Qualitative modelling and indicators of exploited ecosystems 4.1... · exploited ecosystems. The...

Qualitative modelling and indicators of exploited ecosystems

Jeffrey M. Dambacher1, Daniel J. Gaughan2, Marie-Joelle Rochet3, Philippe A. Rossignol 4 & Verena M. Trenkel3

1CSIRO Mathematical and Information Sciences, GPO Box 1538, Hobart Tasmania 7001, Australia; 2Department of

Fisheries, Government of West Australia, PO Box 20, North Beach, Western Australia 6920, Australia; 3IFREMER, Rue de

l’Ile d’Yeu, B.P. 21105, 44311 Nantes Cedex 03, France; 4Department of Fisheries and Wildlife, Nash Hall, Oregon State

University, Corvallis, OR 97331, USA

Introduction 2

The problem of complexity 2

Traditional approaches 2

A qualitative approach 3

Methods 4

Signed digraphs 4

Qualitative modelling tools 4

System feedback and dynamics 4

Shift in equilibrium 5

Time averaging 6

Model formulation 6

Perturbation scenarios 7

Results 7

Abstract

Implementing ecosystem-based fisheries management requires indicators and models

that address the impacts of fishing across entire ecological communities. However,

the complexity of many ecosystems presents a challenge to analysis, especially if

reliant on quantification because of the onerous task of precisely measuring or

estimating numerous parameters. We present qualitative modelling as a comple-

mentary approach to quantitative methods. Qualitative modelling clarifies how

community structure alone affects dynamics, here of exploited populations. We build

an array of models that describe different ecosystems with different harvesting

practices, and analyse them to predict responses to various perturbations. This

approach demonstrates the utility of qualitative modelling as a means to identify and

interpret community-level indicators for systems that are at or near equilibrium, and

for those that are frequently perturbed away from equilibrium. Examining the

interaction of ecological and socio-economic variables associated with commercial

fisheries provides an understanding of the main feedbacks that drive and regulate

exploited ecosystems. The method is particularly useful for systems where the basic

relationships between variables are understood but where precise or detailed data are

lacking.

Keywords Community matrix, ecosystem-based management, indicators, press

perturbation, qualitative modelling, time averaging

Correspondence:

Jeffrey M. Dambacher,

CSIRO Mathematical

and Information Sci-

ences, GPO Box 1538,

Hobart Tasmania

7001, Australia

Tel.: +61 3 6232 50

96

Fax: +61 3 6232 50

12

E-mail: jeffrey.damb-

[email protected]

Received 4 June 2008

Accepted 16 December

2008

F I S H and F I S H E R I E S

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Journal compilation � 2009 Blackwell Publishing Ltd DOI: 10.1111/j.1467-2979.2008.00323.x 1

Indicators of exploited ecosystems 7

Example application 9

Expanded fleet-stock model 11

Multi-species fleet-stock model 11

Discussion 13

Utility of approach 13

Characterizing harvest systems and their feedbacks 14

Management strategies 15

Model complexity and utility 15

Limitations 15

Theoretical synthesis 16

Acknowledgements 16

References 16

Introduction

The problem of complexity

When confronting the impact of resource exploi-

tation, such as fisheries, on ecosystems, ecologists

and managers often struggle to identify and

interpret their great complexity. Policies designed

to improve management of fisheries and conser-

vation of ecosystems have been developed and

legislated in many jurisdictions – e.g. Antarctica

(CCAMLR 1982), Australia (Fisheries Management

Act 1991), United Nations (FAO 1995, UN 2002),

and USA (Magnuson-Stevens Fishery Conservation

and Management Act 2007). The compelling need

for conservation of ecosystem services, beyond the

sustainability of individual target stocks alone, is

recognized and broadly accepted by the interna-

tional community. Nonetheless, much recent lit-

erature continues to focus on describing historical

problems and what scientists and managers

should consider to alleviate the unacceptably slow

progress in implementing ecosystem-based

management of fishery resources. We, and others

(Watson-Wright 2005), contend that this slow

progress to operationalize policies is partly because

of the failure of scientists to provide managers

with sufficient insight and understanding of

ecosystems. Management interventions have pro-

ven difficult to justify and monitor because of

confusion and uncertainty regarding how to deal

with their inherent complexity. The causes are

many, and to name a few, include a lack of

precision, natural variability, the interpretation of

indirect effects, and emergent properties. Where

does one act? What does one monitor? How does

one make predictions or interpret an observed

change?

Traditional approaches

One approach to these questions is to rely on tools

developed at a lower level of organization, such as

those for population studies. A widespread example

is the population projection matrix, where a popu-

lation is broken down into time classes, such as

years, and the survival rates and transition rates

from one class to the other are specified (Caswell

2001). From this matrix the growth rate of the

entire population is derived, and indices, such as

sensitivity, can determine targets for intervention.

Sea turtle excluders on shrimp trawlers were

justified in this fashion (Crowder et al. 1994).

However, for this approach to be successful the

data must be both precise and accurate. The focus

and analysis is for a single species, and if there is a

relationship to another species, then its effect is fixed

and implicit within the transition and survival rates,

thus assessing the influence of feedback from other

species is not possible with this approach.

Another approach has been to develop models of

ecological communities based on food webs, and to

simulate the outcomes of management actions

(Hollowed et al. 2000). Constructing quantitative

ecosystem models requires a trade-off between a

simplified representation, in which model parame-

ters are abstractions that cannot be measured in the

field, and detailed mechanistic descriptions, in

which the parameters correspond to reality but

are too numerous to practically measure or esti-

Qualitative models and indicators J M Dambacher et al.

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mate. Because the number of variables and inter-

actions increases when more components and

details of the ecosystem are included, the degree of

parameterization required is high. Consequently,

estimates of species interaction strengths are in

practice generally less reliable and less available

than estimates of parameters required for single-

species analyses. Moreover, simulation results are

conditional on estimates of all model parameters,

and indeterminacy is especially pervasive in models

of complex marine ecosystems (Yodzis 1988;

Hollowed et al. 2000). Application of quantitative

multi-species models for management intervention

has therefore been limited (Bax 1998; Hollowed

et al. 2000). For example, a combination of models

are used to advise management of ground-fish

fisheries in Alaska, but their shortcomings include

a lack validation and limited power to predict effects

of climate forcing (Livingston et al. 2005).

Ecological theory provides another path to the

problem of ecological communities. Rice (2001)

reviewed the major principles that are the basis of

our understanding of ecosystems impacted by fisher-

ies, namely, Liebig’s law of the minimum, competitive

exclusion and ecological succession. From these

principles arise various widely used concepts, such

as density dependence and carrying capacity. How-

ever, these concepts have seldom been quantified in a

practical sense, and arguably, a unified theory to

interpret them is lacking in fisheries management.

Finally, as many community modelling results

have had little practical utility, much effort has gone

into the development of ecosystem indicators (Cury

and Christensen 2005), on the assumption that

these would convey important information in a

lucid way (Degnbol 2005). Here again, ecosystem

indicators have rarely been of practical use because

they are typically provided to decision makers as

unstructured lists that fail to summarize the under-

lying context and complexity of information. What

is especially needed, and often lacking, is a clear

understanding of the causal connections between

the source of disturbance to an ecosystem and the

asset of importance, and which relevant indicator is

to be monitored (Failing and Gregory 2003; Hayes

et al. 2008; Rochet and Trenkel 2009).

