1

Reconstructing trophic interactions as a tool for understanding and

managing ecosystems. Application to a shallow eutrophic lake

Running title: Trophic interactions and the management of a shallow lake

Antonio Bodini

Department of Environmental SciencesUniversity of Parma- Parco Area delle Scienze 33/A

43100 PARMA- Italy

e-mail: bo@dsa.unipr.itTel. +39-521-905614Fax +39-521-905402

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Abstract

Disturbance and human interventions on lake ecosystems often emerge as impacts in the food

web. The consequences of such impacts are difficult to anticipate solely by intuition because of

the complex interactions that arise in lake communities. Understanding how species interact and

how the structure of the interactions buffers or amplifies external impacts is a current focal point

of the ecology of lakes because of its profound practical implications. This paper shows how the

structure of a lake food web can be reconstructed by applying a procedure that uses the results

of an external intervention combined with a qualitative algorithm that of the loop analysis.

Attention focuses here on Lake Mosvatn (Norway), a moderately eutrophic body of water, which

was monitored for two years after a biomanipulation experiment. By reconstructing the main

interactions that are thought to form the food web of this lake it is shown how the effects of

biomanipulation are converted into a number of predictions that can shed light on the observed

patterns of abundance, and provide clues about consequences of external impacts. Emphasis in

this respect is given to nutrient enrichment, and management opportunities are discussed.

Key words: shallow lakes, biomanipulation, loop analysis, complex interactions, qualitative

predictions, lake management.

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Introduction

Human impacts and natural perturbations often take the form of food web alterations.

Species are added or removed, and disturbance may act on populations by changing one or more

parameters that govern their growth rate. The possibility to predict the effects of such impacts

is crucial for the management and conservation of ecosystems. Yet, populations often are the

targets of intentional manipulations. Here managers need to be able to anticipate the effects of

their actions, if they want to achieve the desired goals and avoiding unintended side effects

(Crowder et al. 1996).

The multiple reticulate connections that characterize ecosystem food webs create such

a complex scenario that establishing causal linkages is inherently difficult (Yodzis 1988,

Crowder et al. 1996, Polis and Strong 1996, Schmitz 1997) and this reflects on the ability to

make predictions. Lake ecosystems are not exceptions in this respect. The cascading trophic

interaction theory (Carpenter et al. 1985, Carpenter and Kitchell 1987) and the biomanipulation

idea (Shapiro and Wright 1984, Gophen 1990), although euristically useful, have a limited

potential for predictions. The number of cases in which interventions carried out in lakes

contradicted the general expectation of the cascade hypothesis in fact is noteworthy (Shapiro

1990, Carpenter and Kitchell 1993). The linear sequence of events that form the basis of the

cascade paradigm cannot always provide a correct grasp of how the diverse array of connections

in a lake affects the dynamics of the food web and its response to external impacts. In this

perspective elucidating the pattern of interactions in a lake becomes a major target of

investigation.

Detecting complex interactions requires tracking a signal generated by a change in some

component as it spreads to other parts of the community. So biomanipulation or enrichment

experiments, which are common practice in ecological research, can be used to unravel the

structure of the interactions in a lake. To discuss this opportunity I use here as empirical base the

results of a biomanipulation experiment conducted in Lake Mosvatn (Norway), which have been

previously published by Sanni and Waervagen (1990).

Lake Mosvatn, a shallow, moderately eutrophic lake, appears particularly suitable for this

kind of analysis because of the relative simplicity of its ecological network and the completeness

of observations conducted over a period of at least one year following biomanipulation. The

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model reconstruction is based upon the algorithm of a qualitative technique, loop analysis, under

the hypothesis of moving equilibrium (Puccia and Levins 1985). In moving equilibrium one

assumes that parameters such as mortality rate, feeding rate, and so forth, change slowly enough

for the variables to keep up with them. This approach might seem too restrictive to study natural

systems, as it is commonplace that in nature changes occur rapidly enough to prevent the

temporary establishment of any equilibrium. I have, however, elsewhere offered evidence of its

validity based on experimental observations (Bodini 1988), while in other cases this approach

was successfully applied to practical situations (Lane and Collins 1985).

The food web model obtained from the reconstruction yields qualitative predictions that

are used to shed light on the patterns of abundance observed in Lake Mosvatn as related to

biomanipulation. Also, by comparing model predictions with examples taken from the literature,

potential effects of selected impacts are discussed. Particular emphasis is given to nutrient

enrichment and management opportunities are considered.

Methods

Loop analysis

Loop analysis (Levins 1974, Puccia and Levins 1985) uses signed digraphs to represent

a network of interacting variables. System variables are depicted as nodes in the graph and each

connective between two nodes represents a nonzero coefficient of the community matrix. If the

connection from variable i to variable j is in the form of an arrow (circle head) the effect of i

upon j is said to be positive (negative), and a positive (negative) coefficient (a ) enters theji

community matrix. The diagonal terms of the community matrix are self-effect on the system

variables. In Figure 1 the correspondence between a community matrix and the related signed

digraph is given for a simple predator-prey system.

[FIGURE 1 GOES HERE]

Press perturbations (Bender et al.1984), also called inputs, may act on ecosystems by

changing one or more parameters in the growth rate of the variables. Effects of these parameter

changes may propagate beyond the direct target of the input because of the sequences of biotic

*xj

*c'

jik

*fi

*c× p k

ji × F (comp)n&k

Fn

*f*c

p (k)ij

*f*c

5

interactions that functionally link the components. Loop analysis identifies such pathways and

may tell one whether the equilibrium value of system variables is expected to increase, decrease

or remain the same following the perturbation. For any variable the new level can be calculated

by the loop formula

where c is the changing parameter, such as mortality, predation rate and so on; designates

whether the growth rate of the i-th variable is increasing (+), decreasing (-) or not changing (0)

because of the parameter change; is the pathway connecting the variable that undergoes

parameter change, x , with that whose equilibrium value is being calculated, x . The last factori j

of the numerator is the complementary feedback, while F indicates the overall feedback of then

system. Appendix I illustrates these concepts concisely, while Puccia and Levins (1985) discuss

them in detail.

Responses of abundances or biomass to parameter changes are usually arranged in a table

of predictions. The entries in a table denote variations expected in all the column variables when

parameter inputs affect row variables. Conventionally the calculation is done by considering

positive inputs, = +, those increasing the rate of change of variables. Consequences of

negative inputs can be obtained by simply reversing the signs in the table. A table of predictions

for the model shown in Figure 1 is given in Figure 2.

[FIGURE 2 GOES HERE]

Experimental evidence

In September 1987 the shallow (mean depth 2.1 m; max depth 3.2 m), eutrophic Lake

Mosvatn was treated with rotenone to eliminate planktivorous fish (mainly whitefish, Coregonus

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lavaretus, L.) (Sanni and Waervagen, 1990). Effects on other variables were observed during the

following two years. The time-weighted average summer concentration (May-August) of total

P, which was equal to 42 µg lt prior to biomanipulation, decreased, and reached a value of 29-1

µg lt in 1989. The concentration of particulate P was 34 µg lt in 1987 and 17 µg lt in 1989.-1 -1 -1

Dissolved phosphorus, the form that is readily available to primary producers, showed a different

trend: its value, equal to 8 µg lt in 1987, augmented to 12 µg lt in 1989. -1 -1

The biomanipulation had a very strong restructuring impact on the algal community. The

time-weighted average summer volume of cyanophyceae decreased from 1005 mm m to 7973 -3

mm m . Green algae augmented (from 2510 to 3493 mm m ), but such an increase was due to3 -3 3 -3

gelatinous coated species, while the edible component decreased. After treatment there was a

fivefold increase in the biomass of Daphnia galeata, the main grazer in Lake Mosvatn. Its time-

weighted summer mean increased from 60 to 318 µg DW lt . Before biomanipulation rotifers-1

were a significant part of the zooplankton community, but after fish elimination a notable

reduction of both biomass and number of individuals was observed for this group, which passed

from a mean value of 10,000 ind.lt to less than 4,000 ind.lt . This coincided with a higher level-1 -1

of invertebrate predators, mainly individuals of the species Cyclops abissorum. As for biomass

and number of individuals, however, this species remained well below the values reached by

rotifers and grazers. Macrophytes, mainly Potamogeton pusillus, responded to biomanipulation

by exhibiting a very strong growth, especially in the second year after the intervention, but the

authors did not provide quantification of this evidence.

