Download On the bioeconomics of predator and prey fishing

Fisheries Research 37 (1998) 179±191
On the bioeconomics of predator and prey ®shing
Ola Flaaten*
The Norwegian College of Fishery Science, University of Tromsù, Breivika, N-9037, Tromsù, Norway
Abstract
A brief review of biological, bioeconomic and non-use objectives in ®sheries management is given. It is shown that
management strategies, with respect to optimal long run effort levels and stock sizes, might differ considerably with the choice
of objective, and of a single-species or multispecies framework. Within a deterministic two-species predator±prey biomass
model, equilibrium catch and resource rent of the predator are positively affected by an increased prey stock level. The effects
on the equilibrium harvest and resource rent of the prey of a change in the predator stock level are ambiguous, although a
negative relationship is most likely. In a simpli®ed model with a linear equilibrium relationship between the two stocks this
negative effect is always present. Such a simpli®ed approach is applied to the case of northeast Arctic cod's consumption of its
commercially most important prey species: capelin, herring, shrimp and small cod, i.e. cannibalism. It is shown that on
average for the years 1984±92 capelin constituted about 75% of the feed, on the basis of weight, of age-speci®c cod of age 2
and older. Shrimp is the second most important prey, constituting about 10±20% of the feed; the higher ®gure occurs for
2-year old cod and the lower for the 7‡ age group. Using Norwegian data on prices of ®sh and cost of effort for the years
1991±92, and data on stomach content 1984±92, it is shown that shrimp constitute 40±80% of the feed costs for all age groups
of 2 years and older, whereas capelin constitutes only 15±20%. For the 7‡ age group cannibalism is just above half the feed
costs. This, despite the fact that capelin is by far the major feed in weight. The assumption of the speci®c functional forms in
biological multispecies models are of importance for the management implications derived. It might be that the inclusion of
economic factors will weaken the relative importance of some biological factors for management strategies. The partial
bioeconomic analyses in this paper strongly indicate that biological and economic factors should be considered
simultaneously in management analyses. # 1998 Elsevier Science B.V. All rights reserved.
Keywords: Predator±prey; Management objectives; Functional forms; Predation costs; Northeast Arctic cod
1. Introduction
This paper analyses and discusses some underlying
assumptions and objectives of multi-species management of ®sheries within the framework of simple
predator±prey models. Within ®sheries systems there
*Corresponding author. Tel.: +47 77 64 40 00; fax: +47 77 64 60
20; e-mail: [email protected]
0165-7836/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved.
PII S0165-7836(98)00135-0
are several types of interactions. For the purposes of
bioeconomic modelling the most important are: (1)
biological interactions, (2) harvest technical interactions, and (3) market interactions. The biological
interactions of most importance in boreal ecosystems
are predator±prey relationships and competition
between species. The interactions between the ®sh
community and the oceanic environment may also be
relevant. Harvest technical interactions, such as when
®shing gear targeting one particular species also gives
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O. Flaaten / Fisheries Research 37 (1998) 179±191
a bycatch of other species, are prevalent in many
®sheries in the north Atlantic. Market interactions
are present when the quantity of one species supplied
affects the market price of another species. This paper
focuses on biological interactions. In the case of
predator±prey interactions it is well known from the
ecological literature that the reduction of the predator
stock level may increase the surplus production of the
prey. When it comes to predators like whales and
seals, however, harvesting is often controversial, as the
following quotation demonstrates:
``An early exploration (of multispecies fisheries),
May et al. (1979), has proved very influential, and
now forms the basis for a very controversial piece
of work, a bioeconomical analysis of the Barents
Sea fishery by Flaaten (1988). Flaaten's work is
controversial because of his conclusion that sea
mammals should be heavily depleted to increase
the surplus production of fish resources for man.
This is in harmony with Norwegian government
policy, which, ever since the European Community
ban on the importation of the skins of whitecoats
(nursing harp seals) and bluebacks (hooded
seals<1-year old) rendered sealing commercially
unviable, has subsidized sealing in the Barents
Sea.'' (Yodzis, 1994, p. 51.)
Harvesting is, however, not the only utility
generated from sea mammals. It has long been
acknowledged that non-use values included in the
objective function may have implications for stock
management. The following quotation demonstrates
this:
``It should, however, be stressed that this result
may be somewhat modified if the resource is
assigned an optional value from people's willingness to pay for keeping the stock at higher level.
A biological argument that also may weaken our
result is the eventual existence of critical depensation for lower stock levels.'' (Flaaten, 1988,
p. 114.)
The aim of this paper is twofold. First, to review and
discuss in a simple way bioeconomic objectives for
®sheries management within single-species and multispecies framework. Second, to analyze and discuss,
within the framework of predator±prey biomass
models, the importance of the assumptions regarding
the speci®cation of the model for the conclusions of
multispecies management studies like, e.g. May et al.
(1979); Flaaten (1988); Yodzis (1994). In addition to
the theoretical analysis an empirical example of costs
in¯icted upon the prey ®shing industries by the predator is presented. This is done for the northeast Arctic
cod (Gadus morhua) and its commercially most
important prey species: capelin (Mallotus villosus),
herring (Clupea harengus), shrimp (Pandalus borealis) and small cod, i.e. cannibalism.