A qualitative approach

Ecosystems and the fisheries they support are

complex systems with numerous relationships

operating at multiple scales. While many single

species analyses are powerful and of great value,

most fisheries worldwide do not explicitly consider

direct and indirect effects on non-target species in

an ecosystem. To evaluate the impact of fisheries on

non-target organisms, fisheries scientists and

managers are often presented with only partial

information from single species data or models,

non-testable theoretical concepts, or unstructured

lists of indicators. Knowledge of relationships

between populations in an ecological community

is rarely clear and unambiguous. Resolving this

problem of complexity first requires a rigorous

conceptualization of how the system works. This

conceptualization must then be embedded within a

working model, the analysis of which allows one to

understand and predict the system’s behaviour and

dynamics.

It is this seemingly simple process of model

building and analysis that has often proven intrac-

table, and qualitative models (Puccia and Levins

1985) offer an effective and logical alternative or

complement to quantitative models, particularly for

those fisheries that lack explicit conceptual models.

The high mathematical rigor of the analyses,

testability of their conclusions, and emphasis on

generality and realism (Levins 1966) makes them

broadly useful and readily understandable by both

scientists and managers. The major drawback of

qualitative models is their lack of precision, but this

lack of dependence on precision can be used to

advantage. In qualitative modelling, attention is

diverted away from the precise measurement of

model parameters and indicator variables, which is

rarely possible anyway, and is instead focused on

describing general relationships and trends, which

often falls within the realm of what is practical and

achievable in the modelling and monitoring of a

complex ecosystem.

This work provides a general overview of qual-

itative modelling methods, their application to

resource management, and a discussion of how

these methods can be integrated with, and supple-

ment current approaches. Our objectives are two-

fold. First, we develop models of interacting

populations under various types of fishery exploita-

tion. Through perturbation scenarios, we predict

qualitative change in abundance and life expec-

tancy, and, for frequently perturbed systems,

whether through time variables vary together or

not. These predictions of population response are

determined by whole-system processes, and thus

serve as an indicator at the ecosystem level. Second,


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we examine the feedbacks arising between a fishery

and its stock by proposing models that incorporate

the processing of catch and market forces, thus

integrating ecological and economic subsystems.

The models yield a balance of positive and negative

feedback cycles, and provide a realistic interpreta-

tion of the system-wide forces that regulate and

drive resource exploitation. We provide an example

analysis of actual fisheries within a complex system.

Methods

Signed digraphs

Qualitative modelling proceeds by analysis of eco-

system structure, which is defined by the relation-

ships of interacting variables. The structure of an

ecosystem can be described qualitatively through a

signed directed graph, or signed digraph (Fig. 1),

where links (graph edges) that connect model

variables (graph vertices) represent the sign of

direct effects (Puccia and Levins 1985; Dambacher

et al. 2002). Model variables typically represent

natural populations or life stages, but can also

portray environmental or socio-economic variables.

A signed digraph has an equivalent representation

in the community matrix A, where each aij element

represents the direct effect of variable j on variable i

(Fig. 1). All pairwise ecological relationships can be

represented in a signed digraph: predator–prey

(+ )), competitive () )), mutualistic (+ +), com-

mensal (0 +) and amensal (0 –). Model (i) in Fig. 1,

for example, represents a three-variable system

where variable X is a competitor of Y, and Y is the

prey of Z. The strength of these pairwise relation-

ships is typically considered to be fixed and

independent of population size. However, there

can be interactions that are modified by the

abundance of a third variable, which creates

additional direct effects in the system (Dambacher

and Ramos-Jiliberto 2007). In model (ii), of Fig. 1,

variable X enhances the intensity of the pairwise

interaction of Y and Z, as denoted by the dashed-

lined link. Taking the directed sign product of this

link with each link in the predator–prey relationship

creates, in model (iii), direct effects leading from X to

variables Y and Z.

Qualitative modelling tools

Construction of a signed digraph formally represents

how one thinks an ecological system works, and is

the starting point of qualitative modelling. Analyses

proceed either by graphical algorithms (Puccia and

Levins 1985) or mathematical operations on the

community matrix (Dambacher et al. 2002,

2003a,b, 2005; Dambacher and Ramos-Jiliberto

2007). Here we provide only a general overview of

the method, which can be supplemented with more

detailed and technical presentations in the above

cited references. Additionally, computer programs

to draw signed digraphs and perform qualitative

analyses can be found in the most recent revision

of Supplement 1 of Dambacher et al. (2002) in

Ecological Archives E083-022-S1 at http://www.

esapubs.org/archive/.

System feedback and dynamics

From the signed digraph one can examine the

feedback properties of a system to understand its

governing dynamics and reveal key processes and

interactions that act to maintain or thwart a state of

equilibrium. System feedback is formed by the

products of links in the system. Negative feedback

returns the opposite effect to an initial change in a

variable, and acts to maintain a system’s equilib-

rium. A familiar example can be found in the

Signed digraph

Community matrix

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−a

−a−a−a

−a−a

ZZZY

YZYYYX

XYXX

a0

0

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−a

−a−a−a

−a

ZZZYZX

YZYYYX

XX

aa

00

X X

X

i ii iii

Y Z Y Z Y Z

Figure 1 Signed digraphs of model ecosystems with

corresponding community matrices. Open circles are

system variables and directed links represent direct effects;

solid-lined links ending in an arrowhead (or filled circle)

represent a positive (or negative) direct effect, and links

connecting a variable to itself denote self effects. Non-zero

elements of the community matrix correspond to solid-

lined links in the signed digraph. In model (i) variable X

is a competitor of Y, which in turn is a prey of Z. In model

(ii) variable X enhances the intensity of the pairwise

interaction of Y and Z (denoted by dashed-lined link); this

modified interaction creates in model (iii) direct effects

leading from variable X to variables Y and Z.


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process of predation, as in Fig. 1, where an increase

in prey Y leads to an increase in the predator Z,

which in turn acts to reduce the level of Y.

Conversely, positive feedback magnifies change in

variables and drives runaway growth or collapse. In

model (i), for example, an increase in Y leads to a

decrease in its competitor X that acts to increase Y

even further. Thus, the dynamics of Y in particular,

and the system in general, can be understood to be

governed largely by the opposing processes of

predation and competition. A qualitative analysis

of an ecosystem’s structure and feedbacks (Puccia

and Levins 1985; Dambacher et al. 2003a) permits

a rigorous appraisal of its stability, and enhances

understanding of what drives and regulates the

growth of interacting variables.

Shift in equilibrium

When there is a change in a parameter or factor

influencing a rate of birth, death or migration of a

population (i.e. the input variable), then each

population in the community can be affected. In a

press perturbation, the parameter change is

sustained long enough for the system to approach

a new equilibrium, and the response variables

can experience an observable shift in their abun-

dance and average age (or life expectancy).

Predictions for the direction (+, ), 0) of these

shifts are obtained by a summation of all direct

and indirect effects that lead from the input

variable to each response variable. This summa-

tion of effects can be obtained from a qualitative

analysis of the inverse community matrix

(Dambacher et al. 2002). In model (i) of Fig. 1,

an input to variable X, say through a sustained

increase in its rate of birth, is predicted to cause a

decrease in the abundance of Y and Z. This occurs

because there is only one sequence of links, or

pathway of interaction, with a negative sign,

leading from variable X to the other two variables.