These results form the empirical base upon which the structure of the interactions of Lake

Mosvatn will be reconstructed. According to experimental observations the key variables are

assumed to be: dissolved phosphorus (N=nutrients, in the model), edible green algae (EG), blue-

green algae (BG), inedible green algae (IA), macrophytes (M), grazers (G), rotifers (R), and

invertebrate predators (I). Planktivorous fish (PF) are also included in the model. This choice

depends completely on which variables were considered of interest by the authors who conducted

the experiment.

Table 1 summarizes the quantitative data concerning some of these variables as they are

provided by Sanni and Waervagen in their paper, which also describes experimental details

(sampling, chemical analysis and so forth) for the interested reader.

[TABLE 1 GOES HERE]

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Strategy of reconstruction

In food web modeling usually the ecological knowledge of the system is combined with

general principles such as the size-selective predation (Brooks and Dodson 1965) or the keystone

predator concept (Paine 1966) to postulate a rough skeleton that represents community-level

interactions. The core structure of Lake Mosvatn’s food web is built here using an inverse

approach that uses the existing empirical information about the effects of the biomanipulation

(Bodini 1998). Rotenone increased fish mortality, and loop analysis classifies this intervention

as a negative input on this population. Qualitative changes observed in the other key variables,

as described above, can be used to compile a row in a table of “observations”, that is realized

predictions, around which a signed-digraph for Lake Mosvatn can be assembled. Table 2

summarizes this information.

[TABLE 2 GOES HERE]

If a complete table of realized predictions were available, the associated graph could

easily be obtained through matrix inversion (Puccia and Levins 1985). However, only one row

of this hypothetical table is available. It is unlikely that more than one or two cases of known

parameter change occur in practical situations. To obtain a complete table of realized predictions

one should introduce an input into each variable in turn, while keeping the others under control,

a logistically unfeasible situation.

The procedure used in this exercise consists of tracing pathways that can be responsible

for the observed effects. Links forming such pathways are only of two types, arrows and circle

heads, and are accepted only if they meet the constraints imposed by the loop analysis algorithm.

Yet, as a pathway propagates its effect to the target only if its complementary feedback exists

(Appendix I), other links can be obtained by identifying the complementary feedbacks for the

pathways.

To compose the pathways and their complementary feedbacks the links are chosen

according to the ecological interactions that likely characterize the variables, as they are

documented in the literature. Hypotheses about new interactions eventually may take shape only

through the comparison of model predictions with the observation (see Puccia and Levins 1985

*f*c

8

for a detailed discussion about model validation).

Although ecological realism must guide the reconstruction to avoid absurd situations such

as, say, zooplankton that feeds upon dissolved nutrients or planktivorous fish that eat

macrophytes, in lake ecosystems many organisms show a wide spectrum of alimentary

preference. However they cannot efficiently combine different ways of energy intake (Oksanen

1991) and only their main diet is considered here. For example planktivorous fish may feed on

phytoplankton, but since they have been described as efficient predators of an abundant

population of large herbivores, likely their energy intake from phytoplankton, if any, should be

of minor importance.

The overall feedback, F , is assumed to be negative to reduce the number of unknownsn

during the reconstruction. This assumption is motivated by the fact that a positive overall

feedback would make the equilibrium unstable (although a negative F does not guarantee thatn

the equilibrium is stable, Puccia and Levins, 1985). At the end of the graph reconstruction,

however, the negative F must become a consequence of the loops formed by the links in then

model.

A table of prediction is conventionally constructed assuming positive inputs on system

variables. The biomanipulation, on the other hand, produced a negative input on planktivorous

fish. For the ease of reconstruction a positive input on the same variable is assumed; its expected

repercussions on the system variables are obtained by simply reversing the signs that appear in

Table 2. In fact the only thing that changes in the loop formula is the term that becomes

positive. Consequences of this conventional positive input on planktivores are summarized in

Table 3.

[TABLE 3 GOES HERE]

A core interaction model for Lake Mosvatn

The susceptibility of large-bodied grazers to fish predation is well known (Galbraith,

1967) and the expected diminution in the level of grazers can be attributed entirely to predation

(Sanni and Waervagen 1990). In pictorial terms this corresponds to a circle head link connecting

planktivores to grazers. The effect on edible green algae and cyanobacteria, both expected to

[PF&BG&BEG(BG)]

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increase, can be the result of the suppression of grazers by fish combined with the grazing

pressure that daphnids exert over phytoplankton in lakes. Two positive pathways emerge (see

Appendix I for sign calculation): .

The capability of large-bodied zooplankton to dampen blue-green algae has been

investigated by many authors with contrasting results (see the review by De Bernardi and

Giussani 1990). In many experimental studies and biomanipulations an effective grazing pressure

was observed, but concentration, size and shape of the algae strongly affected their interaction

with herbivores (Gliwicz 1990). Even different strains may stimulate differential response by

large zooplankton (Nizan et al. 1986).

Neither information about herbivores’ feeding activity on BG, nor details concerning

morphology and toxicity of these algae in Lake Mosvatn were provided by Sanni and Waervagen.

Other authors however showed that species that dominated the phytoplankton community of this

water body, namely Microcystis aeruginosa and Anabaena solitaria/spiroides, can be grazed by

large crustacean herbivores (Shoenberg and Carlsson 1984, De Bernardi and Giussani 1990,

Lyche et al. 1990; but see Claska and Gilbert 1998); so, hypothesizing a certain grazing pressure

on blue-green algae in Lake Mosvatn seems reasonable, in the understanding that additional

information should be gathered in this respect.

Table 3 shows that nutrients are expected to decrease. Phosphorus consumption by

phytoplankton, two negative links connecting EG and BG to N, yields two ecologically plausible

pathways. Inedible algae decrease. This may be obtained simply by considering IA as growing

on dissolved nutrients. The arrow connecting N to IA, combined with the links so far identified,

produces two pathways with negative effect, as it has to be according to Table 3.

As macrophytes are not fed upon by any consumer variable in the web, input to

planktivores may affect this component only through interactions involving nutrients and other

primary producers. Luxury nutrient uptake by macrophytes (VanDonk et al. 1989, Meijer et al.

1994) and inhibition of macrophytes growth by abundant phytoplankton (Scheffer et al. 1993)

seem worth discussing in this respect.

The former can be depicted as an arrow from nutrients to macrophytes. This link would

yield two acceptable pathways from planktivorous fish to M. However macrophytes absorb

nutrients from the sediment through their roots, and only when dissolved nutrients reach very

high concentrations in the water column they are exploited by the plants (Carignan and Kalff

1982). Since Lake Mosvatn was described as moderately eutrophic, nutrients trapped in the

[PF&BG&BEG(BG)&BM]

[PF&BG&BEG(BG)&BNµIA&BM]

[PF&BGµI]

[PF&BGµI&BR]

[PF&BI&BR]

[PF&BG&BEGµR] [PF&BG&BBGµR]

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sediment likely represented the main resource for macrophytes, and the contribution of dissolved

phosphorus might be negligible.

