ICES C.M. 1994
C.M. 19941R:2
BLOOM DYNAMICS IN MARINE FOOD CHAIN MODELS WITH MIGRATION
by
William Silvert
Habitat Ecology Division
Biological Sciences Branch
Department of Fisheries and Oceans
Bedford Institute of Oceanography
P.O. Box 1006
Dartmouth, Nova Scotia B2Y 4A2
Canada
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Email:[email protected]
ABSTRACT
Bioenergetic models of marine food chains exhibit dramatic dynamical responses to
changes in primary production, and limnological experiments suggest that these results reflect real
properties of aquatic food webs. The predicted effects are due to growth and mortality of
planktonic grazers, but for fish and higher trophic levels with generation times of several years
the population response is small over annual cycles. In marine ecosystems the abundance of top
predators is more likely to be governed by migration, which suggests that open marine systems
may show qualitatively different responses to seasonal changes than closed lakes. The analysis
of food chain models casts light on the relative importance of food and predation as controlling
mechanisms for zooplankton populations as weIl as emphasizing the possible role of migratory
predators. Under suitable circumstances it may be possible for opportunistic rnigratory fish to
respond to plankton blooms strongly enough to control the bloom dynamics.
Intrpduction
Studies of the dynamic structure of marine food webs are complicated by patterns of
seasonal succession and interannual variability. Highly aggregated trophodynamic models offer
a way to investigate general patterns of population dynamics in a way that can serve to identify
basic control mechanisms, although aggregation also means that particular interspecific
relationships are not adequately described.
An aggregated food chain model is used to investigate the dynamics of marine ecosystems
and their response to changes in primary production. The population dynamics of plankton are
extremely important, but the characteristic time scales of fish and other nekton are so long that
they have little impact on bloom dynamics. For these larger and slower-growing spccies it
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appears that migratory behaviour may bc- most import~t in determining their dynamical role in
the food chain.
Qmdi(ative Responses
(0
Blooms
Earlier studies of energy flow in pelagic marine ecosystems based on size-structured
models described several qualitatively different types of response of higher trophic levels to
changes in primary production (Silvert and Platt 1978; 1980). Because most pelagic predators
are larger than their prey, energy flows from the primary producers, single-celled algae, up
through particles of increasing size by a combination of predation and growth. In the original
foml of the Silvert and Platt (1978) model the flow of energy with time was smooth and regular,
as shown in Figure 1. However, indusion of nonlinear effects coupled to very different time
scales led to a later conjecture that under certain circumstances the energy could accumulate in
intermediate biomasses as is shown in Figure 2. In this later version of the model (Silvert and
Platt 1980) the difference in time scales is crucial. The high grazing rates and fast generation
time means that algal blooms can rapidly be consumed by rapidly growing zooplankton
populations, and thus the energy of primary production can be quickly transferred into a bloom
of zooplanktonic biomass. But higher carnivores that graze on these populations have much
longer generation times and cannot respond rapidly enough to bloom conditions to pass much of
the energy further up the food chain, so biomass accumulates in the absence of an adequate
removal mechanism.
Mathematically this process is analogous to water waves arriving at a sloping beach. The
speed of waves is inversely related to water depth, so energy flows in from the sea faster than
it can be transported by inshore waves, leading to dramatic instabilities associated with large
amplitude wave crests which break. But unlike sloping beaches, processes can occur in
ecosystems which can provide alternate mechanisms for dissipating the energy of plankton
blooms. One is the occurrence of planktonic carnivores with very fast population response times
which can bloom just as fast as the smaller zooplankton on which they fee<!; many gelatinous
zooplankton behave in this way. Another type of ecosysterri response is migration. Many fish
and other nektonic predators are able to horne in on concentrations of prey species either on an
opportunistie basis or as the result of seasonal migrations tied to the population dynamics of their
prey.
Size-Struc(ured Food Chain Model
Marine ecosystems are characterized by a strong correlation between size and trophic level
(Kerr 1974). Sizc is also highly correlated with metabolie rates, so the parameters for a sizestructured food chain model can be derived from allometric relationships (platt and Silvert 1981).
If we let Bi represent the biomass of organisms at the ith trophic level, a simple
representation of a food chain is given by the equation
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(1a)
where 1tj is the grazing coefficient for trophic level i on level i-I, E is the ecological
efficiency (assumed constant for all trophic levels), and Pi is a natural mortality term
(Thingstad and Sakshaug 1990).