In model (iii) increasing the birth rate of X creates

an ambiguous response prediction for Z, as there

are two paths of opposing sign leading to it from

X; here an increase in the abundance of variable Z

depends on the direct path between X and Z being

stronger than the indirect path via variable Y. The

conditions for an increase in variable Z are

described by the inequality aZXaYY > aZYaYX, which

details direct and indirect effects based on the links

corresponding to the aij elements of the

community matrix.

The average age or life expectancy of individuals

in a population is a function of the rates of flow into

(i.e. birth and immigration) and out of (i.e. death

and emigration) the population. These flows are

sensitive to shifts in the abundance of other

variables, and thus a press perturbation to a single

population can produce shifts in the age distribution

of all populations in the community (Dambacher

et al. 2005). In general, increases in variables that

control the rates of birth or death of a population

will cause individuals to move through it more

rapidly, thereby lowering average life expectancy,

and thus mean age and possibly mean size. An

important assumption in this calculation is that all

ages represented by a population variable are

equally susceptible to dying or migration, although

this assumption can be partially relaxed in a model

with discrete life-stage variables (Puccia and Levins

1985, pp. 62–65 and 131–135; Dambacher et al.

2005, pp. 10–11).

Predictions for life expectancy change in a

population are obtained by accounting for expected

changes in abundance, as translated through the

positive and negative links in the system’s structure.

In model (i), for example, a negative input to X will

cause an increase in Y, which controls Z’s rate of

birth. Thus, Z will experience an increased rate of

birth because of increased availability of prey, and

its age distribution will shift to include a higher

proportion of younger individuals, thus decreasing

the average age of individuals in the whole popu-

lation. In model (iii), a negative input to X gives an

ambiguous prediction for Z, and a decrease in Z’s life

expectancy depends on the relative strength of the

alternative paths leading from X to Z; a condition

that is expressed by the inequality aZXaYY < aZYaYX.

Ambiguous predictions with only a few terms are

comparatively simple to interpret, and prediction

signs can be resolved by considering the relative

strength of specific interactions (Puccia and Levins

1985; Dambacher et al. 2002). However, ambigu-

ous predictions with a large number of terms can

exceed our ability to interpret them reasonably. In

such instances, one can rely instead on the ratio or

weight of positive and negative terms in a response

prediction. This ratio can be used as a relative

measure of the degree of ambiguity in a response

prediction, and is formally defined as a prediction

weight by dividing the net number of terms by the

total number of terms. In calculating a prediction

weight, negative and positive terms cancel in

calculation of the net number of terms, thus a ratio


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of 3:1 terms of opposing sign corresponds to a net of

two terms divided by a total of four, giving a

prediction weight equal to 0.5. Simulation studies

by Dambacher et al. 2003b and Hosack et al. 2008

tested the sign determinacy of qualitative predic-

tions across an assortment of signed digraph mod-

els. By randomly allocating within each model the

magnitude of non-zero community matrix elements,

they were able to test how often a qualitative

prediction matched the sign of responses in quan-

titatively specified matrices. These studies estab-

lished that when the number of opposing terms in a

qualitative prediction equals or exceeds a ratio of

3:1, or prediction weight ‡ 0.5, then a relatively

high likelihood (>90%) of sign determinacy can be

ascribed to a response prediction.

Time averaging

The preceding analyses are useful for systems that

are at or near equilibrium, or where parameter

change is slow enough that the system can adjust to

a moving equilibrium. It is sometimes the case,

however, that parameters are affected by a factor

that is changing too rapidly for the system to track,

thus preventing the system’s variables from reach-

ing, or remaining in, an equilibrium. Over time the

variable’s trajectories may appear periodic or cha-

otic, and defy any description in terms of a familiar

level. But if the trajectories are contained within

fixed bounds, then the system can be considered as

exhibiting sustained bounded motion, and the

technique of time averaging applies (Puccia and

Levins 1985).

Time averaging deals with the average abun-

dance of variables and predicts patterns of covari-

ance for variables that are linked via direct effects

(Puccia and Levins 1985). In general, time averag-

ing gives results similar to those for press perturba-

tions. For instance, in model (i) of Fig. 1, if there

were a stochastic input to X’s rate of birth, such that

over time it fluctuated within fixed bounds, then

time averaging predicts that the abundance of

X and Y will have a negative covariance and

Y and Z’s abundance will have a positive covari-

ance. These predictions match the correlation pat-

terns for a press perturbation to X, which is

predicted to cause a change in the abundance of

X that is opposite in direction to Y and Z. However,

time averaging predictions are limited by ambigu-

ities in system structure, and can be confounded

when there is more than one path emanating from

the input variable, or where there are multiple paths

connecting the pair of variables for which the

covariance predictions are being made. These

ambiguities, unfortunately, cannot be resolved by

examination of algebraic arguments. Thus, in model

(iii), unambiguous time averaging predictions are

possible when the source of input is through

variables Y or Z, but when the source of the input

is through variable X all predictions are ambiguous.

Besides changes in abundance, a population

experiencing sustained bounded motion can be

expected to exhibit concomitant change in its

average age. In general, when the source of

stochastic input is from a lower trophic level, then

predators and prey will have a positive covariance.

Peaks in a population’s abundance will therefore be

expected to coincide with peaks in both resource

availability and predator abundance. The ensuing

high rates of birth and death would thus cause

population peaks to coincide with a younger pop-

ulation. Alternatively, if the source of the input is

from a higher trophic level, then the covariance of

predators and prey will be negative, and population

peaks can be expected to coincide with an older

population.

Model formulation

We developed five qualitative models for a range of

fishery-stock systems (Fig. 2). The core model, A,

has two fishing fleets F4 and F5 that selectively

target two fish stocks, a top predator S1 and its prey

S2. Model B includes the effects of omnivory, and in

model C, S2 functions as a wasp-waist variable (Rice

1995), as it is the only variable that links multiple

resources to multiple predators. In model D there is

incidental catch of S2 that provides no benefit to the

fleet F4, while in model E both S1 and S2 are

targeted by F5. These last two models are relevant to

trawl fisheries that target valuable crustaceans but

also catch juvenile teleosts, or larger, highly valued,

teleosts, model E could also represent a fishery that

derives benefit from a by-catch species that is not

specifically targeted.

In these models, the relationship between fishing

fleets and fish stocks is depicted as that of predator

and prey. The negative link from the fleet to the fish

stock is q, catchability, and the positive effect to the

fleet is q again times some measure of economic

efficiency, such as market price of catch. These

models could represent other systems equally well.

For example, F5 could be a disease of S2, and S3


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could be a specific geochemical variable, such as

nutrients, or a more general variable, such as

habitat quality, in which case the definition of the

other variables and links in the system would need

to be recast in light of the new context.

Treating the interaction of fleets and fish stocks as

a predator–prey type relationship permits analyses

of perturbation and response, but it masks impor-

tant bio-economic processes and feedbacks. Expand-

ing the model to incorporate standard components

of a commercially fished system, namely effort,

catch, stock and market, provides insight into the

dynamics of real systems without undue need for

quantification. An expanded fleet-stock model

(Fig. 3a) includes the interactions of fish stocks S,

fishing effort E, market price M, and the processing

of catch C. Fishing effort imparts mortality to the

stock via the link –aSE, and processed catch is

treated as an intervening variable between stock

and effort, thus creating links aCS and aEC. The link

aEC represents the servicing of operating and capital

replacement costs of the fishing fleet and also the

reinvestment of profit into additional units of effort

(i.e. labour, boats, gear, etc.). Through a supply-

and-demand relationship, market price is a decreas-

ing function of catch, giving the link –aMC. Market

price, in return, increases the magnitude of the link

aEC. This enhancement constitutes a modified inter-

action that creates an additional positive direct

effect leading from M to E. Finally, aCE represents

the processing of catch as an increasing function of

effort.