Different kinds of uncertainty are related to the
three types of interactions above. Following Charles
(1992) the three key forms of uncertainty in ®sheries
modelling are:
``(1) random fluctuations (in fish stocks, fish
prices, etc.) within an otherwise fully determined
model, (2) parameter uncertainty due to imprecise
parameter estimates in an established fishery
model, and (3) basic ignorance about the appropriate model to describe the fishery. Indeed, in this
sequence of increasing uncertainty, the initial step
can be considered as that involving deterministic
models, which ignore the uncertainties in fishery
systems but are useful particularly for the qualitative insights they provide. In contrast, the third
degree of uncertainty, basic ignorance, often
reflects the most realistic situation, but such
circumstances are notoriously difficult to model...''
(Charles, 1992, p. 192.)
In recent years new modelling approaches have
been developed by, e.g. the International Council
for the Exploration of the Sea (ICES, 1994, 1997)
for the evaluation of management strategies under
uncertainty. This paper, however, ignores type 1 and
2 uncertainty.
For management purposes the extension of a singlespecies biological model into a bioeconomic model,
by including ®sh prices, operating costs, capital costs,
etc. is a step towards reducing type 3 uncertainty. The
development of a multispecies biological model for an
ecosystem, in addition to single-species models, may
also be a step towards the reduction of type 3 uncertainty, even if the multispecies model is deterministic.
However, in order to keep models simple and operational, additional information, e.g. about biological
interdependence, should perhaps not be included if the
O. Flaaten / Fisheries Research 37 (1998) 179±191
recommendations, the best performing management
strategy or control laws, remain unchanged.
Management objectives and the long run harvest
strategies implied by single-species and multispecies
analyses, and the differences between them, are brie¯y
reviewed in the next section. In Section 3 a general
predator±prey biomass model is designed to analyze
the effects on each species' equilibrium harvest and
resource rent of changes in the other species' stock
level. Section 4 designs a predator±prey model with
linear equilibrium relationship (isoclines) between the
two stocks, and analyses how changes in the slope of
the isoclines affect the harvest and resource rent of
each ®shery. The discussion of the effects on predator
equilibrium harvest and resource rent of changes in the
prey stock level is related to the sea-mammal±®sh
case. Section 5 presents the empirical ®ndings of feed
costs of northeast Arctic cod, using stomach content
data and Norwegian price of ®sh and cost of effort
data. The concluding section discusses the main ®ndings of the paper.
2. Management objectives
In the biology literature objectives for managing
®sh stocks are usually related to the maximum sustainable yield (MSY), yield per recruit (Y/R) or some
related concepts. In cases of two or more biologically
interdependent species, maximum sustainable yield
frontiers (MSF) might replace the single-species MSY
concept (see, e.g. Beddington and May, 1980; Flaaten,
1988). However, the fallacy of biological management
objectives is that they do not consider the economic
bene®ts and costs of ®sheries. Many ®sh stocks are
deliberately not ®shed due to low market price and/or
high catch cost. In the Barents Sea, for example, there
are more than 100 ®sh species, but only about 10 are
commercially targeted.
Some international organizations and agreements
have established their own objectives for ®sheries
management. The Food and Agriculture Organization
of the United Nations (FAO) formulated the following
objective:
``Recognizing that long-term sustainable use of
fisheries resources is the overriding objective of
conservation and management, states and sub-
181
regional or regional fisheries management organizations and arrangements should, inter alia, adopt
appropriate measures, based on the best scientific
evidence available, which are designed to maintain
or restore stocks at levels capable of producing
maximum sustainable yield, as qualified by
relevant environmental and economic factors,
including the special requirements of developing
countries.'' (FAO, 1995.)
Thus, even though the FAO's Code of Conduct
establishes the single-species concept maximum sustainable yield as the main management objective, it is
quali®ed by relevant environmental and economic
factors.
Contrary to the management objectives above,
economic objectives are strongly related to social
welfare theory that emphasizes the net economic
results to society of utilizing natural resources.
`Society' in this context usually means a country,
but it could also mean a group of indigenous people,
a region within a country, or a group of countries. The
resource rent is the gross catch value minus the harvest
costs. Maximizing the long run resource rent, or the
present value of discounted future resource rent, is the
main economic management objective, when harvested ®sh is the only bene®t to society.
If the stock possesses public non-use values, the
related bene®t ¯ow should be included in the management objective. The existence of organizations working to abolish, e.g. whale, seal and dolphin harvesting
re¯ects the importance of non-use values. There exists
a large literature on methods that can be used to
evaluate non-use bene®ts from natural resources
(see, e.g. Hanley and Spash, 1993).
The main results of single-species bioeconomic
analyses are shown in Fig. 1. Panels (a) and (b) show
how the sustainable revenue and the total cost of
harvesting vary with ®shing effort and stock level,
respectively. Generally speaking, the optimal level of
, is less than the open access level,
®shing effort, Ess
oa
, and the optimal stock level, Wss , is higher than the
Ess
open access level, Wssoa . These general results are valid
whether the optimum is derived by maximizing annual
economic rent or the present value of rent. However,
the economic optimal effort and stock levels approach
their open access levels if the interest rate goes to
in®nity.