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Data from Ware (1978) indicate that the predator/prey weight ratio in marine ecosystems
is approximately (0.07)"3';' 3000. Fenchel (1987) suggests 1000. These numbers fall wellwithin
therange recently reported by Hansen et al. (1994), and the size-efficiency hypothesis (Breoks
and Dodson 1965) suggests that the ratio may respond to changes in predation pressure at the
upper trophic levels. The predation coefficient 7t varies by a factor of roughly 5 between trophic
levels, and J.l by a factor of about 10 (Silvert and Platt 1980). This gives rise to an enormous
difference in time scales, as is actually the case in marine food chains which encompass
organisnis ranging from nanoplankton with tumover times on the order of hours to large fish and
marine mammals with tumover times measured in dccades (a decade is approximately 105 h).
Because of this range of &cales, equilibrium analysis of Equation 1 must be examined very
critically, since thc largc top predators take much longer to equilibnitc than do the lower trophic
levels.
Although the population response time of top predators may bc scveral years, the actual
response time as determined by changes in abundance is generally much shorter. Whereas
zooplankton populations usually grow in place, larger predators respond to changes in spatial
distributions of food and migrate to areas where it is plentiful. We can incorporate this into the
model by adding a migration term to the model. If we assurne that thc rate at which predators
move into a region is proportional to prey abundance, this corresponds to adding a term Mi Bi-l
where Mi is amigration coefficient; this gives
(1b)
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for the rate of change of the jth biomass. An interesting feature of this additional term is that
dB/dt can be positive even if Bi is Zero, meaning that migratory predators can appear where
they were not previously present.
There are obvious weakriesses in this model of migration; it includes immigration but not
emigration, and the reason for this is that we cannot fully describe migration without taking into
account thc spatial distribution of prey. What the above model does describe is simply the fact
that predators can appear in an arca much faster than thcir population dynamics would predict.
J::guilibrium Analysis
The differential equation mOdel representcd by Equation 1b describes the dynamical
response of a food chain, but befere investigating the dynamic properties it is informative to
examine the equilibrium solutions.
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. 1
The system of equations given by Equation 1b must be augmented by a suitable equation
for primary production. There are several ways of doing this, depending on how one introduces
nutrient and light limitation; for simplicity we use a modified logistic model of the form
(2)
where BI is the primary biomass, r is the intrinsic growth rate, and K is the carrying capacity
in the absence of grazers; K might be a measure of thc nutrient pool for examplc.
Formal solution of Equations 1 and 2 is straightforward. We obtain a substantial
simplification if we first ignore thc migration term by letting Mi ... O. Setting dB/dt - 0 for all
i givcs
r (1 - B/K) - ~ B2
...
0
(3)
from Equation 2 and
(4)
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far i > 0 from Equation la. Thc structure of the solutions is quite complex, even without the
migration terms, but fortunately the solutions are casy to derive. Equation 3 relates BI and
B 2 , but Equation 4 contains only B iol and Bj +l , with Bi cancelling out of the equilibrium
equation. Thc solutions are further constrained by the requirement that Bi cannot be negative.
A formal exploration of Equations 3 and 4 might lead to somc interesting mathematical
resuIts, but from a biological point of view solution is straightforward. Infinite food chains are
not found in nature, so we can assurne that there is a highest trophic level N such that Bi ... 0 if
i > N. From Equation 4 we have
(5)
since BN +I = 0, and by iterations of Equation 4 for i ... N-2, N-4, etc., we obtain aIl the odd or
cven biomasses, depending on whether N is cven or odd. \Vc can then usc Equation 3 to
relate BI and B 2 (knowing one of them wc can calculatc thc other), which enables us to
calculate the remaining biomasses by using Equation 4 again. A curious feature of this
solution is that Equation 5 comes from thc cquation dBidt ... 0, but BN is actually thc last
biomass that we calculate!
The cquilibrium solutions for this system displaya remarkable dependcnce on N. the
length of the food chain. For evcn values of N we obtain a biomass distribution like that shown
in Figure 3 which is relatively flat with a moderate degrec of structtuc, fairly typical of marine
ecosystems (Sheldon et al. 1972). In this case N = 6. and for other evcn values of N the
distribution is similar. Dut for odd values of N the situation is completely different, with very
large biomasscs at the odd trophic levels as shown in Figurc 4. Thc total biomass in this case
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is much larger than for any of the even N food chains. (These are of course very long food
chains and are used to make the pattern clearer. but the same pattern is seen for shorter chains.)