Phytoplankton concentration increases lake’s turbidity (Scheffer et al. 1993). When algae

attain high abundance, they may prevent the light to penetrate the water column, inhibiting the

growth of macrophytes. This action can be depicted as a negative direct link from the three algal

groups, and four pathways come out. Two of them produce an effect

that agrees with the expectation, whereas the paths tend to

increase the level of submerged plants. The overall effect of the four pathways is qualitatively

ambiguous. Since there is no ecological reason to discard the pathways that do not confirm the

expectation all the four paths will be maintained.

Predation by planktivores can reduce invertebrate predators (De Bernardi 1981, Porter,

1996). Also the pathway , which combines suppression of grazers by the fish with the

positive action of daphnids on invertebrate predators, may contribute to diminish I. In this latter

path the benefit provided by G to invertebrate predators relates to their feeding on grazers that

are smaller in size, such as younger individuals (Kerfoot 1977, Porter 1996).

A negative link depicts the suppression of rotifers by invertebrate predators, a key process

in lake food webs (Williamson and Gilbert 1980, Williamson, 1983) mentioned also by Sanni

and Waervagen to explain the effect of biomanipulation on rotifers. Input on PF has now two

possibilities to percolate to rotifers producing to the expected result: and

. Algae of small size sustain rotifers and an arrow from EG to R must be considered

in the food web. Yet, it is possible that small zooplankton gathers some food from small blue-

green algae (De Bernardi and Giussani 1990). Because of these two links, new connections with

positive effect on R arise: and . According to the ecology

of lake organisms these are the most plausible pathways that can produce the effects listed in

Table 3. They yield the partial skeleton depicted in Figure 3.

[FIGURE 3 GOES HERE]

A pathway with null complementary feedback produces no change in the abundance of

the target variable, that is the ending node of the path, because the numerator in the loop formula

becomes zero. Since there are no zeroes in Table 3, the level of all the variables changed because

of the biomanipulation; accordingly, every pathway in the model must have its own

[PF&BG&BEG&BN]

[PF&BG&BEG&BN]

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complementary feedback.

The path connecting planktivorous fish with nutrients through edible green algae

leaves out the subsystem [IA-M-BG-R-I], whose feedback must be negative

to confirm the sign of the path. A unique circuit that groups all these variables cannot be

identified, so disjunct loops must be combined to produce the complementary feedback. Inedible

algae may compete with blue-green algae for nutrients but these interactions do not contribute

to the feedback as N is on the path. Path and complementary feedback in fact, according to loop

analysis, must not share variables.

Rotifers and invertebrate predators do not interact directly with inedible algae. Nor it

seems that they act as intermediate nodes in circuits that return some effect to IA. Inedible algae

are rather isolated within this complementary subsystem and the only possibility that they

participates in the feedback is through a self-damping. This link may be due to a density

dependent growth rate, as suggested by other authors (Scheffer 1991, Sheffer et al. 1993).

Macrophytes are not exploited by zooplankton and likely they are not involved in a

reciprocal interaction with phytoplankton, although the suppression of algae by submerged plants

through the release of allelopatic substances is widely considered in the literature (Wium-

Anderson et al. 1982, Van Vierssen and Prins 1985). Before biomanipulation macrophytes were

restricted to narrow zones along the shores of the lake, and because of this they did not affect

phytoplankton growth significantly, as demonstrated by the fact that algae throve in the years

before the intervention.

Since no feedback loop involving M seems to arise in the complementary subsystem for

the path macrophytes must be self-damped. This link seems also ecologically

appropriate as it takes into account the fact that these plants acquire phosphorus from the

sediment through their roots (Pelton et al. 1998). Nutrients trapped in the sediment, available at

a rate independent of their level, would be self-damped, but since they are not explicitly included

in the model their self-damping is passed up to the macrophytes (see Appendix II).

It does not seem that blue-green algae establish feedback-producing interactions with the

variables in the complement. Although it has been assumed that some benefit could be provided

to rotifers by blue-green algae, the negative effect typical of a plant-herbivore interactions does

not seem appropriate in this case as the grazing pressure is not strong enough to keep blue-green

algae in check (DeBernardi and Giussani 1990). According to the above discussion about IA and

M, blue-green algae can contribute to the complementary feedback only by a self-damping.

[PF&BG&BEG&BN]

[PF&BG&BEG(BG)µRµI]

[PF&BG&BBG&BN]

[PF&BG&BBG&BN]

[PF&BG&BBG&BM]

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The way invertebrate predators and rotifers participate in the complement for the path

remains to be specified. As they seem not to interact with variables in the

complement in such a way that some feedback arises, either they form a predator-prey circuit or

both are self-damped. A predator-prey relation requires that an arrow from R to I be added to the

graph. This link would create two new pathways of the form with

positive effect on invertebrate predators and the net effect of the input to PF would be

ambiguous. On the contrary, the negative effect would be maintained if R and I were self-

damped. The latter solution is accepted at this stage of the reconstruction, in the understanding

that it must be justified ecologically and that no disagreement between model predictions and

observation will come out. In that case the predator-prey relationship between I and R will be

reconsidered.

The importance of the microbial loop in lake dynamics is widely recognized (Porter,

1996). Invertebrate predators and rotifers rely on this microbial food web as they prey upon

bacterial picoplankton, ciliates and flagellates (Arndt 1993). So their abundance and growth rate

are regulated by other variables not in the food web, and this requires that I and R are self-

damped. In doing so one takes into account the additional channel through which energy flows

to the upper trophic levels (Polis and Hurd 1996) of the main food web.

The complementary subsystem for the pathway so far has no feedback

because EG does not form circuits in the complement, which is constituted by the variables [IA-

M-EG-R-I]. Edible green algae interact with rotifers, but the same argument used to describe the

interactions between rotifers and blue-green algae applies here: a typical plant-herbivore

interaction, necessary to produce a feedback, implies a dampening ability of rotifers on

phytoplankton, which is rather a characteristic of large-bodied grazers (Havens 1993). Thus a

single link, the arrow from EG to R, characterizes this relation. Accordingly, edible green algae

have to be self-limited to enter the complementary feedback for the path .

Any impact entering the system through planktivorous fish percolates down to

macrophytes along four different routes. Consider the path . Its effect

depends on the feedback of the subsystem [IA-N-EG-R-I]. Invertebrate predators and rotifers

can contribute with their self-limiting action, but the way dissolved phosphorus, edible and

inedible algae participate in the feedback remains to be specified. N receives continuous supply

from outside the system, so a negative self-effect on this variable must be included in the graph

[PF&BG&BEG(BG)µR]

[PF&BGµI]

[PF&BI]

[PF&BI&BR]

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(Appendix II). Yet, both edible and inedible algae consume nutrients: this requires that an arrow

connecting N to EG and a negative link from IA to nutrients be added to the graph. The

complementary feedback for the considered path is now complete. It is made of different

contributions: one includes all the self-damping on the variables in the complement; the others

consider the two-node feedback loops between algae (both edible and inedible) and nutrients

variously combined with the self-loops on the other variables.

The links so far discussed also generate the complementary feedbacks for the other

pathways responsible for the effect on macrophytes and the overall effect on this component

remains ambiguous due to the different sign of the paths. Yet, the complementary feedbacks for

the pathways to edible phytoplankton and blue-green algae can be identified without introducing

other links. However, for completeness, the interaction between blue-green algae and nutrients

requires that an arrow from N to BG be added, as blue-greens grow on dissolved nutrients. This

increases the number of loops involved in the complementary feedback for the path to EG.

