BIO-ECONOMIC MODELING OF CONTAMINATED BLUEFIN TUNA AND
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BIO-ECONOMIC MODELING OF CONTAMINATED BLUEFIN TUNA AND ATLANTIC MACKEREL FISHERIES DYNAMICS Michael S. Press, MEM Candidate Dr. Martin Smith, Advisor Master’s Project submitted in partial fulfillment of the requirements for the Master of Environmental Management degree Nicholas School of the Environment Duke University April 2009
Transcript of BIO-ECONOMIC MODELING OF CONTAMINATED BLUEFIN TUNA AND
BIO-ECONOMIC MODELING OF CONTAMINATED BLUEFIN
TUNA AND ATLANTIC MACKEREL FISHERIES DYNAMICS
Michael S. Press, MEM Candidate
Dr. Martin Smith, Advisor
Master’s Project submitted in partial fulfillment of the requirements for the Master of Environmental Management degree
Nicholas School of the Environment Duke University
April 2009
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Abstract BIO-ECONOMIC MODELING OF CONTAMINATED BLUEFIN TUNA AND
ATLANTIC MACKEREL FISHERIES DYNAMICS
by
Michael Press
April 2009 Following the discovery of acute mercury toxicity from seafood consumption in the 1950s and subsequent research into mercury in the environment, scientists and managers now recognize the health threats of mercury poisoning from seafood consumption, especially in fetuses, infants, and children. Unfortunately, consumers remain confused or uneducated about species-specific mercury concentrations, thus perpetuating the risks associated with contaminated seafood. This study models the bio-economic dynamics of a system involving two species consumed by humans: a highly mercury-contaminated predator, bluefin tuna, and a tuna prey fish with low levels of contamination, Atlantic mackerel. Model scenarios evaluate varying levels of mercury pollution, consumer aversion to mercury, and fishes’ biological resistance to mercury poisoning to determine optimal harvest rates and population sizes for both species. The results demonstrate that while the mackerel fishery remains largely unaffected by the influence of mercury, optimal harvest and population of tuna depend greatly upon their biological resistance to mercury and consumers’ aversion to purchasing mercury-contaminated fish. When resistance to mercury is low, both tuna population and harvest decrease. When consumer aversion is high, harvest decreases and population increases. Increased mercury pollution exacerbates both effects. Due to lack of previous such studies and the paucity of empirical data, this research is both exploratory and qualitative in nature. Effective fisheries conservation and management requires understanding the strength of both fish resistance and consumer aversion to mercury. Future research should address the lack of empirical data, both biological and economic, as well as refine the above model in order to assist managers in appropriate consumer education and setting fisheries management goals that couple sustainability and public health.
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TABLE OF CONTENTS
Introduction…………………………………………………………………………………….3
Background and Motivation……………………………………………………………………4
The Contaminated Fisheries Model……………………………………………………………5
Model Parameterization………………………………………………………………………..16
Visualizing Model Dynamics…………………………………………………………………..21
Interpretation of Contaminated Fisheries Model…………………………………...……….....36
Alternative Model Including Stock Effects…………………………………………………….37
Discussion………………………………………………………………………………………40
References………………………………………………………………………………………43
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INTRODUCTION
In recent years, scientists and consumers alike have recognized the increased threat of
mercury poisoning from fish and seafood consumption. The ecological processes involving
mercury are studied widely and a number of studies have attempted to value the damage to
society due to mercury toxicity and the resulting illnesses, loss of IQ, and mental retardation
(Trasande, 2005). This paper attempts a bio-economic analysis of two mercury contaminated
fish species with different mean concentrations of mercury and different market prices within the
confines of an optimal fisheries management system. This technique models how concentrations
of mercury may influence fisheries by depressing market prices or affecting the population
resilience of mackerel, a mid-level predator, and tuna, a top level predator which preys upon
mackerel. Consumers eat both fish frequently, often raw in the form of sushi and sashimi, thus
making them valuable as study species. If the flow of information is sufficient such that
consumers understand the dangers of highly contaminated fish, and are also aware of the
differing levels of mercury in different fishes, prices of highly contaminated species could drop
relative to the prices of other species. Research supports such reduced demand for mercury-
contaminated fish (Shimshack et al., 2007). Also of concern is the population health of fish,
which several studies have shown to be negatively affected by high mercury concentrations
(Latif et al., 1999; Baker Matta, 2001; Kime, 1999; Beckvar, 1996). This project explores how
the amount of mercury pollution, the consumer perception of mercury dangers, and the damage
to fish populations resulting from high mercury bioaccumulation alter predator-prey relations
and the resulting composition of catches.