Perturbation scenarios

We applied three perturbation scenarios to the

models in Fig. 2. First, we examined the consequence

of a change in catchability, which often increases

continuously through technical progress and adap-

tive fishing practices (Branch et al. 2006). Next, we

examined the consequences of environmental

change (e.g. climatic, pollution, eutrophication) that

affects the productivity of basal resources. Finally, we

examined the effect of change in fishing effort to give

insight to the ecosystem-level consequences of

economic influences or management actions.

Results

Indicators of exploited ecosystems

In Table 1, predictions for models A–E are presented

in terms of change in abundance and life expec-

tancy of stocks S1 and S2, following a sustained

increase in the catchability of S1 by fishing fleet F4,

an increase in the productivity of the resource S3, or

an increase in the effort of fishing fleet F5. An

increase in catchability constitutes a simultaneous

press perturbation to both S1 (negative press) and

the fleet F4 (positive press), from which S1 is pre-

dicted to become fewer and younger, while S2 is

predicted to become more abundant and older. In

the second column, a sustained increase in the

productivity of S3 is predicted to increase the

abundance and decrease the average life expectancy

of both S1 and S2. Lastly, an increase in the effort of

F5 is predicted to decrease S1 and S2, but the life

expectancy of S1 is predicted to increase while S2’s

life expectancy is predicted to decrease.

In the first two perturbation scenarios of Table 1,

a predicted decrease in S1’s life expectancy is caused

by a predicted increase in both its rate of birth, from

increased availability of resource S2, and death,

from increased fishing mortality. However, S1’s

S1

S2

S3

F4

Model C

S6

S7

Model B

Omnivory system

Model A

Core model

Model D

Fishery bycatch

Model E

Multi-target fishery

Wasp-waist system

F5

S1

S2

S3

F4

F5

S1

S2

S3

F4

F5

S1

S2

S3

F4

F5

S1

S2

S3

F4

F5

Figure 2 Signed digraph models of fleet F and stock S

variables in different ecosystems and with different harvest

practices. The core model has two fishing fleets F4 and F5

that selectively target stocks, including a top predator S1

and its prey S2. In the omnivory system, S1 is also a

predator of S3. In the wasp-waist system, S2 is the only

variable that links top predators to multiple resources at

lower trophic levels. In the fishery by-catch model, fleet F4

has incidental catch of S2, but derives no benefit from it,

while in the multi-target fishery model, F5 derives benefit

from catch at two trophic levels.


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abundance is predicted to decrease in the first

scenario and increase in the second. The third

perturbation scenario has the opposite effect on S1’s

life expectancy by diminishing both resource avail-

ability and mortality from fishing, thus slowing

rates of flow into and out of its population

and causing a predicted increase in average life

expectancy.

Some of the predictions for models B, D and E are

ambiguous. These ambiguities, detailed as inequal-

ities in the footnotes of Table 1, can be resolved by

considering the relative strength of specific interac-

tions. In model E (multi-target fishery), for instance,

the positive response of S1 to an input to S3 is

determined by the inequality a1,2a5,5 > a1,5a5,2,

meaning that the interaction of fleet F5 with the

system, via links a1,5 and a5,2, should be relatively

weak. Consequently, it can be predicted that S1, a

top-level consumer, will increase in abundance if it

is known or presumed that the growth it derives

from consumption of prey S2 (a1,2), and self-regu-

lation in fleet F5 (a5,5) is greater than the product of

F5’s rate of profit and reinvestment in effort from

catch of S2 (a5,2) and the catchability of the other

target stock (a1,5). For model E, the predicted change

in average life expectancy of S2 because of increased

productivity of S3 is determined by eight terms,

seven negative and one positive. As it is unlikely

that the single positive term will exceed the com-

bined strength of the seven negative terms, one can

proceed with a prediction of a decreased life

expectancy for S2.

The analysis can also be used to identify the

source of input to the system. If a press perturbation

has occurred through one of the three scenarios

considered in Table 1, and if data are available on

trends in the abundance and mean age of S1 and S2,

then the columns of Table 1 provide unique com-

EffortExpanded fleet-

stock model

E

M

E

CC C C

M

EE

S S

C

M

E

S

Capital-ization

Marketregulation

Stockregulation

Scarcity-driveneffort

aCE

aSS

> aCS

aSE

aCE

aSS

< aCS

aSE

Ann

ual c

atch

or

reve

nue

DP = 0 annual cost plus DP

DP =4 / 5

cost

Profit

ZPMP

(a) (b)

(c)

Figure 3 (a) Signed digraph model of a fish stock S and harvest variable that has been expanded to include fishing effort E,

the processing of catch C, and market price M. (b) Relationship between catch and effort, and revenue and effort for a

fisheries. The revenue curve is generated with a fixed market price for catch and is superposed with the catch curve.

However, if market price is sensitive to supply and demand then the height of the revenue curve, but not the catch curve,

would be depressed. The arrow at peak of catch curve denotes a threshold that reverses an inequality which, in the

expanded fleet-stock model, defines the direct and indirect effects of increased effort on catch. The thin solid line shows a

fixed rate of cost plus no added distributed profit (DP), and the dashed line shows a non-zero value of DP that is added to

cost. The intersection of the cost-plus-zero-DP line with the revenue curve defines an area of profit (shaded region), and

determines a point of equilibrium (s) where there is zero profit (ZP). In this particular graph, an equilibrium with

maximum profit is obtained when distributed profits are paid out at a rate that is 4/5ths of the cost; the corresponding effort

required to maintain this equilibrium is ½ of that for ZP. (c) Interactions between model variables give rise to both

positive and negative feedback cycles that regulate system equilibrium.


� 2009 CSIRO


binations of signs to diagnose the likely source of

input. For instance, an increase in the abundance of

stocks S1 and S2 is only consistent with a sustained

change in a parameter that directly increases the

growth rate of the resource S3. Similarly, an

increase in the average life expectancy of the top

predator S1, matched by a decrease in life expec-

tancy of its prey S2, would only be consistent with

increased effort in F5.

Ambiguous predictions prevent distinguishing

with certainty the source of input for models B, D

and E. Our ability to diagnose the source of input

for these models improves, however, if we can apply

prediction weights or knowledge of the relative

strength of interactions to key inequalities. Addi-

tionally, one can take advantage of duplication

among some of the inequalities. For instance, in

model D (fishery by-catch) there are three ambi-

guous predictions in the first column of Table 1,

which makes it indistinguishable from the third

column. Thus, we can only be certain of distin-

guishing input via the resource S3, from the two

other sources. But since all three of these ambiguous

predictions are determined by the same inequality,

a2,1a1,4 + a2,1a4,4 + a2,4a4,1 > a2,4a1,1, which has

a relatively high likelihood for sign determinacy

(i.e. prediction weight equal to 0.5), then it is

possible to postulate a most likely source of

input. Distinguishing among all three sources of

input in model B (omnivory system) is possible if it

was established that a1,2a3,3 > a1,3a3,2, which

would be likely if stock S2, and not S3, were the

principal prey of S1.

