shown to apply widely, but there is
clearly a need for more research into this new approach to
harvesting.
Among the limitations of our analysis that need further investigation, are the following. Marine ecosystems usually contain
many species; linking fishing to productivity should allow for
the different productivities of the species, in addition to different
productivities of different body masses in a single species.
Predation by other species that grow to larger body sizes would
have the additional effect of reducing the yield. Also plankton
have their own potentially complex dynamics, including seasonal
dependence, which we have not investigated. A further important
point is that, in some parts of the world, there is more to harvesting than simply generating biomass; ultimately the fishing industry
and the market-place can have a major input into what species and
sizes of fish are caught. It is also important to keep in mind that
productivity has to be measured in context, and this context
includes the pattern of fishing itself. The productivity of an unfished stock, the yardstick we have used, would not normally be
available. At the same time, productivity measured in an ecosystem
already greatly disrupted by fishing could give a misleading impression about the level of exploitation the ecosystem could
support. What state a marine ecosystem should be in, when
linking harvesting to productivity, is a question that needs to be
resolved. In the longer term, patterns of exploitation act as
selective forces on life histories of fish stocks, and some of
the parameters may themselves undergo change through
fisheries-induced evolution.
While bearing in mind these caveats, the basic principle that
catching small fish is bad, and that catching large fish is good, is
not supported by our analysis. At a time of so much concern
about the high levels of discarding needed to maintain selective
exploitation, it is surely useful to consider alternatives. Our
study supports the view of Garcia et al., (2012) that exploitation
more in balance with the natural productivities of components
of marine ecosystems has merit and is worth investigating further.
Acknowledgements
This paper grew out of discussions at a workshop on “Selective
fishing and balanced harvest in relation to fisheries and ecosystem
sustainability” organized by the IUCN-CEM Fisheries Expert
Group (FEG); we thank the participants for their input. RL and
MJP were supported by RSNZ Marsden grant number
08-UOC-034. We thank M. Canales, S. Datta, J. W. Pitchford,
and S. Zhou and three referees for helpful comments on the
manuscript.
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Downloaded from http://icesjms.oxfordjournals.org/ at University of Canterbury on June 10, 2012
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spectrum, with the additional feature that the bookkeeping of reproduction is explicitly included. The abundance w(w) of organisms of mass w is then:
∂
f(w) =
∂t
Appendix
(A1)
This equation implicitly defines the mass of a predator before
growth (w 2) as a function of its mass after growth (w) and the
mass of its prey (w′ ); we denote this by w− (w, w′ ). The value of
w 2 satisfying Equation (A1) is unique if the right-hand side of
Equation (A1) is monotonic with respect to w 2. Differentiating
with respect to w 2 shows that a sufficient condition for uniqueness is that
dE
. −1/K,
dw−
(A2)
for all values of w 2.
Jump-growth size-spectrum model and diffusion approximation
We follow the approach of Datta et al. (2010), writing down a
jump-growth equation to describe the dynamics of a size
Table A1. Summary of model variables used in the Appendix.
Variable
w(w)
Unit
g21 m23
T(w1 , w2 )
m3 year21
s(r)
–
E(w)
–
b(w)
Rb
x = ln(w/w0 )
u(x) = w0 ex w(w)
up (x)
g(x)
g21
g m−3 year−1
–
m−3
m−3
year−1
G(x)
year−1
m(x)
P(x)
year−1
g m−3 year−1
death due to predation
Comments
Abundance per unit mass per unit
volume
Interaction rate of predators of mass
w1 with prey of mass w2
Preference of predators for prey
with PPMR r
Proportion of biomass intake used
for somatic growth
Density of eggs of size w
Total reproduction rate
Log body mass
Scaled consumer abundance
Scaled plankton abundance
Biomass intake rate relative to body
mass
Body mass diffusion rate relative to
body mass
Death rate
Productivity
+ T(w− (w, w′ ), w′ )f(w− (w, w′ ))f(w′ )
dw′
reproduction
non−predation mortality
b(w)Rb
+
− mn (w)f(w).
w
(A3)
Here, T(w1 , w2 ) is an interaction kernel corresponding to the
feeding rate of predators of mass w1 on prey of mass w2, and
mn (w) is the per capita rate of mortality at mass w due to all
factors other than predation. Rb is the total rate at which reproductive biomass is made, obtained by integrating the contribution
to reproduction from all masses:
Rb = K (1 − E(w))f(w) T(w, w′ )f(w′ )w′ dw′ dw.