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O. Flaaten / Fisheries Research 37 (1998) 179±191
Fig. 1. Open access (OA) and optimal (*) effort (E) and stock level (W) in a single species (SS) model. The arrows in panel (a) indicate that
the optimal long run fishing effort is higher for the predator and lower for the prey when biological interactions are included, than in the single
species case. Correspondingly, the arrows in panel (b) indicate that the optimal long run stock level is lower for the predator and higher for the
prey.
In single-species models, the biological constraint
of the optimization problem is the yield-effort or
yield-stock curve on which the revenue curves shown
in Fig. 1 are based. Moving from single-species to
two-species models changes the biological constraint
to, for example, the MSF shown in Fig. 2. Maximizing
the yield from each of the two species as if it is
independent of the other, gives the combined yields
at point S in Fig. 2. However, this is not a sustainable
combination of yields since it is outside the MSF. Any
Fig. 2. The maximum sustainable yield frontier (MSF) gives the
maximum possible yield of one species for a given yield of the
other, based on a simple analytical model. Each cross might
represent the possible result of a given management strategy for a
complex, detailed simulation model. The *S represents the
combination of the maximum single species yield of the predator
and the maximum single species yield of the prey, but this point is
not sustainable.
point on or inside the MSF would be sustainable (see
e.g. Flaaten, 1988, 1991). The reasoning so far is based
on simple deterministic models. For complex, detailed
simulation models each set of management strategies
gives a combined average annual yield of the two
species that might be represented by the crosses in
Fig. 2. In this case the MSF is based on the most
ef®cient management strategies. Any management
strategy below the MSF is inef®cient, biologically
speaking.
In the case of harvest technical interactions, say
with two selectivity patterns, the results may be illustrated by drawing two intersecting MSFs instead of
the one in Fig. 2.
Which combination of yield should be chosen
depends in general on the management objective
and on the price of ®sh ± cost of effort ratios for
the two stocks. If the stocks are jointly managed, the
objective might be to maximize the combined average
annual resource rent or the present value of the
resource rent obtained from them. Using the present
value of the combined resource rent from the two
stocks as a management objective, or performance
criterion, could give an optimal solution inside the
MSF, e.g. at D in Fig. 2. This is due to the positive
effect on the unit harvesting cost of an increased
stock level, and the discounting of future bene®ts
and costs.
To illustrate the span of possibilities in predator and
prey ®sheries, let me use two simpli®ed examples. In
both cases I assume that the stocks can be harvested
independent of each other.
O. Flaaten / Fisheries Research 37 (1998) 179±191
2.1. Example 1: valuable predator and cheap prey
Let species 2 be a predator of high net value per
unit harvest and species 1 a prey species with a low
net value. In this case the optimal long run combined
equilibrium harvest is in the vicinity of B in Fig. 2,
where the prey is mainly kept in the sea as feed for
the predator. In this case the effort of the predator
®shery does not have to be increased (much)
compared to its single species effort shown in
Fig. 1(a), whereas the effort of the prey ®shery should
be decreased. The effects on the stock levels are the
opposite.
2.2. Example 2: predator of low net value and prey
of high net value
If the predator is of low market value and/or expensive to harvest, its net value per unit harvest is low.
Likewise, if the prey is of high market value and/or
cheap to harvest its net value per unit harvest is high.
In this case the optimal long run combined equilibrium
harvest is in the vicinity of A in Fig. 2, where the
predator stock during a transitional period was ®shed
down to a low stock level to leave more prey to be
harvested by the ®shermen (see, e.g. Beddington and
May, 1980; Flaaten, 1988). In some cases it even pays
to subsidize ®shermen to harvest more predators than
they otherwise would have done. In this case the
optimal effort of the predator ®shery should be
increased and the stock level of the predator reduced
compared to the single species case, as indicated by
the arrows in Fig. 1.
If a species has a public non-use value this might be
represented by adding a concave increasing utility
function of the stock level to the management objective, e.g. U(W) with U>0, U0 >0 and U00 <0. The question of how to estimate people's willingness to pay for
public goods has been extensively dealt with in the
literature (see, e.g. Hanley and Spash, 1993 and
references therein), but will not be pursued in this
paper. The implications for optimal long run effort and
stock levels are, however, similar to that of the prey in
example 1 above. The positive non-use externality
af®liated to the stock implies lower equilibrium effort
and higher stock level compared to the case without
such non-use value. In actual cases non-use values are
mostly claimed for species at or near the top of the
183
food chain (see Kuronuma and Tisdell (1993) for the
case of minke whale).