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Before attempting to interpret these results, it is important to recognize that the model
described by Equation 1 (hoth la and Ib) is an extreme simplification and should be viewed as
a qualitative. not quantitative model. In particular. the predation terms ni+i Bi+) Bi representing
the grazing rate on trophic level i by level i+ 1 does not take into account the functional response
of the grazers. If Bi is very large the grazers cannot process the high food concentration and thus
the model overestimates both the loss term for level i and, more importantly. the growth rate of
level i+1 (this is dealt with in the equilibrium model of Thingsmd arid Sakshaug, 1990). And
of course very few ecosystems are adequa.tely represented by simple food chains. Despite the
observatiori by Pimm (1980) and other theorists that "[omnivores] will not Oe frequent within a
[stable food] web". omnivory seems very common in real ecosystems (Silvert 1983) arid can be
explained by mechanisrris beyond the realm of simple Lotka-Volterra theory (Ohman and Runge
1994).
InterPretation
of Eguilibrium Rcsults
While the behaviour of the food chain model described 'above is mathematically
interesting, the important biological questions relate to whether this behaviour reflects the
dynamics of real marine ecosystems. There is at least some reason to believe that it dces.
From a strictly mathematical point of view the abrupt changes in structure between odd
änd even values of N are easily understood. Equations 1 and 2 have the form of a generalized
Lotka-Volterra system, with a self-limiting term -rB)2/K appearing in Equation 2. In the limit
K. ~ 00 this can be transformed into a strict Lotka-Volterra system for which is easy to prove
that there is no stable solution if N is odd (Gcel et al. 1971).
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Detailed examination of the solutions confirm that for odd values of N the biomasses are
vcry sensitive to the value of K and diverge in the limit K ~ 00. For even values of N the
systein is mathematically well-behaved no matter how large K iso This can be seen by looking
at the role of Equation 3 in thc two cases. For odd N wc begin by calculating BN _), BN _3•••• , B2~
and thus usc Equation 3 to calculatc B) as
(6a)
which shows that B) is proportional to K and thus divcrgcs as K ~ 00, indicating that a feod
chain with an odd number of levels could be subject to massive algal blooms. For even N
the calculation of BN_). BN_3, •••• B) leads to the equation
(6b)
so that iri the limit K ~
00
this simply approachcs r/~.
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In biological terms these results mean that an even number of trophic levels is required
for effective control of phytoplankton biomass. Limnological data are consistent with this,
although there are of course alternative explanations. For example, the experiments of Hurlbcrt
and coworkers on the introduction of mosquitofish to ponds showed that an increase in the
introduction of a new trophic level completely altered the size distribution, lowered the
zooplankton biomass, and increased phytoplankton counts by an order ofmagnitude (Hurlbert and
Mulla 1981).
It should bc emphasized that these bioenergetic factors may not be the only explanation
for the size structure of food chains. For example, DIBrien (1979) argues that the predominance
of small zooplankton in lakes with abundant planktivorous fish simply reflects that fact that small
prey are less likely to bc seen and consumed, which suggests that the small zooplankton should
be seen as part of the prey category one trophic level below the fish rather than as bcing two
trophic levels down. The situation is further complicated by possible changes in predator-prey
size ratios that can significantly affect the size structure of the plankton (Brooks and Dodson
1965, Hall et al. 1976).
Dynamical Behaviour oe Food Chain Models with Migration
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Equilibrium analysis is useful in that it identifies the asymptotic state towards which an
ecosystem evolves in time, but the situation for food chain models is complicated by the fact that
the wide range of time scales means that predators in upper trophic levels do not approach
equilibrium levels as rapidly as their prey. Indusion of migration alleviatcs some of the
problems of unrealistically slow rates of change; for example, the generation time of whales and
some large fish can approach decades, but as predators they appear seasonally and their local
abundances fluctuate far more rapidly than their populations.
Aseries of simulations have bcen carried out to explore the possible types of response
that might bc exhibited in real ecosystems. It is premature to say whether these responses
actually occur in aquatic ecosystems because of the difficulty in estimating the relevant
parameters, especially those describing migration. However, the purpose is to idcntify in a
qualitative way possible patterns in order to find out whether this type of model can shed some
light on at least the grass features of bloom dynamics.
The results of a typical simulation without migration are shown in Figure 5, which
reprcsents the response to a bloom-like increase in primary production. The initial peak in algal
biomass is transferred first to the herbivore population and then to a smaller peak in carnivores.
The more gradual slopes in the biomass curves for higher trophic levels reflect the slower
response times for larger organisms, and there is virtually no change in the biomass of the largest
organisms (i=4,5,6) on the' time scale of this simulation. This is consistent with the pattern
shown in Figure 2, where the energy flow into the plankton community encountcrs a bottlencck
bccause of the slow response time of fish and other large predators.