(A4)
The mass of an egg is distributed according to some birth kernel
b(w), normalized to sum to 1; this function could, for instance,
be a Dirac-d function corresponding to a single egg size.
The jump-growth equation (A3) is not numerically tractable
for realistic parameter values (Datta et al., 2010; 2011). We
begin by noting that when a predator of mass w− consumes a
prey item of mass w′ , its mass increases by:
Dw = Kw′ E(w− )
(Dw)2 E′′ (w)
+ ... .
= Kw′ E(w) − DwE′ (w) +
2
(A5)
Using this forumla recursively and neglecting terms of order
K 3 w′ 3 and higher gives
Dw = Kw′ E(w) − K 2 w′ 2 E(w)E′ (w) + O(K 3 ).
(A6)
−
We now use this expression to substitute w − Dw for w in equation (A3) and hence obtain a Taylor expansion of Equation (A3):
∂
f(w) = − T(w′ , w)f(w)f(w′ ) dw′
∂t
∂
T(w, w′ )f(w)f(w′ )w′ dw′
− KE(w)
∂w
K2 ∂
∂
2
E(w)2
T(w, w′ )f(w)f(w′ ) w′ dw′
+
∂w
2 ∂w
+
b(w)Rb
− mn (w)f(w).
w
(A7)
This second-order approximation with a diffusion term is used
rather than the more widely used first-order approximation (the
McKendrick –von Foerster equation), because of concern that
the first-order approximation may not provide a reliable indicator
of stability (Datta et al., 2010; 2011).
Downloaded from http://icesjms.oxfordjournals.org/ at University of Canterbury on June 10, 2012
The core of a dynamic size-spectrum model is careful bookkeeping
of biomass as it passes from prey mass into predator mass,
assumed here to take place with efficiency K; we set K = 0.2 for
numerical work. (A summary of variables used in the Appendix
is given in Table A1.) The model here applies this bookkeeping
both to somatic growth and to reproduction of the predator;
reproduction uses incoming mass rather than capital (Houston
et al., 2007; Stephens et al., 2009). We assume simply that a proportion KE(w 2) of prey mass goes to somatic growth of the predator, and a proportion K(1 – E(w 2)) goes to reproduction. Note
that the allocation is set by the body mass w− of the predator
before it eats its prey. Hence, a predator of mass w 2 that eats a
prey of mass w′ , grows to mass
w−
growth to larger size
− T(w, w′ )f(w)f(w′ ) − T(w′ , w)f(w′ )f(w)
growth from smaller size
Allocation of mass to growth and reproduction
w = w− + KE(w− )w′ .
613
On balanced exploitation of marine ecosystems: results from dynamic size spectra
It is realistic to impose some structure on the kernel (Benoı̂t
and Rochet, 2004; Andersen and Beyer, 2006), allowing the
volume V searched per unit time to scale with predator mass,
V = Awa , and making the feeding rate a function s of the
predator-to-prey mass ratio (PPMR) (Ursin, 1973). Thus
w
1
(ln(w/w′ ) − b)2
.
s ′ = √ exp −
w
2s2
2ps
(A9)
For numerics, we set b = 5, and s = 2. This value of b is similar to
a value recorded from herring, but the value of s is substantially
greater (Blanchard, 2008). These values give an ecosystem with a
stable steady state in the absence of exploitation.
We make a standard transformation of mass to a logarithmic
scale x = ln(w/w0), where w0, is an arbitrary mass (Benoı̂t and
Rochet, 2004), taken here to be that of a fish egg. To simplify
the subsequent analysis, we in introduce a transformed density
function u(x,t) with dimensions L23, where u(x,t)dx = w (w,t)dw
[similarly b̂(x) dx = b(w) dw], with the transformed parameter
 = A w0 a . Equation (A7) then becomes
∂
∂ u(x) = − m(x) + mn (x) u(x) − E(x)
g(x)u(x)
∂t
∂x
∂
∂
+
E(x)2 e−x (G(x)u(x)) + R̂b e−x b̂(x).
∂x
∂x
(A10)
R̂b =
for x p,min ≤ x , x p,max .