3. The general predator±prey model and the
bioeconomic effects of harvesting
In the population ecology literature there are
numerous deterministic models represented by
non-linear functions. My starting point is the following model, with continuous time dynamics acting
on the total biomass of the prey, W1, and the predator,
W2. Let the biomass be measured in weight, e.g.
tonnes. There are three basic model functions in
the predator±prey biomass model: namely f(W1)0,
the intrinsic growth rate of the prey population;
k(W1, W2)0, the amount of prey consumed per unit
time per unit predator, which is called the predator's
functional response; g(W1, W2)0, the per unit biomass growth rate of the predator population, which
is called the predator's numerical response. For a
review and discussion of the three functions, see,
e.g. Yodzis (1994). Suppressing the arguments in
the various functions, and with hi being the harvest
of species i, the general form of such biomass models
can be represented as follows
dW1
ˆ f ÿ W2 k ÿ h1 ; W1 ; W2 > 0; k 0
dt
2
df W msy ; f 00 ˆ d f < 0
0
if
W
f0 ˆ
1
1
<
dW1 >
dW12
@k
0;
@W1
@2k
0;
@W12
@k
0;
@W2
@2k
0
@W22
(1)
and
dW2
ˆ W2 g ÿ h2 ;
dt
@g
@2g
@g > 0;
0;
0
2
@W1
@W2 >
@W1
msy
if W2 < W2 jW1ˆconstant ;
@2g
0
@W22
(2)
By adding the price of harvest, cost of harvesting,
discount rate, non-consumptive bene®ts and costs,
etc., the model in Eqs. (1) and (2) is extended to a
bioeconomic model. In accordance with the aim of this
paper, it is suf®cient to include the bene®ts and costs
of the harvesting. To simplify the analysis I assume
that for each species the net value per unit of catch
depends on the respective stock level only. The net
184
O. Flaaten / Fisheries Research 37 (1998) 179±191
value (bene®t) per unit harvest is
bi ˆ bi …Wi †;
b00i ˆ
…i ˆ 1; 2† b0i ˆ
d2 bi
0
dWi2
dbi
< 0;
dWi
(3)
An in®nitely elastic demand, implying a constant
price of harvest, and a unit cost of harvest which
increases with the stock level, is the rationale for
Eq. (3). Assuming harvest technology that allows
independent harvesting of the two species, the
resource rents are
i …Wi † ˆ bi …Wi †hi ;
…i ˆ 1; 2†
(4)
for a given catch, hi.
A common approach in bioeconomic analysis of
optimal harvesting is to assume a sole owner maximizing the present value of the total resource rent
from the two species. This allows the optimal long run
steady state to be derived (see, e.g. Clark, 1990, ch. 9;
Hannesson, 1983; Flaaten, 1991). For short run analysis, and actual management of ®sh stocks, agestructured biological models would be of great advantage compared to simpler biomass models. However,
for the purpose of illustrating the economics and
management of biologically interdependent species,
biomass-based bioeconomic models are useful. A ®rst
step is to study the marginal effects of changes in the
other stock level on the harvest rate and the resource
rent of each species in our model Eqs. (1)±(4).
4. The effect on prey harvest and resource rent
of changes in the predator stock
For dW1/dtˆ0 Eq. (1) gives the equilibrium catch of
the prey, and it will depend on the prey stock level as
well as the predator stock level. The effect on the
equilibrium catch of the prey of a marginal change in
the predator stock level is
@h1
@k @k W2 ÿ 1 (5)
ˆ ÿ k ‡ W2
0 if
@W2
@W2 >
@W2 k <
Thus, the effect on the equilibrium harvest rate of
the prey of an increase in the predator stock level is
ambiguous. To get the positive sign in Eq. (5) it is
necessary that the relative change of per unit consumption, ÿ@k/k, of a marginal increase in the pre-
dator stock, is greater than the relative change of the
predator stock level, @W2/W2. The equilibrium harvest
of the prey is negatively affected by an increased
predator stock level if, and only if, the predator's total
consumption moves together with its stock level. This
is valid for partial changes in the predator stock with
the prey stock unchanged.
To summarize, the equilibrium catch of the prey is
negatively affected by an increased predator stock
level when (@/@W2)(W2k)>0, which usually is biologically plausible. However, as seen in Eq. (5), there
might be cases where an increase in the predator stock
level also may increase the equilibrium harvest of the
prey.
The effect on the resource rent is derived by substituting the equilibrium harvest rate from Eq. (1) for
h1 in Eq. (4):
@1
@k @k W2 ÿ1
ˆ ÿb1 k ‡ W2
0 if
@W2
@W2 >
@W2 k <
(6)
The effect on the equilibrium resource rent of the
prey ®shery of an increase in the predator stock level is
ambiguous. The sign is the same as for the effect on
the harvest rate in Eq. (5), and the conditions are the
same. As is seen from Eqs. (5) and (6), the effect on
the harvest and the resource rent in this case depends
on the predator stock level, W2, and the predator's
functional response, k, but not on the predator's
numerical response, g. In order to estimate the economic impact, data on the net value per unit harvest for
the prey ®shery, in addition to estimation of W2 and k,
are required. Uncertainty about the predator's numerical response, g, does not matter for the assessment of
the predator's marginal impact on the prey ®shery,
since the latter is independent of g.