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BACKGROUND & MOTIVATION
Since the characterization of “Minamata Disease” in the 1950s (Harada, 1968), many
studies such as those conducted in the Faroe Islands (Budtz-Jorgensen E., et al., 2004), have
detailed the loss of IQ and increased incidence of mental retardation caused by mercury-
contaminated seafood, primarily in high trophic level predators such as fish, seabirds, and
mammals. Methylmercury, a compound resulting from bacteriological interactions with the
environment once emitted, represents the vast majority of mercury in fish and is highly toxic
(Clarkson, 2002; Wiener, 2003). Its relative chemical stability results in very long halftimes for
detoxification and bioaccumulation in marine predators (Wiener, 2003). Mercury readily crosses
the placental barrier resulting in both fetal blood concentrations 5-7 times higher than those in
maternal blood and subsequent degradation of fetal brain development despite the absence of
noticeable effects in the mother (Cernichiari, 1995). In response, health ministries around the
world including the United States Food and Drug Administration (FDA), the United States
Environmental Protection Agency (EPA), and the Agency for Toxic Substances and Disease
Registry (ATSDR) have set maximum standards for the presence of mercury in food (Clarkson,
2002). Recently, studies sampling fish in US sushi restaurants have uncovered mercury
concentrations higher than recommended intake guidelines and the FDA actionable limit
(Burros, 2008; Saddler, 2006).
About 30% of mercury deposition results from natural environmental events such as
volcanic activity, while the remaining 70% results from anthropogenic pollution (Wiener, 2003).
The United Nations Environment Programme (2002) estimates that human sources emit
approximately 5,500 metric tons of mercury every year. While the US currently only contributes
about three percent of total global anthropogenic mercury emissions (EPA, 2007), scientists and
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policymakers nonetheless fiercely debate policies to reduce US mercury emissions (Trasande,
2005; Trasande, 2006; Griffiths, 2007).
Some studies have attempted to address the economic and behavioral aspects of the
mercury problem. Trasande et al. (2005) used the environmentally attributable fraction model to
estimate that the costs of mercury in terms of lost lifetime earnings due to lowered IQ lie
between $0.7 billion and $13.9 billion. Booth and Zeller (2005) modeled the flow of mercury
throughout the Faroe Island ecosystem to determine mercury concentration effects based on the
consumption of pilot whales and cod. They found that once warned of high mercury levels,
women consumed less whale meat and more cod, thus reducing their exposure. Model
simulations showed increased mercury concentrations over time and changing biomass for
different marine species due to changing fishing pressure. However, neither of these studies
combines the biology, chemistry, and economics of mercury together in a dynamic fashion.
Other studies have modeled the bioeconomic dynamics of predator-prey management
systems (Hannesson, 1983; Ragozin and Brown, 1985; Kaplan and Smith, 2000). This paper
addresses a system in which fishers harvest both fish species and incorporates the additional
dynamics of mercury concentrations that vary between fish species and depend upon species
interactions. This approach allows the market to dictate the optimal extraction for both species
based on the concentrations of mercury in each species. Unfortunately, modeling the influence
of mercury in the environment complicates an interactive predator-prey model to an analytic
degree beyond the scope of this study. Consequently, the direct biological interactions between
tuna and mackerel stocks do not appear in this model. Instead, the species are linked through
their mercury concentrations.