(A15)
In Equations (A12) and (A13), the prey abundance u(x′ ) is now
replaced by the sum of the consumer abundance and the plankton
abundance, u(x′ ) + up (x′ ).
The range of plankton extends to sufficiently small body masses
x p,min for there to be suitable food for newly born larvae, and to a
mass that can be greater than that of fish eggs, i.e. xp,max .x0. The
parameter sets the overall abundance of plankton. From an interpolation of Figure 2c in San Martin et al. (2006), we chose a value
100 m23 at x0.
As with most recent size spectra models (e.g. Blanchard et al.,
2009; Hartvig et al., 2011; Rochet et al., 2011), there are several
sources of discontinuity in the model. For instance, all eggs have
the same size x0, the plankton resource spectrum is truncated at
x = xp,max and the feeding kernel s is truncated, so that predators
are never smaller than their prey. It would be possible to introduce
smoothing functions to remove these discontinuities, but the
choice of these smoothing functions would be arbitrary and the
substantial increase in model complexity would not be justified.
Furthermore, it should be noted that the size spectrum and the
functions defined in Equations (A11) –(A13) are smooth away
from the boundaries and hence the partial differential equation
(A10) is well posed.
We adopt a function for reproduction similar to that used by
Hartvig et al. (2011), E(x) being defined by
−1 r(x−x1 ).
1 − E(x) = 1 + exp −rm (x − xm )
e
(A11)
(A12)
ÂK 2 (a−1)x 2x′ x−x′
e s(e )u(x′ ) dx′
e
2
(A13)
Rb
= K (1 − E(x))u(x)g(x)ex dx,
w0
(A14)
G(x) =
up (x) = u0,p e(1−g)x ,
Reproduction
All the calculations in this paper use this log-transformed equation. The birth kernel b̂(x) is assumed to be a Dirac-d function:
b̂(x) = d(x − x0 ), which means that all eggs are the same log mass
x0 . The other functions in Equation (10A) are:
′
′
m(x) = Â eax s(ex −x )u(x′ ) dx′
′
′
g(x) = ÂKe(a−1)x ex s(ex−x )u(x′ ) dx′
In line with some previous models (e.g. Law et al., 2009; Datta
et al., 2010), we make a simplifying assumption that there is a
fixed spectrum of plankton, with abundance given by a power-law
function of mass with exponent -g:
with biological meanings as follows. m(x) is the average death rate
due to predation for organisms of size w0 ex . g(x) is the average rate
of biomass intake for a predator of size w0 ex , expressed as a proportion of predator biomass. G(x) has a less intuitive meaning,
but is effectively a coefficient of diffusion of abundance with
respect to body size, which arises from the demographic stochasticity in predators’ growth rates contained in Equation (A3).
(A16)
The first factor in this expression is a maturity ogive, i.e. the proportion of organisms of that body mass that have reached reproductive maturity. The parameters in this are: xm defining
midpoint of maturity (i.e. the body mass at which 50% of organisms are mature), and rm controlling the width of the transition
from immaturity to maturity (smaller values of rm mean that
the transition to maturity is spread over a wider range of body
masses).
The second factor in Equation (A16) is the proportion of
biomass intake allocated to reproduction by a mature organism.
This is assumed to be an exponentially increasing function of x
(i.e. a power-law relationship with body mass w). The value of x
where this proportion reaches 1 defines a maximum size x1 to
which a mature organism can grow: once it reaches this size, all
biomass intake is allocated to reproduction and somatic growth
ceases. The reason for choosing this second factor to be an exponential function is that, in the case where u(x) / e(1−g)x , the
biomass intake function g(x) / e(a+1−g)x and so the fecundity of
a mature organism is proportional to e(a+1−g)x er(x−x1 ) . If r is
chosen to be equal to g − a − 1, then the fecundity of a mature
organism will be approximately constant with respect to body
mass. Since g is known empirically to take values close to 2, and
we assume that a = 0.8, this motivates a choice of r of 0.2.