We next consider that the nature of the pertur-

bation is not a sustained directional change in a

parameter, but rather one that causes the param-

eter to vary randomly or periodically within limits

over time (such as cyclical climate changes that

alters a population’s carrying capacity). Through

time-averaging (Puccia and Levins 1985) we gain a

set of predictions for covariances between model

variables (Table 2), which again affords unique sign

combinations to discern the source of the perturba-

tion. For instance, a negative covariance between S2

and the fleet F5 is only consistent with variation

entering through the fleet, and is unambiguously

predicted in all models except E. Similar to predic-

tions for press perturbations, time averaging pre-

dictions suffer increased ambiguity with increased

model complexity. From the combination of covari-

ance signs (Table 2), it is possible to distinguish the

source of the input for models A (core model) and C

(wasp-waist system). However, for models B (omni-

vory system) and D (fishery by-catch) only F5 can be

distinguished as a unique source of input, while

for model E (multi-target fishery), which is

the most complex model, no input source can be

distinguished.

Example application

A predator, anglerfish (Lophius piscatorius, Lophii-

dae), and its prey, horse mackerel (Trachurus

trachurus, Carangidae), in the Bay of Biscay, North-

east Atlantic, illustrate the use of qualitative mod-

elling to interpret bottom-trawl time-series data

(Poulard et al. 2003; Trenkel et al. 2007) for mean

length, a surrogate for life expectancy, and

Table 1 Sign of predicted change in abundance and life

expectancy for variables S1 and S2 in models A–E (Fig. 2)

for a sustained increase in the catchability q of S1 by fleet

F4, an increase in the productivity of S3, or an increase in

effort for fleet F5. Models are listed where response

predictions are ambiguous, and footnotes refer to

inequalities composed of aij interactions that determine the

sign of the predictions. Asterisks denote inequalities that

have at least a three-to-one ratio in the number of terms,

which coincides with prediction weights ‡ 0.5 that in

simulation studies (Dambacher et al. 2003b; Hosack et al.

2008) have been shown to have a high probability of sign

determinacy. Model B: omnivory system, model D: fishery

by-catch, E: multi-target fishery.

Response/

variable

Source of sustained input

Increased q of

S1 by F4

Increased

productivity of S3

Increased

effort for F5

Abundance

S1 ) + (Ea) ) (Db)

S2 + (Dc*) + (Bd) ) (Ee)

Life expectancy

S1 ) (Bf*, Dc*) – ) (Bb, Ee)

S2 ) (Bg, Dc*) + (Eh*) + (Ee)

aa1,2a5,5 > a1,5a5,2.ba1,2a3,3 > a1,3a3,2.c*a2,1a1,4 + a2,1a4,4 + a2,4a4,1 > a2,4a1,1.da2,3a1,1a4,4 + a2,3a1,4a4,1 > a2,1a1,3a4,4.ea2,5a1,1a4,4 + a2,5a1,4a4,1 > a2,1a1,5a4,4.f*a2,1a1,2a3,3a5,5 + a3,1a1,2a2,3a5,5 + a1,3a3,1a2,2a5,5

+ a1,3a3,1a2,5a5,2 > a1,3a3,2a2,1a5,5.ga3,2a2,1a5,5 > a3,1a2,2a5,5 + a3,1a2,5a5,2.h*a1,2a2,1a4,4a5,5 + a1,1a2,2a4,4a5,5 + a1,4a4,1a2,2a5,5 + a1,5a5,1

a2,2a4,4 + a2,5 a5,2a1,1a4,4 + a1,4a4,1a2,5a5,2 + a1,2a2,5a5,1a4,4

> a1,5a5,2a2,1a4,4.


� 2009 CSIRO


ln-abundance. The system is probably well described

by model A (Fig. 2), as the anglerfish (S1) and horse

mackerel (S2) species are fished by separate fleets,

and anglerfish are a known predator of horse

mackerel (Quero 1984). Time variations of abun-

dance and mean length in Fig. 4 suggest the system

exhibits sustained bounded motion. In most years

the abundances of both species vary in synchrony,

cov(anglerfish, horse mackerel) = 0.51, while

abundance and mean length are negatively related

within both species, cov anglerfish(ln-abundance,

length) = )0.73 and cov horse mackerel(ln-abun-

dance, length) = )0.72.

For brevity we limit ourselves to consideration of

predictions for model A and the three perturbation

scenarios in Table 2. Here, a positive covariance

between anglerfish and horse mackerel is inconsis-

tent with a predicted negative covariance of S1 and

S2 when input enters through S1. We are left to

decide among input via a lower trophic level S3 or

the fishing fleet F5, both of which predict a positive

covariance between anglerfish and horse mackerel.

If the system were driven by changes in the

fishing effort of the fleet F5, then this input would

essentially function as top-down, with a cascading

effect on the horse mackerel S2, and its resource

S3. Peaks in horse mackerel abundance would

therefore coincide with lows in both its fishing

mortality and resource availability, both of which

would increase the average age of horse mackerel

and, counter to the time series in Fig. 4, give a

positive covariance between its abundance and

mean length. However, this predicted effect is not

clear-cut as increases in fishing mortality are

predicted to decrease the abundance of anglerfish,

resulting in a partial or full compensatory reduc-

tion in horse mackerel mortality. This reduction

will diminish or eliminate, but most importantly

for our purposes not reverse, the predicted positive

covariance of horse mackerel abundance and

average age or length.

If input to the system is via S3, then peaks in the

abundance of horse mackerel are predicted to

coincide with high levels of resources and predation,

both causing a younger population of horse

mackerel with a concomitant reduction in mean

length. Thus, we conclude that the dynamics of

both the anglerfish and horse mackerel populations

were most likely driven by variations in system

productivity rather than variations in the mortality

of top predators or fishing pressure. This conclusion

could be further corroborated by time-series data on

the fishing effort of fleet F5. A positive covariance

with S2 would support the above conclusion, while

a non-positive covariance would cast doubt on the

source of the input. However, effort data for fleet F4

would not help to distinguish among the three

scenarios, but if they revealed a negative covariance

with S1, then they would cast doubt on model A.

Here we have considered only model A and the

perturbation scenarios of Table 2, a more compre-

hensive analysis, however, could include additional

model structures and scenarios, such as recruitment

of anglerfish and horse mackerel both being driven

by a shared factor or resource.

Table 2 Sign of covariance between variables in models

A–E (Fig. 2), as calculated by time-averaging with sto-

chastic input to parameters associated with S1, S3, or F5.

Signs of covariances are unambiguously predicted for the

listed models, while models with ambiguous predictions

are not listed. Model A: core model, model B: omnivory

system, model C: wasp-waist system, model D: fishery

by-catch, model E: multi-target fishery.

Covariance

Source of stochastic input

S1’s survival S3’s productivity F5’s effort

S1 & S2 ) (A, C) + (A, C) + (A, C)

S1 & F4 + (A–E) + (A–E) + (A–C, E)

S2 & F5 + (A–D) + (A–D) ) (A–D)

Year

Nor

mal

ised

inde

x

1990 1995 2000 2005

0

4

8

12

ln-abundanceMean lengthAnglerfish

Horse mackerel

Figure 4 Normalized time series of ln-abundance (solid

line) and mean length (dashed line) for a predator,

anglerfish (Lophius piscatorius, Lophiidae), and prey, horse

mackerel (Trachurus trachurus, Carangidae), based on

1988–2006 bottom-trawl survey data in the Bay of Biscay.