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By comparison, Figure 6 shows a simulation with migration taken irito account. The most
signific3.nt difference between this simulatiori and that shown in Figure 5 is the speed with which
the top predator responds to. abundarit food in lower trophic levels. Because of the migration
parameters usCd iri this simulation the response times of the top predators are comparable.
Sh!ruficance for Marine Ecosystems
It is difficult to test a modellike this directly in marine ecosystems. Limnologists work
mostly with relatively closed ecosystems, and it is relatively easy to modify feod chains and
iritroduce or remove trophie levels. Marine ecosysteins are more open, and the length of oceanic
feod chains is thus more likely to be regulated by biological dynamics rather than by physical
constrairits. Many ecosystems fall in between the two extremes, and even relaiively isolaied
lakes may contain riligratory predators like birds and anadromous fish .
.
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Massive algal blooms are common in lakes and reservoirs. These often appear as thc
resuIt of hyperriutrificcltion of ponds too small to support secondary carnivores (Le. N-3), often
leading to anoxie conditions (N-=l). They are not common in marine ecosystems except for short
periods of iriiense upwelling. It is unlikely that a food chain with an odd value of N could
persist for long in a marine ecosystem because the high biomass at the top trophic level, BN,
would be very anraciive io the large predators which foam the world's oceans. Thus, food chains
with odd values of N are inherently unstable iri open systems, not in a stdctly matherriatical sense
but in the sense of being vulnerable to invasion by rriigratory top predators. The apparent
constancy of biomass in the upper trophic levels for the simulations is misleading because the
abundance of top predators is generally ,dCtermined more by migration than by population
dynamics.
These results do not provide any universal answer to the question of what role fish and
other large predators play in controlling plankton blooms. Migration plays cl critical role in the
simulation results, and we suspect that it plays an equally important role in many marine
ecosystems, but it is far easier to estimate the parameter values for population dynamics than for
fish migration. There are certainly many cases in which algal blooms are followed by massive
zooplankton blooms, and there are also cases in which the zooplankton are grazed down by fish
or invertebrate carriivores which appear suddenly. Sometimes both situations cOcxist side by
side, as concentrated patches of planktivores depletc plankton densities only in local regions.
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The simulations indicate that the dynamical response of the system to an algal bloom is
aseries of Lotka-Volterra-like oscillations between phytoplankton and herbivores which are
controlled by the more gradual response of the primäry camivores. To the extent that sccondary
camivores respond on time scales consistent with allometrie rates, they represcnt a constant
mortality term and cannot control plankton blooms. Howevcr, these conclusions can be
invalidated by a riumber of factors, including both invertebrate camivores capable of very rapid
response to changes in food availability and to migratory invasions of migratoTy species.The
balance between these two effects plays a major role in determining the dynamics of plankton
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blooms. Under suitable circumstances a large migratory stock of fish can playadominant role
in regulating plankton densities. However, if migration is not a significant factor, perhaps
bec..use of largc-scale synchrony in plankton blooms, the grazing pressure cannot respond to
changes in plankton abundance and thus the control of plankton populations must depend either
on food limitation or on invertebrate predation.
The size of individual organisms plays an important role in determining the dynamics of
ecosystems, and models which assurne a fixed size for each trophic level are thus based on a very
restrictive assumption. It is possible to inc1ude a range of sizes within each trophic level in order
to consider parallel transfers of energy along separate food chains as described by Armstrong
(1994), but it is probably more important to deal with the dynamic changes in size structure due
to shifts bctwccn groups of organisms of different size (Sprules and Holtby 1979).
An important considcration which is suggested by this model but not inc1uded in the
model formulation is the variety of ways in which the ecosystem might respond to the changes
in trophic biomass. The equilibrium results for an odd number of trophic levels indicate that the
odd trophic levels have high biomasses with low tumover while the even levels have very low
biomasses with high tumover. The degree to which this can happen in areal ecosystem is of
course constrained by physiologicallimitations on grazing rates, but in a situation where a given
trophic level is subject to high grazing rates there would be selective pressures that could lead
to the replacement of species with long tumover times by species with fast tumover times. Thus
changes at the top of thc food chain can lead to major shifts in the relative biomasscs of fish and
zooplankton, with significant consequcnces for the species composition of both groups.
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References
Annstrong, R. A. 1994. Grazing limitation and nutrient limitation in marine ecosystems: steady
state solutions of an ecosystem model with multiple food chains. Limnol. Oceanogr.
39: 597-608.
Brooks, J. L., and S. 1. Dodson. 1965. Predation, body size, and composition of plankton.
Science 150: 28-35.