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Bluefin tuna (Thunnus thynnus) and Atlantic Mackerel (Scomber scombrus) comprise the
species modeled in this study. Both fish are common on sushi menus and data on
methylmercury concentrations exist for both species. According to Nakagawa et al. (1997), the
mean concentration found in bluefin was 1.11 ppm and the mean concentration in mackerel was
0.27 ppm. These samples of fish eaten in Japan correspond similarly to FDA data for mackerel
indicating a mean mercury concentration of 0.05ppm (FDA, 2006) and Tyrrell’s (2004) findings
of <0.03ppm; no information specific to bluefin tuna is currently available from the FDA, but
other studies have found average concentrations of mercury in bluefin from 0.899 to 3.03 ppm
(Srebocan, 2007; Licata, 2005, Storelli, 2001). The FDA sets health standards at a maximum of
1 ppm, but has never enforced them (Oceana, 2008). It is likely, however, that because mercury
emissions continue to increase, mercury concentrations are constantly increasing and are likely
to be higher than the data indicate (Booth, 2005). Because Bluefin tuna readily consume
Atlantic mackerel (Chase, 2002), there results an interesting dynamic choice between eating the
two fish as fishing pressure on one species may affect mercury concentrations in the other.
The roles of these two fish species in the current market carry broader implications than
the dangers of methylmercury toxicity. Environmental Defense Fund (EDF), the Natural
Resources Defense Council (NRDC), and the Blue Ocean Institute, among others, have listed
Bluefin tuna as threatened by overfishing (Blue Ocean Institute, 2004; NRDC, 2006;
Environmental Defense Fund, 2008). However, Atlantic mackerel stocks rebounded after
overfishing in the 1970’s and the same environmental organizations above list this species as
ecologically safe to eat (Environmental Defense Fund, 2008). The implications of the model
results may then affect other critical choices about which fish to eat for ecological reasons, not
just those of personal health.
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THE CONTAMINATED FISHERIES MODEL
Initial research concentrated on the biological interactions between tuna and mackerel.
However, mathematical complexities inherent in modeling a predator-prey system combined
with the influence of mercury render the analytics prohibitively difficult and time-consuming for
the scope of this study. Consequently, while the model specifies the flow of mercury between
tuna and mackerel, trophic interactions affecting stocks are omitted for the sake of simplicity.
Logistic growth for each species is bounded by terms describing interactions between the
two species. The model for these interactions relies heavily on Kaplan and Smith (2000).
However, the model discussed below differs because both prey and predator are harvested rather
than just the prey and, as specified above, the state equations for the species omit inter-species
interactions. Because some studies have implicated mercury in reduced efficacy and
reproduction of fish (Baker Matta, 2001; Latif et al., 1999; Beckvar, 1996; Devlin, 1992; Khan
and Weis, 1987), the model includes these effects as well, with mercury represented as m. The
state equations for tuna, represented as x, and mackerel, represented as y, are
= x (a1 – a2x – Mtuna ) – htuna (1)
= y (b1 – b2y – Mmackψ) – hmack (2)
where h is harvest, a’s and b’s are growth parameters (Kaplan & Smith, 2000), and and ψ are
the reproductive damage to the respective fishes as a result of mercury loading. A third state
equation is necessary to model the flux of methylmercury in the environment
= F – φ m (3)
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where F is the flow of mercury into the marine environment and φ is the demethylation or rate of
flow of mercury out of the marine environment. Ambient, inorganic mercury enters the aquatic
system through atmospheric and terrestrial deposition and then, among many other processes,
bacteria methylate the mercury through interactions still not completely understood (Wiener,
2003). Methylmercury bioaccumulates readily and consequently, concentrations of mercury in
fish depend on the environmental concentration. The literature suggests that mercury which
enters the food web is passed upwards through trophic levels, and that most, but not all, mercury
passes upwards from one trophic level to another (Wiener, 2003). For simplicity’s sake we shall
assume that the change in concentration of methylmercury in mackerel is directly proportional to
the change in concentration of methylmercury in the environment. The following state equation
dictates the concentration of mercury in mackerel
mackM& = ζ (4)
where ζ is the parameter guiding the proportional flow of mercury into mackerel. This term
accounts for the rate of methylation and the almost complete transfer of mercury with ingestion.
The concentration of mercury in tuna relies on both the concentration of mercury in the mackerel
and how much mackerel the tuna consume. Therefore, the time derivative of the variable for
mercury concentration in mackerel is included in the state equation describing the concentration
of mercury in tuna
tunaM& = η
mackM& (5)
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Where η denotes the rate of mercury bioaccumulation in the tuna.