Identifying the complementary feedback for the negative action by PF on grazers, the

positive action on rotifers mediated by grazers and phytoplankton and the

inhibition of invertebrate predators mediated by the grazers requires no further

addition of links to the model. The path does not have complementary feedback; G, so

far, does not enter any feedback loop with other variables in the complement [G-EG-IA-BG-R-I-

M-N]. This inconsistency can be resolved by introducing a positive link from EG to G. It also

completes in pictorial terms the interaction between edible green algae and grazers, a typical

resource-consumer relationship, and, as such, increases the realism of the model. Because of this

addition a two-node loop takes shape that can be variously combined with other one- or two-node

loops in the complement to yield the complementary feedback for the path. Also, this feedback

makes the pathway active.

The overall feedback is the feedback of the highest level in a signed-digraph; it must be

associated to a circuit or a combination of disjunct loops that comprises all the variables in the

graph. No feedback of this type can be calculated from the graph so far obtained, and this means

that links are still missing. A closer examination of the structure reveals that it excludes PF from

the computation of the overall feedback: neither it establishes feedback loops with other

components, nor it is self-damped. But planktivores and grazers interact as predators and prey

and their feedback plays a key role in the dynamics of lakes (Kerfoot and Sih 1987, Carpenter

1988). To make the structure realistic in this respect an arrow from G to PF must be added to the

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graph. An overall negative feedback can now be identified.

Another possibility is to introduce an arrow from I to PF so that planktivores and

invertebrate predators interact as a predator-prey pair. However, the benefit that planktivores

obtain by feeding on copepoda might be negligible because of the low abundance of these

organisms. So this link is omitted in the core structure of the food web.

The interaction between grazers and blue-green algae is considered as unidirectional;

these algae are of poor nutritional value for zooplankton (DeBernardi and Giussani 1990) and

the positive effect on the growth rate of grazers seems negligible. Yet, in representing the

interaction between G and I the negative impact of the consumer will not be considered.

Predatory copepods can feed on small individuals but this may not be enough to control the

population of grazers (Carpenter and Kitchell 1993), also because of the low abundance of I. The

core structure of the food web seems now complete and takes the form of the graph of Figure 4.

[FIGURE 4 GOES HERE]

Some of the main paths described in Figure 3 generate ambiguities about the effect of

input to PF on the level of model variables. In addition, some links that could not be deduced

solely from pathway reconstruction have been added to guarantee ecological realism to the

structure. Because of this new pathways may emerge as unexpected combinations of links and

the graph can yield predictions not confirmed by field observations. To assess the plausibility of

the structure depicted in Figure 4 the graph must be qualitatively analyzed and its predictions

compared with the signs listed in Table 2 or Table 3. Model predictions are reported in Table 4.

One must enter the last row of this table and compare the signs along the columns with those

reported in Table 3. The comparison can be done also by reversing the signs in Table 4 and using

Table 2. Model predictions agree with the signs in Table 2 and 3 and the structure of Figure 4

seems thus a plausible skeleton of the food web of Lake Mosvatn.

[TABLE 4 GOES HERE]

Incorporating nutrient recycling in the food web of Lake Mosvatn

Although the structure presented in Figure 4 explains how effects observed after

15

biomanipulation could arise through food web interactions, in a shallow lake all sorts of different

processes may play a significant role. Consumer-mediated nutrient recycling deserves particular

attention. Considerable evidence exists that this process can influence lake community structure

and dynamics (Boers et al.1991, De Angelis 1992, Vanni 1996, Vanni and Layne 1997), so that

it must be incorporated into general models of food webs.

Nutrient excretion could be considered since the beginning in the reconstruction as arrows

connecting upper trophic levels to nutrients, but it would have increased dramatically the number

of combinations of links to be discussed. Including recycling at this point requires that alternative

structures are built and their predictions compared with the effects of biomanipulation.

Recycling may come from planktivorous fish and zooplankton. Excretion by herbivorous

zooplankton has long been recognized as a potentially important source of nutrient for

phytoplankton (Sterner 1986, 1990). Among the herbivores rotifers dominated the community

in terms of biomass, and their contribution to phosphorus recycle could be relevant; also, owing

to their smaller size, they have higher mass-specific rates of nutrient excretion (Bartell 1981,

Peters 1983). Planktivorous fish, by reducing the mean size of daphinds, could increase their

recycling rate (Carpenter and Kitchell 1984).

Excretion by fish seems to affect phytoplankton community composition and standing

stocks (Schindler 1992). Yet, fish species can recycle phosphorus at rates exceeding external

loading (Brabrand et al. 1990, Reinertsen et al. 1990). In Lake Mosvatn invertebrate predators

seem not to contribute much, because of their low abundance (Sanni and Waervagen 1990).

According to these observations the core structure of Lake Mosvatn can be modified as shown

in Figure 5.

[FIGURE 5 GOES HERE]

The qualitative analysis of this model (Table 5) shows that all predictions concerning the

effect of impacts entering through planktivorous fish are ambiguous (Table 5, last row), and

relative strengths of links determine the sign of the predictions. Making hypotheses about these

strengths requires that constraints are identified so that they impose conditions that links must

satisfy. The local stability of the equilibrium could be one such constraint, as it requires that

negative feedbacks must be stronger than the positive ones. However, the number of feedbacks

in the model is so high that conditions for stability can be met in both cases of positive and

16

negative sign for each ambiguous entry in the last row of Table 5.

Table 6, which lists the number of pathways and feedbacks associated with the

predictions of Table 5, can reveal a potential tendency for the signs.

[TABLE 5 AND 6 GO HERE]

A positive input to planktivores percolates down to nutrients through eight different

routes: five yield a positive effect on this variable, whereas three reduce its level. Accordingly,

N shows a tendency to increase that is not confirmed by the data. Contradiction between

predicted tendencies and results of biomanipulation can be observed also for inedible algae, as

a tendency to increase for this component emerges from model predictions. As for grazers the

positive terms equal the negative ones: compensation between opposite effects is possible so that

moderate or null variation would be expected from the model, whereas this variable showed a

very strong growth after the treatment. Predicted tendencies based on the number of pathways

suggest that this model seem not appropriate to describe the structure of the interactions in Lake

Mosvatn.

Other investigators hypothesized that recycling by fish and zooplankton have different

importance with respect to phosphorus budget (Vanni 1996). The different architectures

presented in Figure 6 describe two situations in which, in turn, one effect is reputed unimportant

and neglected.

[FIGURE 6 GOES HERE]

The table of predictions of Model 6a (Table 7) shows the same type of ambiguity observed in

Model 5, and same tendency of signs (Table 8). Thus, the model seems not appropriate to explain

variations induced by biomanipulation. Instead the model that includes only recycling by fish

(see Table 9) predicts effects that agree with the signs listed in Table 2 and 3, although some

uncertainty is possible. These models suggest that in Lake Mosvatn fish recycled much more than

zooplankton, a result that agrees with the hypothesis that in shallow lakes, when fish biomass is

abundant, recycling by these organisms is more important than excretion by zooplankton (Boers

at al. 1991, Carpenter et al. 1992).

17

[TABLE 7, 8 AND 9 GO HERE]

Model predictions, lake dynamics and management

Rotenone treatment in Lake Mosvatn has been considered a successful example

of biomanipulation. Changes observed in the phytoplankton groups, especially the reduced

volume of cyanobacteria and edible green algae, and the increase of macrophytes, have been

associated to planktivores elimination, and considered as signs of improved trophy conditions

for the lake, although the inedible component of the phytoplankton augmented (but chlorophyll

concentration decreased). However a careful examination of the patterns of abundance reveals

that blue-green algae started to decrease before biomanipulation (see Table 1), and this change

was accompanied by an increase of inedible green algae. Moreover, the decline in the abundance

of blue-green algae that took place between 1986 and 1987 was more pronounced than that

assumed to be caused by fish elimination.