5. The effects on predator harvest and resource
rent of changes in the prey stock
For dW2/dtˆ0, Eq. (2) gives the equilibrium catch
of the predator. From this, the following is obtained
@h2
@g
ˆ W2
>0
@W1
@W1
(7)
The effect on the harvest rate of the predator of an
increase in the prey stock level is unambiguously
O. Flaaten / Fisheries Research 37 (1998) 179±191
positive, and consequently, the effect on the resource
rent of the predator ®shery of an increase in the prey
stock level is unambiguously positive:
@2
@g
ˆ b2 W 2
>0
@W1
@W1
(8)
As is seen in Eqs. (7) and (8), the effect on the
harvest and the resource rent depends on the predator's
numerical response, g and on the predator stock level.
The predator's functional response does not matter in
this case.
According to Eqs. (7) and (8), resource managers
concerned with the negative effect on the predator
®shery of a declining prey stock have little cause for
concern if they do not know how the predator affects
the prey. However, they should worry about how the
prey stock affects the predator, i.e. the predator's
numerical response.
Note that the results in Eqs. (7) and (8) are valid for
any stock level of W1 and W2, given the assumptions in
Eq. (2). Therefore, within the model described in
Eqs. (1)±(4), the positive marginal effects on the
predator harvest and resource rent of an increase in
the prey stock level is a robust result. For actual
®sheries the magnitude of the impact will obviously
depend on the empirical formulation of the model and
the stock levels. Also, more complex ecological models with three or more interdependent ®sh stocks could
give ambiguous results compared to Eqs. (7) and (8)
(see, e.g., Flaaten, 1988; Bogstad et al., 1998).
6. Results in four typical models
A broad class of models can be represented by
Eqs. (1) and (2). One speci®c class of models comprises those with functional and numerical responses
that give linear isoclines, i.e. with a linear equilibrium
relationship between the stocks. Such models are
simple to handle and easy to depict graphically. They
have been used to analyse the management of marine
mammals±krill ®sheries (see e.g. May et al., 1979;
Nicole and de la Mare, 1993) and marine mammals±
®sh ®sheries (see e.g. Flaaten, 1988; Yodzis, 1994).
The previous section revealed that the predator's
numerical response is of importance for the harvest
rate and the pro®tability of the predator ®shery. The
management of whales, seals and other sea mammals
185
in relation to their prey is often controversial, as
demonstrated by the Yodzis (1994) quotation in Section 1. A central point of focus in such controversies is
the competition between sea mammals and ®shermen
for commercially valuable ®sh stocks. Given a predator±prey model with linear isoclines, this section
analyses how the slope of the predator isocline, which
is determined by the numerical response, affects
management conclusions. However, the analysis starts
with the effects on prey equilibrium harvest and
resource rent of marginal changes in the predator
stock.
The population dynamics of the prey in the linear
isocline predator±prey model is described by
dW1
ˆ r1 W1 …1 ÿ W1 =K† ÿ aW1 W2 ÿ h1 ;
dt
r1 ; a > 0
(9)
The ®rst term of Eq. (9) is the growth rate of the
prey population in the absence of the predator. The
predator's functional response, aW1, is proportional to
the prey stock. Note, however, that in this case the
functional response, i.e. the amount of prey consumed
per unit time per unit predator, is independent of the
predator stock level. If the growth Eq. (9) is substituted for the general prey growth Eq. (1), the results in
Eqs. (5) and (6) are reduced to
and
@h1
ˆ ÿk ˆ ÿaW1
@W2
…50 †
@1
ˆ ÿb1 k ˆ ÿb1 aW1
@W2
…60 †
Thus, if we can neglect the fact that the predator's
per capita consumption of the prey decreases when the
predator stock level increases, it becomes easy to
calculate the effects on prey harvest and resource rent
of changes in the predator stock level. The results in
Eqs. (50 ) and (60 ) will be applied to the case of northeast Arctic cod in the next section.
The population dynamics of the predator is
described by
dW2
ˆ r2 W2 …1 ÿ W2 =…W1 ‡ L†† ÿ h2 ;
dt
r2 ; > 0
(10)
The predator's numerical response, r2 …1 ÿ W2 =
…W1 ‡ L††, decreases with the predator stock and
186
O. Flaaten / Fisheries Research 37 (1998) 179±191
increases with the prey stock. It also increases with the
unspeci®ed resources given by the independent part of
the carrying capacity, L. The case of a negative sign in
front of L will also be considered.