Time series are shifted along a normalized y-axis to

separate species.


� 2009 CSIRO


Expanded fleet-stock model

The expanded fleet-stock model in Fig. 3a repro-

duces the same basic dynamics of a two variable

predator–prey type model, but adds greater resolu-

tion to the system’s dynamics. As in a two-variable

system, a press perturbation to effort in the

expanded model leads to a decline in the equilibrium

level of the stock, but in the expanded model the

response of catch is ambiguous, and its increase

depends on aCEaSS > aCSaSE. The tipping point of this

inequality coincides with the maximum of a catch-

and-effort curve (Fig. 3b), where catch initially

increases with effort and then declines. The curve

also describes the revenue obtained from the catch,

which, when costs are subtracted out, defines the

general scope for profit in the fleet, and thus

capacity for reinvestment into fishing effort. The

model considered here, then, applies only to the

profitable region of effort and catch that is to the left

of the point of zero profit in Fig. 3b. Describing

dynamics outside this region would require a

different model where the links aEC and aEM were

both negative.

Innovations in gear or tactics that increase the

catchability of a stock can be treated as a simulta-

neous negative input to the stock and a positive

input to catch. Here again the predicted response of

catch is ambiguous, and its increase depends on

aSS > aCS, which requires that the catch rate does

not exceed the regenerative capacity or self-regula-

tion of the stock. This inequality also determines, in

part, the predicted response of effort to an increase

in catchability via (aSS – aCS)(aECaMM – aEMaMC),

which additionally depends on the balance between

aECaMM, the product of market self-regulation and

the amount of reinvested profit, and aEMaMC, the

extent to which market price is affected by the

change in supply of catch.

In the expanded fleet-stock model, there are four

feedback cycles that drive (via positive cycles) and

regulate (via negative cycles) the system (Fig. 3c).

The positive feedback cycle between effort and catch

fuels capitalization of the fleet, and is the main driver

of the system. Reinvesting all profits into effort leads

inexorably to an equilibrium level where revenues

equal costs, and profits are zero (Fig. 3b). If instead,

profits are distributed from the system, say as a fixed

proportion of costs, as in the dashed line in Fig. 3b,

then the limitation placed on reinvestment will lead

to a lower equilibrium level of effort. This limitation

on reinvestment can be treated as a negative input to

effort, which is predicted to decrease the equilibrium

level of effort, increase the abundance of the stock,

and either increase or decrease the level of catch

depending on the relative strength of aCEaSS vs.

aCSaSE. In the particular graphical example presented

in Fig. 3b, the maximum amount of profit that can

be sustainably generated occurs when distributed

profits are set at 4/5ths of costs. The level of effort

at this equilibrium is equal to 1/2 of that for the

zero-profit equilibrium.

The expanded fleet-stock model is regulated by

two negative feedback cycles involving the fish stock

and the market. These cycles contain the links

associated with capitalization feedback, and both act

to limit the equilibrium level of effort in the system,

either by diminishing the abundance of the stock or

market price. These feedback cycles are also related

to the shape of the catch and revenue curve in

Fig. 3b. Strong stock or market regulation effec-

tively limits the height of the curve for catch or

revenue, respectively, leading to a reduced equilib-

rium level of effort. Lastly, there is a single positive

feedback cycle that embodies scarcity-driven effort,

which is fuelled by high consumer demand, and

thus market price, for a diminished fish stock.

Multi-species fleet-stock model

An important aspect of fleet-stock dynamics is that

they can be affected by the management of other

fishing fleets, as well as consumer demand for other

species in the community. To understand the

feedbacks that can arise in an ecosystem when

there is harvest of multiple species, both harvest

variables were expanded in the core model in

Fig. 5a. A critical assumption here is that the top

predator S1 is the preferred choice of consumers,

and that consumer purchases of S2 are contingent

on either S1’s availability or its price.

In the negative link to E5 from C4 (Fig. 5a), we

consider the possibility that the capacity of a fishing

fleet is apportioned according to the level of catch of

the preferred stock. Here, a high catch of S1 leads to

suppression of effort for S2, or oppositely, a reduced

catch of S1 would cause effort for S2 to rise. The link

between C4 and E5 creates a positive feedback cycle

that drives the serial depletion of S1 and S2 (Fig. 5b)

– i.e. capacity-driven serial depletion: a decrease in

abundance of the preferred stock S1 leads

to reduction in its catch C4, causing effort E5


� 2009 CSIRO


to increase, leading to reduction of S2, a prin-

ciple prey of S1, resulting in further diminishment

of S1.

Next we consider that consumer choice between

S1 and S2 is based either on S1’s availability or its

price relative to S2. The first dynamic is achieved in

Fig. 5a through a negative link from C4 to M5,

where demand, and thus market price, for S2 drops

in proportion to the availability of catch of S1. Here,

consumer choice for S2 is based only on the

availability of S1, and not its relative price. Where

price is the deciding factor, then an increase in the

price of S1 leads to an increase in the demand, and

hence price, for S2. This effect is represented by a

model structure that includes a positive link leading

from the market price of S1 to the market price of S2.

Either manner of consumer choice creates a positive

feedback cycle that drives serial depletion (Fig. 5b) –

i.e. availability- or price-driven serial depletion.

Note that serial depletion essentially describes a

state of disequilibrium that is driven by positive

feedback, thus press perturbation and time-averag-

ing analyses do not apply. However, the arrange-

ment and order of the links within the positive

feedback cycles of Fig. 5b suggest likely indicators

for monitoring. All three positive feedback cycles

include E5, which implies that decreased landings at

a higher trophic level will always coincide with

increases in fishing effort for species at a lower

trophic level. Moreover, if an increase in E5 precedes

an increase in M5, then it supports the existence of

the C4 to E5 link and capacity-driven serial deple-

tion. The market price of the less preferred stock M5

is in two of the positive feedback cycles, while M4 is

only in one, which suggests they will have a either a

negative or zero correlation with diminished land-

ings of top-predator species.

While examining an expanded depiction of a fleet

and its stock in a multi-species system has revealed

important feedbacks and processes, the essential

dynamics of the system can nonetheless be depicted

by a simpler model with fewer variables. In Fig. 5c,

a negative link between fleets F4 and F5 has been

added to the original core model, which creates a

single positive feedback cycle that drives serial

depletion of S2 and S1.

Price-drivenserial depletion

Capacity-drivenserial depletion

Availability-drivenserial depletion

General model ofserial depletion

S1

C4

M4

M5

C5S3

S2

M5

E4

M

(a) (b)

(c)

4 S1

C4

E 5

S2

E5

M

S

5

3

S2

S1

S1

S2

C4

E5

S1

S2

C4

E5

F4

F5

Figure 5 (a) Signed digraph model

of a two-fleet-two-stock fishery using

the expanded fleet-stock model of

Fig. 3a, with three dashed-lined links

showing possible interactions be-

tween variables in the fleet-stock

subsystems. (b) Each of these inter-

actions form a positive feedback cycle

that can drive serial depletion of S1

and S2, the essential dynamics of

which are captured in (c) by a

general model with reduced

dimension.