Confronted with this evidence one could argue whether biomanipulation was effectively

responsible of the changes observed between 1987 and 1989. An input could have entered the

ecosystem before the intervention (1986), imposing trajectories that the variables could have

followed during the successive two years, showing variations that have been interpreted as

consequences of fish elimination, being the changes of the same type.

To tackle this problem one first has to show that the variations observed in the level of

the components between 1986 and 1987 could be the result of some input. Second, evidence

must be found that changes detected after 1987 could be due uniquely to rotenone treatment, in

order not to confound causes and effects. Model predictions can help in this respect as patterns

of co-variation can be diagnostic of the source of change.

The input that supposedly affected Lake Mosvatn in 1986 (or before) must have altered

the growth rate of a variable for which the model predicts a negative co-variation between BG

and IA, as they changed in opposite directions. As for Model 6b input to nutrients, macrophytes,

and zooplankton, yields positive or no co-variation between blue-green algae and inedible algae

(see Table 9, row # 1, 2, 6, 7 and 8). Consequences of input on edible green algae are not

predictable (Table 9, fourth row). So far input to cyanophyceae (third row), planktivorous fish

(last row) and green inedible algae (row # 5) seem equally acceptable.

18

To decide which variable could be the target of the parameter change predictions must

be compared with the patterns of abundance of variables other than phytoplankton. The table of

predictions suggests that large herbivores change only for input to planktivorous fish.

Unfortunately no information about abundance or biomass of daphnids before 1987 is available

(see Table 1) and attention must focus on other components. Nutrients decreased between 1986

and 1987 (see Table 1) and the only acceptable hypothesis seems that the input entered through

the inedible component of phytoplankton. In fact a negative input to BG, necessary to reduce the

level of the variable itself and to increase inedible algae, would increase also N. The same holds

true for input to planktivorous fish.

Model predictions thus suggest that an input prior to biomanipulation could have entered

the lake producing the variations observed in 1986 and 1987. Moreover, this input must have

increased the growth rate of inedible algae. This hypothesis is supported by the observation that

strong variations in phytoplankton growth and biomass due to meteorological conditions

characterized Lake Mosvatn in that period (Sanni and Waervagen 1990). However it is dubious

that such climatic changes affected only inedible algae. A more realistic hypothesis considers that

the impact affected BG and EG as well and that the observed patterns of abundance were the

result of the net effect of such inputs.

To clarify this point, further indications provided by Sanni and Waervagen in their paper

are useful. Water temperature seems to play a crucial role in phytoplankton development in lakes.

In particular the growth of cyanobacteria is enhanced by high temperature (Reynolds 1984, Olsen

1989). In summer 1986 the temperature in Lake Mosvatn reached values well above 20°C (T

max= 24°C), favoring the growth of blue-green algae. This may explain the observed dominance

of this group over green algae. Similar conditions did not occur in 1987 as the temperature never

reached 20 °C (T max = 18°C). Lower temperature could have reduced the growth rate of BG

while conditions for green algae improved, and the authors considered this event responsible for

the observed drop in the blue-green algae and the raise of green algae. These variations of the

temperature might have produced a negative input to blue-green algae and a positive impact on

green algae.

In Model 6a both the positive input to IA (Table 9, fifth row) and the negative input to

BG (Table 9 third row, reversed signs) are expected to reduce blue-green algae, while increasing

inedible green algae. Nutrients however are affected in opposite way.

The temperature drop observed between 1986 and 1987 must have produced a positive

[EG&BN]×[PF&BG] [EGµG]×[PFµN]

[EG&BN]×[PF&BG] [EGµG]×[PFµN]

19

impact on the edible component of phytoplankton, as it is composed of green algae. Predictions

about blue-green and inedible algae are ambiguous (row #4) and nothing can be said about the

effects of this positive input. Considering the pathways involved in these predictions, if

prevails over blue-green algae decrease, and so do

inedible algae, but no evidence exists that the inequality >

is satisfied. However, excretion by planktivores must not be too strong for the system to be stable

(this link form a destabilizing positive feedback, see discussion below) while the control over the

grazers by PF was extremely efficient in the lake before biomanipulation. So in Lake Mosvatn

likely conditions were such that the above inequality could be satisfied and the positive input to

EG produced a diminution of both blue-green and inedible algae.

In the period 1986-1987 a strong decrease of blue-green algae was observed, while the

increase of green algae was less pronounced. This may be the consequence of the hypothesized

inputs, as all three parameter changes (IA, BG, EG) are expected to reduce BG, whereas two out

of three increase IA and EG. The same links discussed above determine the sign of the prediction

about nutrients. If the inequality is met, the effect on nutrient is negative. Two inputs (EG and

IA) out of three act to reduce the concentration of dissolved nutrients, and this tendency agrees

with what observed in the Lake.

The hypothesis that an input entered the ecosystem through phytoplankton before

biomanipulation seems plausible. It remains to show that the intervention, carried out in

September 1987, determined what observed in the lake after 1987. According to model

predictions, in fact, changes in the level of BG and IA described as effects of biomanipulation

could be produced by the input on phytoplankton imposed by temperature fluctuations. Although

no data about grazers prior to 1987 are available, the pattern of abundance for this group showed

a significant increase after the rotenone treatment; in the model this can be caused only by a

negative input to planktivores. Nutrients were observed to increase: if they simply followed the

trajectory imposed by the input to phytoplankton they should have continued to decrease or they

could remain at the same average value calculated in 1987. So biomanipulation really affected

the level of the variables in the lake, but the scenario seems more complicated than presented by

the authors. In 1988 the temperature raised again, and contemporary to biomanipulation an

additional input to phytoplankton must be considered.

According to the above discussion the overall effect of this input would be an increase

of dissolved nutrients, a decrease in green algae and an augmented abundance of blue-green

20

algae. On the contrary cyanobacteria decreased in 1988, and this may be taken as an evidence that

biomanipulation was successful. Moreover, if the temperature did not rise with respect to 1987

the effect of fish elimination upon BG could be even stronger: likely the effect of

biomanipulation was partially compensated by the positive input on BG due to the higher

temperature.

The biomanipulation and the increased temperature are predicted to act in the same way

on edible green algae. It follows that the observed decline of this component could be caused by

two inputs instead of biomanipulation only. Inedible green algae are expected to decrease because

of the combined effects of the inputs to phytoplankton but their abundance augmented,

suggesting that the effect of fish elimination was strong enough to overcome that of temperature.

As with edible green algae it seems that biomanipulation and temperature acted in a synergistic

way to set the new level of dissolved nutrients.

This discussion highlights that in ecosystems several mechanisms may act simultaneously

to produce a certain effect, and explaining their relative contribution is very difficult. Without

even a rudimentary knowledge about the forces that shape the ecosystem causes and effect can

be confounded with profound practical implications. In Lake Mosvatn if no change were detected

in the abundance of blue-green algae after biomanipulation planktivore elimination could be

judged ineffective to reduce blue-green algae, and, as such, useless for managing eutrophication.

Instead it simply could not compensate for the input to phytoplankton. The structure of the

interactions may provide insight about causal connections among ecosystem components and as

such it may help to understand the patterns of abundance as produced by ecosystem responses

to external events.

The nutrient status of Lake Mosvatn before biomanipulation was described by Sanni and

his co-worker as moderately eutrophic and the intervention produced quasi-oligotrophic

conditions. Eutrophication, caused by excess input of nutrients, is a widespread problem of lakes

(Carpenter et al.1999). In shallow lakes this excess of nutrient can cause a substantial or complete

loss of macrophytes and their replacement by dense phytoplankton population (Phillips et al.