Assuming catch per unit of effort (CPUE) proportional to the stock level, the isoclines are easily
derived. The proportional CPUE implies the Schaefer
harvest function
hi ˆ ri Fi Wi ; …i ˆ 1; 2†
(11)
which may be substituted for the hi's (iˆ1, 2) in
Eqs. (9) and (10). Fishing effort, Fi, has, for simplicity,
been normalized such that the catchability coef®cients
equal the intrinsic growth rates, ri, in Eq. (11). The
isoclines are now derived for dWi/dtˆ0 in Eqs. (9) and
(10). The prey isocline is given by
r1
W2 ˆ …1 ÿ F1 ÿ W1 =K†; for dW1 =dt ˆ 0 (12)
a
Fig. 4. The linear isoclines of four submodels, of which the
predator isoclines intersect at an assumed known equilibrium point
…W1F ; W2F †.
and the predator isocline by
W2 ˆ …1 ÿ F2 †…W1 ‡ L†; for dW2 =dt ˆ 0
(13)
The isoclines are shown in Fig. 3 for Fiˆ0 and Fi>0
(iˆ1, 2). Increasing the ®shing effort of the prey
®shery, F1, moves the isocline to the left, and the
equilibrium stock levels, W1F and W2F , are both
reduced. Increasing the ®shing effort of the predator
®shery, F2, moves the predator isocline downward. As
is clear from Fig. 3, the prey stock level increases,
whereas the predator stock level decreases for a partial
increase in F2. This is because the equilibrium point in
Fig. 3. The linear isoclines without and with harvesting.
this case moves along the downward sloping prey
isocline.
The slope of the prey isocline in Eq. (12) depends
on only one parameter, a, in addition to r1 and K,
whereas the predator isocline in Eq. (13) depends on
two parameters, L and . Different combinations of L
and will describe different numerical responses of
the predator. The following four combinations are
considered of greatest interest with respect to the
different biological information they contain:
…L; † ˆ ‡‡; 0‡; ÿ‡; ‡0
The four types of predator isoclines in Fig. 4 are
drawn through the same equilibrium point …W1F ; W2F †
by using different combinations of L and . In all four
cases the predator is extremely opportunistic; it takes a
constant portion of the prey, through the functional
response term aW1 in Eq. (9). Assuming we know the
stock level of the predator, Fig. 4 illustrates how
uncertainty about the predator's numerical response,
given partly by the isocline, may affect management
decisions. One extreme is submodel ‡0, where the
predator stock level is not affected by changes in the
harvesting of the prey. Increased ®shing effort of the
prey moves the prey isocline downwards to the left
towards the origin, reducing W1F . However, W2F
remains constant because of the horizontal predator
isocline.
O. Flaaten / Fisheries Research 37 (1998) 179±191
At the other extreme is submodel ÿ‡. If this is the
correct description of the predator's numerical
response, the predator is very vulnerable to changes
in prey harvesting and stock level. If the prey stock is
F1 , the predator becomes
reduced below the level W
F
extinct. Thus, W 1 is a critical prey level for the
predator in model ÿ‡. The predator is then highly
specialized and goes extinct when not enough food is
available.
When L>0 the predator can persist without the prey
with a minimum carrying capacity L. This means that
even in the extreme case where the prey becomes
extinct, the predator may survive. In the limiting
model ˆ0, the predator develops independently of
the prey. It has its own independent carrying capacity,
L, yielding the horizontal isocline at W2F in Fig. 4.
The results of this analysis can be summarized by
using the expressions Eqs. (7) and (8) to study the
effects on the equilibrium harvest and resource rent of
partial changes in the prey stock. From Eqs. (2) and
(10) the following is derived for given levels of W1
and W2
@g
r2 W2
ˆ
@W1 …W1 ‡ L†2
(14)
Substituting for @g/@W1 from Eq. (14) into Eqs. (7)
and (8) gives the effect of a change in the prey stock
level on the equilibrium harvest and resource rent of
the predator ®shery. Note that the unambiguously
positive results in Eqs. (7) and (8) were based on
the assumption that @g/@W1 is strictly positive in
Eq. (2). When ˆ0, however, @g/@W1ˆ0. This
implies that @h2/@W1 and @2/@W1 both are equal
to zero in this case.
The numerator of Eq. (14) equals the square of the
carrying capacity of the predator, given in Eq. (10).
The effect on the equilibrium harvest of the predator of
an increase in the prey biomass, @h2/@W1, is, of
course, nil when ˆ0, and positive for >0. From
Eqs. (14) and (7) it increases with L. This, simply
because the increase in reproductive capacity of the
predator by an increase in the prey stock, increases in
L. Sea mammals that are opportunistic feeders, e.g.
minke whale (Balaenoptera acutorostrata) and harp
seal (Pagophilus groenlandicus) (see Haug et al.,
1995; Nilssen, 1995) most likely have >0 and L>0
relative to a given prey species. Only very specialized
predators would have L<0.
187
7. An example of predation effects ± the case of
northeast Arctic cod
The analysis in the previous sections includes one
predator and one prey. However, the analysis will also
be valid in the case of several prey, if the biological
interaction between them is negligible or their stock
levels are kept constant by adjusting ®shing effort
(Flaaten and Stollery, 1996). The basic ideas presented
above may also hold in the case of a predator consisting of several cohorts, as long as there exists a long
run joint stationary state for the cohorts. In such cases
the expressions derived in Eqs. (5)±(8) and (60 ) are
applicable. To illustrate the theoretical analysis above,
data from the Barents Sea ®sheries will be used.
For many years, Russian and Norwegian researchers have conducted studies on `who eats whom' in the
Barents Sea area and have modelled these and other
multispecies interactions. The model MULTSPEC
from the Institute of Marine Research (IMR), Bergen
is a biological multispecies model for the Barents Sea
®sh/sea-mammal system (see, e.g. Ulltang, 1995;
Bogstad et al., 1998 and references therein). The
MULTSPEC model now includes cod, capelin, herring, minke whale, harp seal and species of zooplankton, but not shrimp.