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w
T(w, w′ ) = Awa s ′ ,
(A8)
w
where the function s is referred to as the feeding kernel. A value for
a in the neighbourhood of 0.8 can be derived from the literature
(Ware, 1978). Typically a predator is larger than its prey and
we therefore assume that s(w/w′ ) is zero for w , w′ . Following
(Ursin, 1973; Andersen and Beyer, 2006; Blanchard et al., 2009;
Hartvig et al., 2011), the feeding kernel is assumed to be a
Gaussian with respect to the ln(PPMR), having a mean b and
standard deviation s:
Plankton spectrum
614
R. Law et al.
For the parameter values chosen, the reproduction function
(A16) comfortably satisfies the constraint (A2).
Non-predation mortality
In addition to mortality from predation, there are three further
causes of death: background intrinsic death at rate mi (x), senescent
death at large body sizes at rate ms (x), and death caused by fishing
at rate mf (x):
mn (x) = mi (x) + ms (x) + mf (x).
(A17)
Intrinsic mortality at all body sizes takes into account the likelihood of natural causes of death other than predation. Following
the literature on scaling of mortality with body mass, we use a
function of the form:
(A18)
We assume that predation is the main cause of death and make the
intrinsic death small compared with predation, setting mi,0 = 1.0
year−1 at the egg size x0, and use an exponent ji = 0.25 (Brown
et al., 2004). Senescent mortality assumes that organisms above
a certain body size xs are subject to further risk of mortality that
increases exponentially with x:
ms (x) =
ms,0 ejs (x−xs ) ,
0
x . xs
,
otherwise
(A19)
where ms,0 = mi,0 e−ji xs . We use a large exponent, js = 5.0, to
prevent a build-up in density near the maximum body mass x1 ,
which would otherwise occur because of the low predation mortality near the asymptotic mass. In a real community, there would
typically be other species with larger body masses that would continue to act as predators, and such a build-up in density would not
be expected.
Fishing mortality depends on the scheme of fishing, and is
described in the “Methods” section. Productivity P(x) as a function of body mass x is needed as a precursor to make fishing mortality proportional to productivity, and is defined as:
P(x) = w0 ex E(x)g(x)u(x);
(A20)
here both g(x) and u(x) are measured in an unexploited ecosystem
at steady state. Note that E(x)g(x) describes the rate of incoming
mass to somatic growth, i.e. after it has been partitioned
between growth and reproduction.
Growth trajectory
To obtain growth trajectories, we solved the differential equation:
dx
= E(x)g(x).
dt
Numerical calculations
Numerical calculations were carried out by discretizing Equation
(A10) with respect to x. The convective derivative was approximated by a forward difference approximation and the diffusive derivative by a central difference approximation. The step size used
for discretization was dx = 0.05 and results were checked for sensitivity to reducing dx.
For a chosen rate of fishing mortality mf (x), we obtained the interior steady state of Equation (A10) ûf (x), i.e. the steady state at
which the fish population has a positive density. This was done iteratively using the Newton– Raphson algorithm (as in, for
example, Law et al., 2009), where possible preceded by a numerical
integration of (A10) to get close to the steady state. If there was no
interior steady state, we proceeded no further with this pattern of
fishing. If there was an interior steady state, the biomass yield Y at
steady state is
Y = w0 ex mf (x) ûf (x) dx
(A22)
The Newton–Raphson algorithm also returns the Jacobian matrix
of the discretized system at steady state. The eigenvalue of the
Jacobian with largest real part l∗ gives a measure of the local
asymptotic stability (resilience) of the steady state.
To measure the disruption of the steady state from the unexploited ecosystem, we obtained a dimensionless deviation of the
logarithm of the harvested steady state ûf (x) from logarithm of
the unharvested steady state û(x) at each log body mass x, then
summed the absolute value over all body masses to get a total
deviation Du:
Du = log ûf (x) − log û(x) dx.
(A23)
(A21)
Handling editor: Verena Trenkel
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mi (x) = mi,0 e−ji x .
Equation (A3) is the continuum limit of a stochastic, individualbased model of growth due to predation (Datta et al., 2010). In
the individual-based model, demographic stochasticity means
that different individuals that are initially in the same size
cohort will encounter different prey items and will, therefore,
grow at different rates. As a result, the distribution of body sizes
in the cohort spreads out over time and there is no single, welldefined growth trajectory. However, previous numerical results indicate that the trajectory defined by the mean growth rate, as in
Equation (A21), gives a good approximation to the mean body
size of a cohort (Law et al., 2009; Datta et al., 2010).