1978, Jeppsen et al. 1990a, Moss 1990). Model 6b confirms that this is expected also in Lake

Mosvatn, as a positive input to N decreases the abundance of macrophytes whereas it increases

all phytoplankton groups.

Other predictions are in agreement with general trends observed in lakes during

eutrophication. A typical transformation that takes place is the shift in dominance from large-

[PFµNµEGµGµPF]

21

bodied zooplankton to small grazers (Gliwicz 1969). This shift is predicted also by this model,

as for positive input to nutrients grazers are expected not to change while rotifers increase.

Enrichment experiments conducted in enclosures with and without fish (Parson et al. 1972,

Levitan et al. 1985) showed that in the absence of planktivores large zooplankton increased,

while phytoplankton remained unaffected; only in the presence of fish did the enrichment lead

to an increase in phytoplankton density, while zooplankton remained largely unchanged. The

variations predicted by the model seem to be in agreement with those findings: algae increase and

no change is expected for grazers following positive input on N.

Other authors found that fish density increased in response to enrichment (McAllister

1972) and was positively related to the nutrient status of lakes (Gascon and Legget 1977,

McQueen 1986). The model seems to agree with these findings as it predicts that fish density

becomes higher for positive impact on nutrients; also, except for input to PF, when the level of

nutrient augments, so does that of the planktivorous fish.

A key issue of the ecology of shallow lakes concerns the possibility that a progressively

increasing nutrient load may induce a switch from one clear, macrophyte-dominated state to a

turbid phytoplankton-dominated state; both are described as alternative stable equilibria

(Scheffer 1989, Blindow 1992, Blindow et al. 1993, Scheffer et al. 1993). Continuing enrichment

is thought to cause the stability of the clear state gradually to decrease, making it more vulnerable

to perturbations, until a catastrophic transition occurs that drives the system into the turbid, stable

state. The alternative stable state model implies that the system, in each state, is resistant to

disturbance and tends to return to equilibrium when perturbed. This may explain why shallow

lakes that have undergone this transition may stay turbid despite reduced nutrient loading and

only a severe reduction of nutrient level can result in an effective improvement.

The models presented here predict that this bi-stability is possible also in Lake Mosvatn.

In loop models if the overall feedback is negative regardless of parameter values, that is in the

graph all cycles produce negative feedback, the equilibrium values of the variables are

continuous functions of the parameters. If some positive feedbacks exist and they become

stronger relative to the negative feedbacks when some parameter change, then the overall

feedback decreases in magnitude and, when it nears zero, a small shift of parameter causes a

discontinuous change that falls into the category of catastrophic events (Levins 1998).

Again consider Model 6b. Its overall feedback is composed of four negative and one

positive feedback. The loop in combination with the self-damping on I, R,

22

BG, M and IA forms this positive part. Because of nutrient enrichment (positive input to N, Table

9 first row) fish density is predicted to increase and this may reflect on the amount of phosphorus

made available through excretion. This would make the positive part of the overall feedback

stronger. Rotifers increase as well, and their contribution to recycling may become more

important and, accordingly, no longer negligible in the model. The standing crop of grazers

remains the same: as there is more food (algae increase) more individuals are produced, but

because of the increased population of predators (planktivores) more grazers are eaten by fish.

However the turnover rate of G is expected to increase with two interlinked consequences: the

age distribution shifts toward younger age-classes, and the body-size becomes smaller (Lane and

Levins 1977). As recycling rate is inversely related to mean body size it turns out that also

recycling by grazers increases with eutrophication. According to these observations it seems that

in more eutrophic conditions Lake Mosvatn resembles much more to Model 5 rather than Model

6b.

Increased nutrient excretion by fish reinforces the positive feedback in which this link

takes part; yet, as recycling by zooplankton becomes stronger other positive feedbacks may

become active in the food web. In Model 5 the overall feedback combines equal numbers of

positive and negative contributions. Thus, in comparison with Model 6b, the number and

magnitude of positive feedbacks have increased, and the compensation between positive and

negative feedbacks may reduce the magnitude of F so that it becomes closer and closer to zeron

as eutrophication proceeds, creating the conditions for a discontinuous change, likely the

transition to the turbid state.

Planktivore population was reduced by rotenone treatment. This would be the effect of

any negative input to PF, such as restocking with piscivores. As planktivores diminish, their

contribution to phosphorus budget may decrease too. Rotifers also decrease, while large

zooplankton would increase in body-size (this was observed in Lake Mosvatn after the negative

input to PF). All these effects can lead to a situation in which recycling play a marginal role in

comparison with trophic interactions active in the food web. So the core structure given in Figure

4 may be more appropriate to describe the system in the more oligotrophic conditions that

characterized the system after biomanipulation; in such conditions it is likely that strong short

feedback loops due to predator-prey and plant-herbivore interactions dominate (Neill 1988).

As nutrient load progressively increases, these stabilizing forces are overcome by positive

feedbacks, some due to recycling, which becomes more and more important according to changes

23

predicted in the levels of consumer populations. In this framework Model 4, 5 and 6b, rather than

representing alternative structures, seem to describe three different situations in a gradient of

progressively increasing eutrophication, which causes the stability of the clear state to shrink to

nil as F approaches zero. Functional to this interpretation is the consistency of the three modelsn

in predicting consequences of enrichment (see Tables 4, 5 and 9). In fact they all point to the

same final state characterized by phytoplankton dominance and macrophyte reduction. In terms

of management these graphs predict that improving trophic conditions in Lake Mosvatn is

possible by reducing nutrient load, as a negative input to N lowers phytoplankton, and increases

macrophytes.

Passing from more oligotrophic to eutrophic conditions, the magnitude of the overall

feedback diminishes. Since F is the denominator in the loop formula it follows that the intensityn

of variations imposed to system components by inputs is higher in more eutrophic conditions.

This agrees with the observation that changes are more dramatic in highly eutrophic systems than

in oligotrophic ones (Jeppsen et al. 1990b, Lyche et al. 1990). Recycling, by introducing positive

feedbacks, seems to reduce system resistance, that is the ability to resist perturbations.

Bottom-feeding (benthivorous) fish seem to play an important role in the transition from

clear to turbid waters in shallow lakes. They may damage plants mechanically during feeding and

resuspend sediment (Blindow 1992, Blindow et al. 1993, ten Winkel and Meulemans 1984).

These actions would produce, respectively, a negative input on macrophytes and a positive input

on dissolved nutrients. Model predictions (first row, second row reverse the signs) suggest that

introducing this type of fish in Lake Mosvatn (they were not present) would create the conditions

that favor the transition toward the turbid state.

The feedbacks responsible for the bi-stability are however much more complex then one

can deduce from these diagrams. They involve factors inherent the biochemistry of the water-

sediment surface, sedimentation rate, competition between phytoplankton and macrophytes for

dissolved nutrients, balance between nutrient excretion and luxury uptake by macrophytes, and

release of allelopatic substances by submerged plants (Carpenter 1981, Van Vierssen and Prins

1985, Reinertsen et al. 1990, Scheffer et al. 1993, Meijer et al. 1994). The influence of these

factors however could be investigated by qualitative models in two ways: using the table of

predictions, whenever they take the form of positive or negative impacts on the rate of change

of system variables, or by including them as new variables or links in the model. It is always the

comparison of model predictions with the observation, however, that makes the model itself a

24

plausible description of the system, and, as such, a useful tool for predictions.

Concluding remarks

In this paper a digraph diagram of the food web of Lake Mosvatn is presented as the

outcome of a reconstruction based on effects of biomanipulation. Most limnetic food webs are

more complicated than the simple planktivore-zooplankton-phytoplankton-nutrient chain and

identifying the structure of the interactions may be useful to predict effects of environmental

disturbance or anticipate the outcomes of manipulative experiments.