Fenchel, T. 1987. Ecology - Potentials and Limitations. Ecological Institute, Germany.
Goel, N. S., S. C. Maitra and E. W. Montrol!. 1971. On the Volterra and other nonlinear
models of interacting populations. Rev. Mod. Phys. 43: 231-276.
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Hall, D. J., S. T. Threlkeld, C. W. Bums, and P. H. Crowley. 1976. The size-efficiency
hypothesis and the size structure of zooplankton communities. Ann. Rev. Ecol. Syst.
7: 177-208.
Hansen, B., P. F. Bj0msen, and P. J. Hansen. 1994. The size ratio between planktonic predators
and their prey. Limnol. Oceanogr. 39: 395-403.
Hurlben, S. H., and M. S. Mulla. 1981. Impacts of mosquitofish (Gambusia affinis) predation
on plankton communities. Hydrobiologia 83: 125-151.
Kerr, S. R. 1974. Theory of Size Distribution in Ecological Communities. J. Fish. Res. Board
Can. 31: 1859-1862.
O'Brien, W. J. 1979. The predator-prey interaction of planktivorous fish and zooplankton. Am.
Scientist 67: 572-581.
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Ohman, M. D., and J. A. Runge. 1994. Sustained fecundity when phytoplankton resources are
in shon supply: omnivory by Calanus finmarchicus in the Gulf of St. Lawrence. Limnol.
Oceanogr. 39: 21-36.
Pimm, S. L. 1980. Propenies of food webs. Ecology 61: 219-225.
Platt, T., and W. Silvert. 1981. Ecology, physiology, allometry and dimensionality. J. Theor.
Bio!. 93: 855-860.
Silvert, W., and T. Platt. 1978. Energy Flux in the Pelagic Ecosystem: A Time-Dependent
Equation. Limnol. Oceangr. 23: 813-816.
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Silvert, W., and T. Platt. 1980. Dynamic cnergy-flow model of the particle size distribution in
pelagic ecosystems. In Evolution and Ecology 01 Zooplankton Communities, W. C.
Kerfoot, Ed. Univ. Press of New England, Hanover, N. H., pp. 754-763.
Silvert, W. 1983. Is dynamical ecosystems theory the best way to understand ecosystem
stability? In Population Biology, H. I. Freedman and C. Strobeck, Eds. Springer-Verlag
Lecture Notes in Mathematics 52, pp. 366-371.
Sprules, W. G., and L. B. Holtby. 1979. Body size and feeding ecology as alternatives to
taxonomy for the study of limnetic zooplankton community structure. J. Fish. Res. Board
Can.36: 1354-1363.
Thingstad, T. F., and E. Sakshaug. 1990. Control of phytoplankton growth in nutrient recycling
ecosystems. Theory and terminology. Mar. Ecol. Prog. Sero 63: 261-272.
Ware, D. M. 1978. Bioencrgetics of pelagic fish: theoretical change in swimming speed and
ration with body sizc. J. Fish. Res. Bd. Canada 35: 220-228.
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Figure Captions
Figurc 1. Propagation of energy through a size-structured marine ecosystem, based on a simple
linear model of continuous energy transfer; after Silvert and Platt (1978).
Figure 2. Propagation of energy through a size-structured marine ecosystem, with nonlinearities
and time-scale differences taken into account; after Silvert and Platt (1980).
Figurc 3. Equilibrium biomass distributions of a food chain with an cven number of trophic
levels, in this case 6.
Figure 4. Equilibrium biomass distributions of a food chain with an odd number of trophic
levels, in this case 5.
Figure 5. Dynamic response of a food chain to a sudden peak in primary production when
migration is not included in the model. The lower four trophic levels are labelIed.
Figure 6. Dynamic response of a food chain to a sudden peak in primary production when
migration is taken into account.
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·e
. Figure 1
After Silvert and Platt (1978)
250~-------------------------.
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start
200
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time 1
....... ......
150
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o
time 2
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50
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time 5
Figure 2
After Silvert and Platt (1980)
250
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200
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time 1
time 2
150
co
time 3
t~
0
h:l
100
time 4
timeS
50
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Size (Iogarithmic scale)
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Figura 3
100
90
•
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ro
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80
70
60
50
40
30
20
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10
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3.
4
Trophic Level
6
Figure 4
100
90
•
80
70
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60
E 50
0
Co
40
30
20
10
0
1
2
3
4
Trophic Level
5
6
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Figure 5
•
cn
~
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in
2
•
i=1
i=2
i=3
i=4
i=5
i=6
1
Time
Figure 6
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i=i
i=2
i=3
i=4
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•
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Time