A standard fisheries optimal control problem attempts to maximize the present total value
net benefits of fishing—defined as the revenue from fish sales (price of the fish times the number
of fish sold)—minus the cost of fishing for the amount of fish sold. Constructing the objective
function for this problem involves the crucial assumption that the price at which fish are sold is a
function of the mean mercury concentration in an individual of that particular fish species.
Therefore, the goal of maximizing overall benefits in terms of fishing profits remains the same,
but is further influenced by prices which fluctuate based on the concentration of mercury in the
different fishes. Denoting the price of the fish in the market as p, the cost of catching the fish as
c, the amount of fish harvested as h, time as t, and the discount rate as δ, the objective function is
max∫∞
0
[( ptunahtuna – ctunahtuna2) + (pmackhmack – cmackhmack
2] dt (6)
subject to
ptuna = tunatunam
eγ−
(7)
pmack = mackmackm
eγ− (8)
and (1), (2), (3), (4), and (5) above. denotes the unaffected price and γ denotes the rate of price
decay with an increase in mercury. We shall assume that the price of each fish decays at the
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same rate for a given increase in mercury content. The Current Value Hamiltonian, where λ1, λ2, λ3, λ4, and λ5 denote the co-state variables for x, y, m, Mtuna, and Mmack, respectively, is
= [(ptunahtuna – ctunahtuna2) + (pmackhmack – cmackhmack
2)]
+ 1[x(a1 – a2x – Mtuna ) – htuna]
+ 2[y(b1 –b2y – Mmackψ) – hmack]
+ 3[F – φ m]
+ 4 [η mackM& ]
+ 5 [ζ ] (9)
If we assume that there are only two controls, htuna and hmack, and that the parameter for the flow
of mercury into the environment, F, is constant, then the First Order Conditions for this system
are
tunah
H
∂
∂~
= Ptuna – 2Ctunahtuna – 1 = 0 (10)
mackh
H
∂
∂~
= Pmack – 2Cmackhmack – 2 = 0 (11)
x
H
∂
∂−
~= 1 – 1 = – 1[a1 – 2a2x – Mtunaω] = 0 (12)
y
H
∂
∂−
~ = 2 – 2 = – 2[b1 – 2b2y – Mmackψ] = 0 (13)
m
H
∂
∂−
~ = 3 – 3 = 3 φ = 0 (14)
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–tunaM
H
∂
∂~
= 4 – 4 = γ tunatunam
eγ− = 0 (15)
–
mackM
H
∂
∂~
= 5 – 5 = γ mackmackm
eγ−
= 0 (16)
= x[a1 – a2x – Mtunaω] – htuna (17)
= y(b1 – b2y – Mmackψ) – hmack (18)
= F – φ m (3)
mackM& = ζ (4)
tunaM& = η
mackM& (5)
While the flux of mercury is irregular and currently escalating as humans increase
mercury-emitting processes such as burning coal, should mercury emissions level off or be offset
by the rate of demethylation, the system may reach a steady state in which the variables do not
change from one time period to the next. A steady state such as this might also represent any
period in instantaneous time while the system is in equilibrium. Thus all state equations will
have a net change of zero. The steady state can be written as
= x[a1 – a2x – Mtunaω] – htuna = 0 (1’)
= y(b1 – b2y – Mmackψ) – hmack = 0 (2’)
= F – φm = 0 (3’)
mackM& = ζ = 0 (4’)
tunaM& = η
mackM& = 0 (5’)
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The above set specifies that the stocks of fish and the stocks of mercury do not change over time.