Other studies as well have proposed strategies to describe the patterns of interactions in

ecosystem food webs (Yodzis 1988, Wootton 1994, Schmitz 1997). The one presented here,

besides its qualitative nature, differs from the others because no a priori hypotheses about the

whole structure of the web and the form of the equations describing the interactions are made.

The links are obtained by integrating the results of bimanipulation with existing knowledge on

lake organisms as suggested by the scientific literature and the rules of loop analysis. The inverse

method of reconstruction used for Lake Mosavtn can be applied to many situations in which

experiments are conducted on ecosystems. However it may not be generally useful for webs that

are very large or have many connections.

The structure of this food web leaves out some links that can be important in lake

ecosystems. Specifically, the interaction between plantktivores and invertebrate predators, that

between invertebrate predators and rotifers, and the relationship between rotifers and edible algae

have been depicted as unidirectional. The effect of the missing links on model predictions were

tested on the core structure and on the models including recycling, but the results produced

several disagreements with respect to Table 3. Thus the model has been maintained as it has

come out from the reconstruction described here.

Results of biomanipulations have been used under the hypothesis of a system in moving

equilibrium. Whether the situation in the two years after the intervention represented a new

steady state or only a temporary one is questionable. Observations conducted on Lake Mosvatn

in 1989 (Lyche et al. 1990) revealed that macrophyte beds increased dramatically. This suggests

that in some years the feedbacks activated by these plants (Meijer et al. 1990, Blindow et al.

1993) might have stabilized the system in the direction of changes imposed by biomanipulation.

The moving equilibrium approach seems thus appropriate, also considering that lakes exhibit

25

very different behaviors in this respect: some show stable changes after two or three years since

the manipulation, whereas for others one year seems sufficient to reach a new equilibrium

(Scheffer et al. 1993).

The table of predictions can be used to forecast qualitatively the consequences of human

interventions or external perturbations, and attention has focused here on nutrient enrichment.

The table however, by predicting the effects of inputs entering the lake through any variable,

contributes also to understand lake dynamics as it is shaped by the many forms of stress that acts

continuously on the rate of change of components that are not the targets of human manipulation.

Such impacts often produce unexpected effects that may obscure the results of deliberate

interventions and can confound our perception of cause and effects in the lake, with profound

practical consequences. In this perspective one can use the table of predictions as a diagnostic

tool that can help to explain the patterns of abundance.

The ecology of shallow lakes as it emerges from the literature is characterized by the shift

between two extreme conditions represented by a turbid state and a clear state. Such transition

seems to center around the interaction between submerged vegetation and turbidity (Blindow et

al. 1993, Sheffer et al. 1993). The model proposed here suggests that the positive feedbacks

introduced by nutrient recycling are destabilizing forces that can contribute to the shift. This leads

one to think that the main feedback loop thought to be responsible for the existence of alternative

equilibria in shallow lakes may be far more complicated than often represented.

26

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*f*c

p (k)ij

*f*c

35

Appendix I

The loop formula allows one to calculate expected changes in the equilibrium level of

variables in response to parameter input. Besides the sign of the input, indicated by the term

, the loop formula makes use of the concept of path, complementary feedback and overall

feedback. They refer to structural elements that can be identified in any graph. Their meaning can

be fully understood by referring to the correspondence between matrix algebra and the formalism

of loop analysis and that can be found in Puccia and Levins (1985). Instead, in what follows

criteria to identify such elements in a graph are provided by using the scheme depicted in Figure

A1.

Circuits and Feedbacks. In loop analysis a pathway that starts at one node and, by following

the direction of links, returns to it without crossing intermediate nodes more than once is called

loop, or circuit. Any circuit produces a feedback that can be positive or negative depending on

the product of the signs of the links that form the loop. As there may be circuits of different

length (1, 2, 3,...variables involved) in a system there are as many levels of feedback as variables.

Each level of feedback considers all the circuits (feedbacks) involving that particular number of

variables. In the system of Figure A1 there are 3 levels of feedback.

Overall Feedback (F ). It is computed only once and corresponds to the highest possible leveln

of feedback in a system. In model A1 it is the third level of feedback and includes all the

feedbacks of the circuits involving the three components. No disjunct loop of length three exists

in the graph, and in this case the overall feedback comprises all the product of disjunct loops that

have a combined number of variables equal to 3. As disjunct loops are those that share no

variables, F is composed by the self-damping on A (a self-effect link is a loop of length 1) plusn

the two-node loop [B-C]. Its sign is obtained by multiplying the signs of the links involved,

further multiplied (-1) , where m is the number of disjunct loops entering the feedback. As them+1

links involved are two negative and one positive, and there are two disjunct loops, the overall

feedback is negative.

Path . A path is a series of links starting at one node and ending on another without crossing

any variable twice. Suppose a positive input occurs on A (its rate of change increases, >0).

To predict the new equilibrium of, say, C, the path along which the effect travels is the positive

link from A to B plus the arrow from B to C. It involves three variables (k) and its sign, given by

the product of the signs of the links that form the path, is positive.

dNdt

'MN&PN

**N

dNdt

(

'&Mn

N (

36

Complementary feedback (F ). If the k variables in the path are ideally excluded from then-k

graph, what remains is called complementary subsystem. The complementary feedback is the

highest possible level of feedback that can be found in the complementary subsystem. For

positive input on A and effect on B, what remains is C and the highest possible level of feedback

is level 1. As C has no self-effect link, there will be a null (0) complementary feedback. For

completeness, it has to be noted that a path from a variable to itself is equal to 1, while if all the

variables are in the path (i.e. input to A and effect on C) there is no complementary subsystem,

but the complementary feedback is equal to -1. These are two algebraic conveniences. The

summation sign in the loop formula considers the fact that two variables can be connected by

more than one path.

Appendix II: self-damping

If a variable is not self-reproducing it is usually self-damped. For instance if nutrient enters a

lake at a rate Φ , because there are tributaries of the lake, and is removed by some consumerN

plant P , the concentration of usable nutrient in the lake may follow the equation

By taking derivative with respect to nutrient itself one obtains

and the diagonal term of the community matrix is negative, which translates into a self-loop.

If a model excludes variables that are self-damped, mostly those at lowest trophic levels

such as inorganic nutrients, then their self-damping property is transferred to variables which

interact with the self-damped ones. Consider a nutrient N which enters a system from outside and

is consumed by a species S. This interaction can be represented by the following equations

dNdt

'Mn&aNS

dSdt'S(bN&$)

N ('Mn

aS

dSdt' b

Mn

aS&$ S

37

where a is the rate of nutrient uptake by S, b the rate of conversion of N into S and β is the death

rate of S. In this system N results self-damped, as shown before, but S is not as taking partial

derivative with respect to S yields zero. If N is not recognized as a distinct variable then it can

be replaced in the second equation by its expression obtained at equilibrium from the first

equation

and the equation for the consumer becomes

whose partial derivative with respect to S is negative and the variable is self-damped.

Also, a self-damping may originate from a density-dependent growth rate. The demonstration is

not given here as it follows the same procedure shown for the two previous cases.

38

Table 1. Quantitative changes observed in the main variables of Lake Mosvatn during the period 1986-1989.Original data are in Sanni and Waervagen (1990). Numbers about model variables are time-weighted averagesummer values. Biomanipulation was carried out in September 1987.