The most important ®sh preying predator is cod
which seems to eat any available prey of the right size.
Fig. 5 shows the northeast Arctic cod's age-dependent
average annual consumption of some commercially
important prey species. Species included are shrimp,
capelin, herring and cod (cannibalism) above 5, 10, 10
and 20 cm, respectively.
The ®gures are in grammes of prey per kg cod, for
each age class of cod from 1 to 7‡ years. Fig. 5 shows,
for example, that 1 kg of 2-year old cod annually
consumed 2000 g of prey from these four species
above the given size, and that about 75% of this
was capelin. For all age classes, capelin is the main
prey among the species and size groups included in
Fig. 5.
Taking the net value per unit prey species into
consideration (see Flaaten and Stollery (1996) for
details) gives the results shown in Fig. 6. The net
value per unit prey, which equal b1 in Eq. (60 ), is the
net contribution which the ®sh in the sea could have
given for the prey harvesters if they had less competition from the predator, the cod. The net value per unit
188
O. Flaaten / Fisheries Research 37 (1998) 179±191
8. Discussion
Fig. 5. Northeast Arctic cod's age-dependent average annual
consumption of some commercially important prey species.
Species included are shrimp (Pandalus borealis), capelin (Mallotus
villosus), herring (Clupea harengus) and cod (Gadus morhua)
above 5, 10, 10 and 20 cm, respectively. In gram prey per kg cod,
1984±92. Sources: Eide and Flaaten, 1998.
prey catch was found to be 30% of the landing price in
these ®sheries. Fig. 6 shows, for example, that 2-year
old cod had an annual feed cost of NOK 1.50 (ECU
0.19) per kg of biomass, and that about 75% of this
was in¯icted on the shrimp ®sheries. Apart from age
class 7‡, the feed costs of shrimp dominates the
economic ®gures, whereas capelin dominated the
biological results in Fig. 5.
Fig. 6. Age-dependent average annual feed cost of northeast Arctic
cod's consumption of some commercially important prey species.
Species included are shrimp (Pandalus borealis), capelin (Mallotus
villosus), herring (Clupea harengus) and cod (Gadus morhua)
above 5, 10, 10 and 20 cm, respectively. In NOK per kg cod, at
1991±92 prices. Consumption data from 1984±92. Sources: Eide
and Flaaten, 1998.
The uncertainty about the effects on predator and
prey stocks and yield when ®shed, correspond to
uncertainty about the functional forms of model
equations, in addition to random ¯uctuations and
parameter uncertainties. There are several ways to
analyze the effects of such uncertainty on derived
management strategies and the performance of
®sheries, e.g. by use of scenario modelling, decision
analysis, baysian expert modelling, risk and uncertainty analysis and stochastic programming. However,
it is also of interest to study the importance of the
choice of functional forms for management strategies
and performance in simple deterministic multispecies
models.
This paper has brie¯y reviewed some connections
between bioeconomic objectives, management strategies with respect to effort and stock size, and performance within single-species and multispecies
frameworks. When the objective is to maximize
resource rent from harvesting a prey the optimal long
run effort level is lower and the stock level higher
within a predator±prey framework than in a singlespecies one. For a predator ®sh the opposite result
appears ± the optimal long run effort level is higher
and the stock level lower within a predator±prey
context compared to the single-species case. When
the prey is valuable and inexpensive to catch and the
predator has a low market value, it may even bene®t
the total resource rent to subsidize predator harvesting.
When managing an abundant species at, or near, the
top of the marine food chain, we should include an
evaluation of the predation effects on commercially
important prey species. The use of simple deterministic models could well be a ®rst step in such an
analysis. Experimentation with functional forms may
reveal whether they are important or not for management strategies.
The MULTSPEC model, which was designed and
implemented in Norway, (see Ulltang, 1995; Bogstad
et al., 1998) has minke whale and harp seal as the top
predators. In relation to the predator±prey models
above they have carrying capacities independent of
their prey stocks, i.e. (L, )ˆ(‡0). The commercially
most important ®sh species of MULTSPEC are cod,
capelin and herring. Models like MULTSPEC are
useful for studying the implications of changes in
O. Flaaten / Fisheries Research 37 (1998) 179±191
the stock levels of the top predators for the prey
harvest. However, they are of no use in studying
how changes in the prey stock affect the predator
harvest and ®shing industry. Thus, sea mammals in
MULTSPEC are one type of extreme predators.
At the other extreme is the highly specialized
predator, with (L, )ˆ(ÿ‡), which does not have
other sources of food than the modelled prey. The
more extreme it is the steeper is the isocline, and the
prey harvesting may have a strong impact on the
predator stock and harvest. The minimum stock level
of the prey that can sustain the ÿ‡ type of predator is a
critical boundary (see Fig. 4) for the predator's survival. However, with respect to opportunistic feeders
like minke whale and harp seal in the northeast
Atlantic (see Haug et al., 1995; Nilssen, 1995) it is
highly unlikely that their survival depends critically on
any single prey.