Likewise, because the stocks do not change, the shadow values for these stocks also do not
change over time
1 = δλ1 – λ1[a1 – 2a2x – Mtunaω] = 0 (12’)
2 = 2 – 2[b1 – 2b2y – Mmackψ] = 0 (13’)
3 = 3 + 3φ = 0 (14’)
4 = 4 + γ tunatunam
eγ− = 0 (15’)
5 = 5 + γ mackmackm
eγ−
= 0 (16’)
The goal now is to solve for the variables in steady state. This will help uncover the
structure of the system as it operates dynamically. From (3’) we obtain the steady state stock of
mercury in the environment at time τ
mss = ϕ
F (19)
Because also appears in the function for mackM& , we can substitute into (4) to obtain
mackM& = 0 (21)
Integrating from 0 to τ and solving yields
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Mmackss = ζ (F/φ – m0) (23)
Similarly, when substituting this expression into (5), integrating and solving yields
Mtunass = η [ζ (F/φ – m0) – Mmack
0] (24)
The previous steps provide the solutions in the form (19), (23), and (24) for the evolution of the
stock of mercury in (3), (4), and (5). Because the model assumes a constant flow of mercury into
the environment, no controls exist in the model for these variables. The interesting aspects of the
steady state are then the expressions defining the harvests, stocks, and co-state variables for tuna
and mackerel. From (10) and (11) we know that
λ1 = Ptuna – 2Ctunahtuna (25)
λ2 = Pmack – 2Cmackhmack (26)
Substituting into (12) and (13) yields
1 – (Ptuna – 2Ctunahtuna)δ =
– (Ptuna – 2Ctunahtuna) (a1 – 2a2x – Mtunaω) = 0 (27)
2 – (Pmack – 2Cmackhmack)δ =
– (Pmack – 2Cmackhmack) (b1 – 2b2y – Mmackψ) = 0 (28)
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This provides, in conjunction with (1’) and (2’), two systems of equations that can be solved for
the steady-state values of x and htuna, and y and hmack. Solving for htuna and hmack in (1’) and (2’)
results in
htuna = x[a1 – a2x – Mtunaω] (29)
hmack = y[b1 – b2y – M mack ψ] (30)
By substituting into (27) and (28) we obtain
0 = (2Ctuna (x (a1 – a2x –Mtunaω)) – Ptuna) (a1 – 2a2x – Mtunaω – δ) (31)
0 = (2Cmack (y (b1 – b2y –Mmackψ)) – Pmack) (b1 – 2b2y – Mmackψ – δ) (32)
Solving these cubics for x and y yields expressions defining the steady-state stocks of tuna and
mackerel.
tuna
tunatunatunatunatunatunatunass
Ca
MCCaPCaMCax
2
2121
4
)22(82 ωω −+−+−−= (33.1)
tuna
tunatunatunatunatunatunatunass
Ca
MCCaPCaMCax
2
2121
4
)22(82 ωω −+−−−−= (33.2)
2
1
2a
Max tunass ω−
= (33.3)
mack
mackmackmackmackmackmackmackss
Cb
MCCbPCbMCby
2
2121
4
)22(82 ζψ −+−−−−= (34.1)
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mack
mackmackmackmackmackmackmackss
Cb
MCCbPCbMCby
2
2121
4
)22(82 ζψ −+−+−−= (34.2)
2
1
2b
Mby mackss ψ−
= (34.3)
The controls for this system result in three solutions for each of the steady-state fish stocks, up to
two of which can be valid depending on the values of the parameters. The negative root solution
will never be optimal and so must be rejected as a steady-state equilibrium solution. We can
mitigate the complexity of these equations by visualizing the system in phase-space. Such
diagrams require an expression for the evolution of the harvests over time. By taking the time
derivatives of (25) and (26) and rearranging, we obtain
tuna
tunatuna
C
Ph
21λ&&
& −= (35)
mack
mackmack
C
Ph
22λ&&
& −= (36)
In turn, we can then rearrange (27) and (28) and substitute for 1 and 2 yielding
tuna
tunatunatunatunatunatuna
C
PMxaahCPh
2))(2( 21
&& +−−−−
=δω
= 0 (37)
mack
mackmackmackmackmackmack
C
PMybbhCPh
2))(2( 21
&& +−−−−
=δζ
= 0 (38)
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wheretunaP& and
mackP& necessarily equal zero in the steady-state. Graphing these expressions in x
versus htuna and y versus hmack phase-space illustrates where any steady-state equilibria exist.