1986 1987 1988 1989

pH 8.0 8.0 7.9 no data

Secchi depth (m) 1.6 1.7 >2.3 no data

Maximum temperature (°C) 24 18 22 no data

Total P (µg lt )-1 45 42 no data 29

Particulate P (µg lt )-1 32 34 no data 17

Dissolved P (µg lt )-1 13 8 no data 12

Cyanophyceae (mm m ) 3446 1005 797 no data3 -3

Chlorophyceae + 1345 2510 3493 no data

µ algae (mm m )3 -3

Daphnia ( µg DW lt )-1 no data 60 318 no data

Rotifers (ind. lt ) no data 10,000 4,000 no data-1

Table 2. Qualitative changes observed in the main variables of Lake Mosvatn after biomanipulation.

Nutrients Edible Blue-green Inedible Macrophytes Grazers Rotifers Invertebrate

(N) (EG) (BG) (IA) (M) (G) (R) (I)

green algae algae green algae predators

+ - - + + + - +

Table 3. Qualitative changes that would be observed in the main variables of Lake Mosvatn following a positiveinput on fish population.

Nutrients Edible Blue-green Inedible Macrophytes Grazers Rotifers Invertebrate

(N) (EG) (BG) (IA) (M) (G) (R) (I)

green algae algae green algae predators

- + + - - - + -

39

Table 4. Table of predictions for the core structure of Lake Mosvatn (Figure 4). A parenthesis around the + meansthat the pluses outweigh the minuses by two to one. A parenthesis with an asterisk (+)* means greater than or equalto a three to one ratio in favor of the pluses. The same is true for (-) and (-)*, but in favor of the minuses. A questionmark means that nothing can be said about the direction of change for the variable, as the numbers on the left andon the right are the same.

N M BG EG IA I R G PF

N + - + + + - + 0 +

M 0 + 0 0 0 0 0 0 0

BG - ? + - - + ? 0 -

EG - ? - + - - (+)* 0 +

IA - ? - - + + - 0 -

I 0 0 0 0 0 + - 0 0

R 0 0 0 0 0 0 + 0 0

G 0 0 0 0 0 - + 0 +

PF - (-) (+)* (+)* - (-)* (+)* - (+)*

Table 5. Table of predictions for the model with complete recycling (Figure 5).

N M BG EG IA I R G PF

N + - + + + - + 0 +

M 0 ? 0 0 0 0 0 0 0

BG ? ? ? ? ? ? ? 0 ?

EG (+)* (-)* (+)* (+)* (+)* (-)* (+)* 0 (+)*

IA - + - - ? + - 0 -

I - + - - - ? (-)* 0 -

R + - + + + - (+)* 0 +

G + - + + + (-) + 0 (+)

PF ? ? ? ? ? ? ? ? ?

Table 6. Table that provides the number of positive and negative factors responsible for the expected changes inTable 5. Each entry has two numbers: that on the left indicates active pathways that decrease the equilibrium level

40

of the column variable, while the number on the right indicates active pathways that increase that variable.

N M BG EG IA I R G PF

N 1- 0 0- 3 1- 0 1- 0 1- 0 0- 1 3- 0 0- 0 1- 0

M 0- 0 4- 4 0- 0 0- 0 0- 0 0- 0 0- 0 0- 0 0- 0

BG 1- 1 5- 5 3- 3 1- 1 1- 1 1- 1 3- 3 0- 0 1- 1

EG 3- 1 3- 9 3- 1 3- 1 3- 1 1- 3 7- 1 0- 0 3- 1

IA 0- 1 6- 3 0- 1 0- 1 3- 4 1- 0 0- 3 0- 0 0- 1

I 0- 1 3- 0 0- 1 0- 1 0- 1 4- 3 1- 4 0- 0 0- 1

R 1- 0 0- 3 1- 0 1- 0 1- 0 0- 1 4- 1 0- 0 1- 0

G 2- 0 0- 6 2- 0 2- 0 2- 0 2- 4 6- 0 0- 0 4- 2

PF 5- 3 11- 17 7- 5 5- 3 5- 3 6- 8 14- 8 4- 4 5- 3

Table 7. Table of prediction for the model with recycling by zooplankton (Figure 6a).

N M BG EG IA I R G PF

N + - + + + - + 0 +

M 0 ? 0 0 0 0 0 0 0

BG ? ? ? ? ? ? ? 0 ?

EG (+) (-) (+) (+)* (+) (-)* (+)* 0 (+)*

IA - ? - - ? + - 0 -

I - + - - - (+) - 0 -

R + - + + + - + 0 +

G + - + + + (-) + 0 (+)

PF ? ? ? ? ? ? (+) ? ?

Table 8. Tendency of signs related to predictions given in Table 7.

N M BG EG IA I R G PF

N 1- 0 0- 3 1- 0 1- 0 1- 0 0- 1 3- 0 0- 0 1- 0

M 0- 0 4- 3 0- 0 0- 0 0- 0 0- 0 0- 0 0- 0 0- 0

BG 1- 1 4- 5 3- 2 1- 1 1- 1 1- 1 3- 2 0- 0 1- 1

EG 2- 1 3- 7 2- 1 3- 1 2- 1 1- 3 6- 1 0- 0 3- 1

IA 0- 1 5- 3 0- 1 0- 1 3- 3 1- 0 0- 3 0- 0 0- 1

I 0- 1 3- 0 0- 1 0- 1 0- 1 4- 2 0- 4 0- 0 0- 1

R 1- 0 0- 3 1- 0 1- 0 1- 0 0- 1 4- 0 0- 0 1- 0

G 1- 0 0- 3 1- 0 1- 0 1- 0 2- 4 4- 0 0- 0 4- 2

PF 4- 3 10- 15 6- 4 5- 3 4- 3 5- 8 13- 6 3- 4 5- 3

Table 9. Predictions of the model with recycling by fish (Figure 6b).

N M BG EG IA I R G PF

41

N + - + + + - + 0 +

M 0 (+)* 0 0 0 0 0 0 0

BG - ? (+)* - - + ? 0 -

EG ? (-) ? + ? - (+)* 0 +

IA - ? - - (+)* + - 0 -

I 0 0 0 0 0 (+)* (-)* 0 0

R 0 0 0 0 0 0 (+)* 0 0

G + - + + + - + 0 +

PF (-) (-) (+) (+)* (-) (-)* (+) (-)* (+)*

Captions for Figures

Figure 1. Community matrix and signed-digraph of a resource-consumer interaction

(X =resource, X =consumer).1 2

Figure 2. A predator-prey model: signed-digraph and table of predictions.

Figure 3. Structure that includes only the pathways thought to be responsible for the observed

effects of biomanipulation

Figure 4. Graph representing the core structure of the food web in Lake Mosvatn as deduced by

the reconstruction.

Figure 5. Core structure with the addition of recycling by consumers.

Figure 6. Alternative graphs including nutrient recycling by upper trophic levels. Recycling by

zooplankton (a) and excretion by fish only (b).

Figure A1. Signed-digraph of a three-level trophic chain. Paths, complementary subsystems, and

feedbacks used to calculate expected changes in the equilibrium level of the variables in

response to a positive input on A.

=0

-a12

d -a12-a11

a21

a21

dt

-a11

-a12

0 +- +

Effect of parameter changeon the abundance of

Increase in therate of change of

-a11

a21

Figure 1

Figure 2

N

PF

G

I

R

BG

M

EG

IA

G

N

PF I

R

BG

M

EG

IA

Figure 3

Figure 4

G

N

PF I

R

BG

M

EG

IA

G

N

PF I

R

BG

M

EG

IA

a)

G

N

PF I

R

BG

M

EG

IA

b)

Figure 5

Figure 6

CBA

A B C

BA

B C

CB

+1

C

δf(A)

δc

Path

>0Positive input to A

FeedbackEffect on

A

none

(-aBC)(aCB)

none -1

+aBA

+aCB+aBA

-aBC

+aCB

Figure A1