The 0‡ type predator, with a carrying capacity
proportionate to the prey stock level, is a compromise
between the ‡‡ type predator, with its partly prey
independent carrying capacity, and the ÿ‡ type predator. The analyses of sea-mammals ± ®sh interactions
and harvesting in May et al. (1979); Flaaten (1988) are
based on sea-mammals as a 0‡ type predator. As
shown above the effect on the equilibrium harvest of
the 0‡ type predator of a marginal change in the prey
stock level falls in between the corresponding effects
of the ‡‡ type and the ÿ‡ type predators. Contrast
this with Yodzis (1994), who
``... draw the conclusion that the form of model
inherited by Flaaten (1988) from May et al. (1979)
biases the case against marine mammals, without
justification in terms of the underlying biology.''
(Yodzis, 1994, p. 52.)
Note, that uncertainty about the predator's numerical response to changes in the prey stock level does
not matter for the effects on the prey equilibrium
harvest and resource rent of marginal changes in
the predator stock level, as shown in Eqs. (5),(6),(50 )
and (60 ). Also note that these effects on the prey
harvest and resource rent can be decomposed into
two parts: one containing the predator's average per
unit time consumption of the prey, and one containing
the density dependent change in this of a marginal
change in the prey stock level. The latter effect
vanishes in the model with linear isoclines, as seen
189
in Eqs. (50 ) and (60 ), which makes it simple to apply to
actual cases (see Flaaten and Stollery (1996) for such
an application).
Within the framework of a simple predator±prey
biomass model it has been shown, in Eqs. (7) and (8),
that the effects on the predator harvest and resource
rent of a marginal increase in the prey equilibrium
stock level is unambiguously positive for all prey
stock levels. However, the magnitude of these
effects might be less for high prey stock levels than
for low levels. This happens if there is a decreasing
marginal biological productivity of the prey, i.e.
@ 2 g=@W12 < 0.
Some stocks possess public non-use values that
makes them a mixed good. They are private goods
in that it is possible to harvest them, and public goods
due to their amenity value. Including the positive nonuse value of the stock level in the management objective implies a lower long run effort level and higher
stock level compared to the private good only case.
Since non-use values of environmental goods, and
evils, are subjective by de®nition, the methods used
to estimate bene®ts and costs are not trivial. However,
there are numerous examples where such methods
have been used (for a review see Hanley and Spash,
1993). In the case of marine predators like whales and
seals there are few examples of such studies (Kuronuma and Tisdell, 1993). From a scienti®c point of
view it is important to distinguish between management strategies derived from purely biological objectives and strategies derived from economic objectives
that include resource economic aspects of harvesting
and/or non-use values. The former need only input
from biologists, in a broad sense, whereas the latter
also require input from economists.
To illustrate the use of the theoretical analyses of
this paper, data on the northeast Arctic cod's consumption of some commercially important prey species have been presented graphically. It is shown, in
Fig. 5, that of the four species capelin, herring, shrimp
and cod, above 5, 10, 10 and 20 cm, respectively,
capelin is the most important prey for cod. On average
for the years 1984±92 capelin constituted about 75%
of the feed of cod of age 2 and older. Shrimp is the
second most important prey, constituting about 10±
20% of the feed, highest for 2-year old cod and lowest
for the 7‡ age group. These ®gures do not fully
acknowledge the importance of capelin as prey since
190
O. Flaaten / Fisheries Research 37 (1998) 179±191
they are based on weight. The caloric content of ®sh
species varies with the season, but the annual averages
of capelin, herring, shrimp and cod are 1.58, 1.74, 1.39
and 1.12, respectively (adapted from Nordùy et al.,
1995). The prey species have alternative value as catch
for ®shermen. Knowing landing prices and harvest
costs, the feed costs, i.e. the net value per unit harvest,
of the prey species were calculated. It is shown in
Fig. 6 that shrimp constitute between 40% and 80% of
the feed costs for all age groups 2 years and older, with
the highest percentage for the age 2 group. Capelin is
by far the major feed in weight. For the oldest age
group, 7‡, cannibalism is just above half the feed
costs.
From a feeding ecology point of view capelin is the
most important prey of cod in the Barents Sea. However, the economic ®gures shown in this paper indicate
that the costs per kg of cod biomass in¯icted on the
prey ®shing industries is much higher for shrimp than
for capelin and herring. Even though this is a partial
bioeconomic analysis it strongly indicates that biological and economic factors should be considered
simultaneously in management analysis. The assumption of functional forms of biological multispecies
models are of importance for the management implications derived. However, it might be that the inclusion of economic factors will weaken the relative
importance of some biological factors for management strategies.
Acknowledgements
I gratefully acknowledge the generous help from
S. Mehl for providing the cod consumption data and
E. Kolsvik for research assistance. I am indebted to
two anonymous referees and co-editor T. Schweder for
their valuable comments. This work has been partly
funded by the Norwegian Research Council's Program
for Marine Resource Management (grants no. 108164/
120 and 108158/120).
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