� 2009 CSIRO


Discussion

Utility of approach

We have examined the utility of qualitative model-

ling for predicting change in populations belonging

to complex exploited communities under various

pressures, and investigated the main feedback

controls in commercial fisheries. Analysing press

perturbations across a suite of community models

allows us to identify predictions that are robust

to uncertainty in system structure. In addition,

patterns of covariance can provide insight into

communities that are in a state of disequilibrium,

often thought to be unpredictable. Whether there is

omnivory or a wasp-waist structure in a community,

or whether exploitation is selective or not, change in

food availability to stocks or in the intensity by

which they are harvested can have similar conse-

quences in terms of stock abundance or life expec-

tancy. Where there is ambiguity in these predictions,

then the use of qualitative analysis in combination

with knowledge of other aspects of the system

identifies a subset of key parameters to be measured,

or critical tipping points in the system, thus focussing

research and progressing management without full

parameterization of a model. Moreover, different

sources of perturbation often have a unique signa-

ture of impacts across the system. We maintain that

qualitative modelling can aid a formal process for

selecting between potential indicators, by overcom-

ing subjective- or intuition-based approaches that

may provide no useful information or defensible

advice within the context of a particular problem.

Moreover, they provide a relatively rapid, albeit

imprecise, alternative or complement to simulation

models (Fulton et al. 2005; Hall et al. 2006), to

evaluate the usefulness and sensitivity of potential

indicators (Hayes et al. 2008).

The combination of qualitative predictions has

several advantages. First, predictions from qualita-

tive models can be readily tested without requiring

the often onerous task of model parameterization. In

ecosystems where the source of perturbations are

well known, expected directional changes in abun-

dance and life-expectancy at different trophic levels

can be predicted, and checked for, in time-series

data. In doing so, we recognize the importance of

validating models, like many fisheries scientists

(Sparre and Hart 2002), but also argue that it is

not necessary to fully parameterize models to test

them. For example, the relevance of the models

presented in Fig. 2 will be tested in a large-scale

analysis of trends in survey-based population and

community indicators, by assessing whether or not

combined changes in indicators follow changes in

environmental drivers in a way that conforms to

predictions in Table 1 (M.-J. Rochet and V.M.

Trenkel, unpublished data). Validation of qualitative

models can also be facilitated by the use of Bayesian

belief networks, allowing one to judge relative

consistency of model predictions with data (Hosack

et al. 2008). If a given model is validated, then it

can be used in ways detailed below. If not, then the

flaw may well be in the model’s structure, not in

parameters or estimation techniques, and the

structure can be revised and tested again. This

process of falsification and revision leads to a deeper

understanding of the system, and is a significant

step forward compared to the difficult problem of

validating numerical models in natural sciences

(Oreskes et al. 1994).

Second, in complex systems, direct and indirect

interactions can create feedbacks with counter-

intuitive effects that make ecological predictions

difficult (Dambacher et al. 2002). The perturbation

scenarios of Tables 1 and 2 reveal important, often

opposite, predictions in the response of S1 and S2.

These predictions could scarcely be developed by

intuition alone or analyses of single species models.

There are numerous examples in the literature

where narrative or conceptual models offer a general

understanding of a system’s dynamics – e.g. through

conceptual models Bakun (2001, 2006) and Bakun

and Weeks (2006) described several feedback pro-

cesses that are known to operate between marine

predators and prey, and which reinforce alternative

ecosystem states. While these models are certainly

useful in describing the dynamics of some well-

known systems, their utility is largely limited to the

contexts and environments in which they were

developed. Between the narrative approach taken by

these authors, and fully parameterized models

designed to seek unique sets of solutions, qualitative

modelling provides a complimentary method that,

while imprecise, is nonetheless rigorous and flexible.

Third, qualitative predictions have a high poten-

tial for ecosystem-based fisheries management. To

support ecosystem-based fisheries management,

Rochet and Trenkel (2009) proposed a conceptual

framework to use known interactions in an ecosys-

tem to connect the problems to be managed with

possible management actions. Here we propose that

this framework can be implemented via qualitative


� 2009 CSIRO


modelling. A validated model structure could then

be used to interpret combined trends in indicators,

whether they have been suggested by population-

level analyses (Trenkel et al. 2007), or those at the

community level (Rochet et al. 2005).

The Example Application shows how examining

trends in population abundances and mean lengths

at two trophic levels has the potential to identify the

origin of an ecosystem’s major driver, whether from

top-down or bottom-up processes. The analysis also

illustrates how qualitative models can be used to

focus selection of an indicator where acquired data

are more likely to be interpretable in the context of a

specific problem. However, compared to previous

applications (Rochet et al. 2005; Trenkel et al.

2007) the interpretation of combined trends in

indicators based on qualitative analysis would rely

on ecological theory and a rigorous analysis of

feedback cycles, instead of expert opinion. Here

again, the potential for qualitative models in

practical application is shown to be midway

between discursive–conceptual models and quanti-

tative models that are fully parameterized.

Characterizing harvest systems and their

feedbacks

While the expanded fleet-stock model of Fig. 3a has

been developed within a fisheries context, it is

generally applicable to the commercial exploitation

of any renewable natural resource. It provides a

general understanding of the feedbacks that drive

and regulate a fishery, is consistent with general

bio-economic theory (Clark 1990), and recognized

phases in the development of a fishery (Hilborn and

Walters 1992). In the early phases of a fishery,

increases in effort can be fuelled by reinvestment of

profits, but also by subsidies and other fishers

joining the fleet. As fishing effort increases a peak

in catch is reached, beyond which the system settles

upon an equilibrium with a limited profit margin

that is determined by the regulatory effects of the

market and the stock’s capacity for regeneration.

There are few documented cases for the occurrence

of feedback based on scarcity-driven effort, where, in

the face of stock collapse, effort is buoyed by

elevated prices for rare catch. However, Sadovy

and Cheung (2003) described such a case for a

highly valuable species, the Chinese bahaba (Bahaba

taipingensis, Sciaenidae), which has been driven to

near extinction by the spiral of increasing prices in

conjunction with diminished landings. Atlantic

bluefin tuna (Thunnus thynnus, Scombridae) could

be another candidate, as it is undergoing heavy

overfishing while market demand maintains an

exorbitant price (Fromentin and Powers 2005).

Our modelling has treated fishing fleets as preda-

tors, which impart a rate of mortality to their prey

and in return react by reinvesting profits from catch

back into effort. Walters and Martell (2004), Chap-

ter 9 argued that this view is no longer appropriate

for modern fisheries for three main reasons: (i)

broad availability of venture capital, (ii) global

mobility of accumulated capital stock, and (iii)

predators (fleets) not depending on a single prey

(stock). Arguments (i) and (ii) imply positive input

to effort from outside the model system, while

argument (iii), as well as (ii), could be interpreted as

a shift in the targeting of stocks within the model

system. The two-fleet-two-stock model of Fig. 5a

accounts for shifts in effort between stocks with a

link from C4 to E5. Additional ways of modelling

multi-species harvesting include, but are not limited

to, models D and E of Fig. 2.

The two-fleet-two-stock model represents feed-

backs between the fishery, market, and ecosystem,

and reveals the potential for serial depletion via

three feedback cycles that are created by the

transfer of effort between fishing fleets, or a dual

aspect of consumer choice. Serial depletion describes

a system in a state of disequilibrium, but our

analysis of its feedback cycles nonetheless suggests

useful indicators to support understanding of its

dynamics. For instance, the serial depletion of fish

stocks from higher to lower trophic levels has been

termed ‘fishing down the food web’ by Pauly et al.

(1998). It has been documented in the Celtic Sea by

Pinnegar et al. (2002), who found pelagic fisheries

expanded simultaneously with decreasing landings

of higher trophic level species, for which prices

increased. Subsequently, the price for species from

lower trophic levels also increased (Pinnegar et al.

2006). The immediate increase in fishing effort at

the lower trophic level suggests the existence of the

link from C4 to E5, and capacity-driven serial

depletion. A concomitant increase in price for

higher trophic level species, and a delayed increase

in price for lower trophic level species supports the

existence of the M4–M5 link and price-driven serial

depletion, but not necessarily the C4–M5 link and

availability-driven serial depletion. Essington et al.

(2006) documented this phenomenon across multi-

ple fisheries, but also described an alternative

mechanism termed ‘fishing through the food web,’


� 2009 CSIRO


whereby fisheries at low trophic levels increase

independently of declines in the catches of apex

predators.

Serial depletion, as described here, is caused by

consumer preference for species at higher trophic

levels. Switching this preference to a lower trophic

level would create negative feedback between the

variables in the expanded fleet-stock model. This is

easily seen in Fig. 5b by switching subscripts 1 and

2, subscripts 4 and 5, and the sign of the link

connecting the two stocks. Alternatively, the same

can be observed in Fig. 5c by simply reversing the

direction of the negative link between the two fleet

variables. The negative feedback cycles created by

the preference shift would act to stabilize the system,

in accordance with the hypothesis that it might be

more sustainable to be consuming lower in the food

chain (Hall 2007; Halweil and Nierenberg 2008).

Management strategies

Qualitative modelling is a practical and rigorous

tool for management strategy evaluation because it

identifies important interactions and feedbacks of

exploited ecosystems. The approach can comple-

ment, or in some cases substitute for, quantitative

simulation tools currently in use or development.

Recognizing that year-to-year decision-making

using stock-assessments encourages a short-term

view for a high cost (Butterworth and Punt 1999), a

substantial effort is currently invested in the devel-

opment of simulation-based evaluations that

account for the long-term consequences and uncer-

tainty when choosing a management strategy.

However, simulation-based approaches suffer vari-

ous challenges including uncertainty in model

structure (Butterworth and Punt 1999), for which

the proposed solution is to use several model

formulations. Qualitative analyses provide the

means to specifically address this problem by

investigating the consequences of different hypoth-

eses about the structure of an ecosystem, and

identify ways of falsifying them. As shown above,

qualitative analysis can indicate which predictions

are robust to different model formulations. Once a

model, or suite of models, is validated, and before

undertaking costly or time consuming simulations,

qualitative analysis can be used to identify a range

of management strategies that are likely to be

effective through adding or subtracting specific

variables and links according to envisaged manage-

ment rules and modes of governance.

Model complexity and utility

In the expanded fleet-stock model of Fig. 3, the

increased resolution of the system’s details comes at

the cost of ambiguity in qualitative predictions.

While interpretation of algebraic arguments associ-

ated with ambiguous predictions can deepen our

understanding of a system and define thresholds in

its dynamics, we are faced with the dilemma that as

model size and complexity increase the algebraic

arguments quickly become too large to reasonably

interpret (Dambacher et al. 2003b). In qualitative

modelling, a trade-off naturally exists between large,

highly realistic and complicated models that are

difficult to interpret, and small, simple models that

lack sufficient detail to be useful. A practical goal is to

not try to account for all species in an ecosystem but

rather to identify a relevant subsystem that captures

the essential dynamics of the system and is tailored

to the problem at hand. The focus then becomes a

representation of the feedbacks that include and

encompass the variable(s) of concern, and the source

of the perturbation, sensu Punt and Butterworth’s

(1995) ‘minimal realistic model.’ A complementary

approach to this problem is to increase the reliability

of model predictions by aggregating populations into

groups based on common trophic and functional

relationships (Hulot et al. 2000). It is expected that

members of a group that are more or less the same in

terms of their links to other species or model

variables will exhibit similar responses to a pertur-

bation. The extent to which the link topology of

species in a group differ will erode the similarity of

their individual responses, thus creating aggregation

error (Metcalf et al. 2008). As with quantitative

ecosystem models (Fulton et al. 2003), qualitative

models will generally be most useful when they are

constructed at an intermediate level of complexity.

Limitations

While this work has stressed the advantages of

qualitative modelling, there are limitations to bear

in mind. Most notably, press perturbation analyses

require that a system is at or near a stationary

equilibrium, or at least tracking a gradually moving

one. This is a particularly strong assumption for

exploited ecosystems that may not be near equilib-

rium for various reasons, (e.g. environmental

change, shifting financial or management con-

straints, or technological innovations). However,

this limitation is pervasive in much of ecological


� 2009 CSIRO


modelling and is not assuaged by quantification.

The challenge can be partly addressed by the choice

of an appropriate time scale. Predicting a change in

equilibrium requires that all variables have adjusted

to the perturbation, thus, a sufficient period needs to

elapse before all of the changes can develop and be

observed (i.e. several generations of longest-lived

species). Time-averaging provides another way to

address this problem by providing predictions of

covariance that can be monitored in a fluctuating

system. Time averaging is especially useful in

diagnosing the influence of weather-driven pertur-

bations, because they typically are system bounded.

Length- or age-structured models are not neces-

sary to calculate changes in average life expectancy.

However, these calculations assume that all ages

represented by a population variable are equally

susceptible to mortality and migration. Where

critical processes or interactions are involved, such

as size-structured ontogenetic diet shifts, then vari-

ables representing specific life stages can be included

within the signed digraph.

Finally, a qualitative model is constructed from

interaction links that have been linearized around a

specific equilibrium state. Non-linear interactions

create the possibility of the system transitioning

across a threshold, whereby new links are created or

the sign of links are possibly reversed. These thresh-

olds require consideration of multiple signed digraph

models to represent different conditions and states of

a system (Dambacher and Ramos-Jiliberto 2007).

Theoretical synthesis

Signed digraph models can represent ecological

and socio-economic relationships in complex sys-

tems, and their qualitative analysis addresses Rice’s

(2001) call for a synthesis of basic ecological

theory to address the impact of fisheries on

ecosystems. The qualitative approach presented

here is meant to balance and augment quantitative

techniques, allowing one to highlight the impor-

tance of ecosystem structure in the early stages of

model building; a step typically eclipsed by the

problem of parameter measurement and estima-

tion. Qualitative models allow one to address

model structure uncertainty separately from para-

metric uncertainty; additionally they provide an

ideal means to disentangle the relative importance

of direct and indirect effects. In so doing they

highlight key parameters and interactions, thereby

focusing research on critical uncertainties in a

system. Qualitative models also provide a logical

way of recognizing the effects of human interven-

tions and environmental change, and can provide

insights into the dynamics and feedbacks of human

and ecological systems.

Acknowledgements

We thank Keith Hayes and two anonymous review-

ers for comments that improved the clarity of this

work. This study received financial support from the

EU funded project IMAGE (FP6 – 044227) and the

project CHALOUPE funded by the French Agence

Nationale de la Recherche, additional funding was

provided by the Western Australian Marine Science

Institution. This research was carried out as part of

CSIRO’s Wealth from Oceans National Research

Flagship.

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