MODEL PARAMETERIZATION
Using appropriate parameter values poses a challenge. Highly variable fish market prices
compound difficulties caused by the paucity of empirical data regarding stock sizes, intrinsic
growth rates, or the carrying capacity of either species. Therefore, in order to obtain meaningful
results, parameter values must be constructed through a combination of empirical data,
calculation, and estimation. Maintaining a base case scenario in which the static profit
maximizing harvest level remains higher than the maximum intrinsic growth of the fish is a
significant part of a relevant and interesting outcome. Otherwise the model fails to capture many
of the tradeoffs between price decrease and growth decrease. Ensuring that base cases operate in
this manner with limited empirical data requires that some parameters and variables do not
correspond with observed data. The assumptions this dictates are unfortunate, but necessary
given the complexity of the model and present fewer problems in a qualitative model such as this
compared with modeling for quantitative results.
Because some literature explores population variables, we begin with the parameters in
the logisitic growth equations by using the formula a2 = a1 / ktuna, where a1 is the intrinsic rate of
growth of the species and k is the carrying capacity. McAllister and Carruthers (2007) estimated
the intrinsic rate of growth for Western Atlantic Bluefin tuna at 0.1667. We will assume this is
similar to the growth rate for the Eastern Atlantic population and use this value for a1. In the
same study, McAllister and Carruthers estimated a carrying capacity for the Western stock at
approximately 131,500 tons. No such number exists for the Eastern Atlantic, but by taking the
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total biomass estimate for the Eastern Atlantic (International Commission for the Conservation
of Atlantic Tuna, 2008) and dividing by the International Commission for the Conservation of
Atlantic Tuna (ICCAT) (2008) estimated percentage of spawning stock biomass (SSB) necessary
to support maximum sustainable yield (MSY)—less than 40%—we can obtain a rough estimate
for the MSY stock, which when doubled yields the carrying capacity for the Eastern stock.
Adding 131,500 results in the total carrying capacity for the Atlantic of ktuna = 1,375,740.
Unfortunately, using the percentage of SSB necessary to support MSY (SSBmsy) is not an ideal
proxy for what percentage of MSY current biomass represents. Nor is ICCAT confident in their
estimates. Indeed, they suggest that the current SSB may be significantly less than 40% of that
necessary to support MSY and current population trends suggest SSB is declining further relative
to the population due to targeted fishing of mature individuals (ICCAT, 2008). Consequently,
assigning reliable values for ktuna and a2 remains problematic. The next best approach is to
structure these parameters in such a way as to obtain the greatest amount of useful information
from the model, which in this case requires maintaining the base case intersection of the profit
maximization level and the alternate harvest solution line above the logistic growth curve. This
most accurately simulates reality because current fishing of bluefin occurs at an unsustainable
rate. Therefore, fishing at a rate that maximizes profits necessarily results in harvesting beyond
the capacity of fish to reproduce, that is, harvest is higher than logistic growth for all given
population sizes. Simply using the numbers above results in a carrying capacity of about
1,375,740 tons and a value for a2 of 1.21171 * 10-7. These are suitable for our purposes.
The same logistic growth equation is used for mackerel with growth parameters labeled
differently: b2 = b1 / k. Fishbase.org lists the intrinsic rate of growth as a range from 0.33 – 0.56
(Collette, 2009). Because of model calibration needs, we use a slightly low number so that b1 =
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0.3. In the latest stock assessment, the Northeast Fisheries Science Center calculated SSBmsy at
644,000mt, current SSB at 2.3 million tons and total biomass at approximately 2.9 million tons
(2006). Multiplying the ratio of total biomass to SSB (1.26087) by the SSBmsy of 644,000
results in a value for biomass necessary for MSY of 812,000mt. Doubling MSY amounts to a
carrying capacity of kmack = 1,624,000mt. Applying this to the logistic equation and solving
yields b2 = 1.84729×10-7.
For δ, we assume a constant discount rate of 5%, so δ = 0.05.
Assigning values for ω and ψ is a greater challenge. No studies have examined the
effects of mercury on the reproductive capacity of tuna or mackerel, so we cannot say with
certainty how significant these effects might be. However, given the number of studies
demonstrating adverse effects on reproduction in a variety of different fish species (Baker Matta,
2001; Latif et al., 1999; Devlin, 1992; Khan and Weis, 1987; Beckvar, 1996), we assume that
such effects occur in tuna and mackerel. Studies of Coho salmon, killifish, carp, and walleye
demonstrate that increases in mercury concentrations limit such reproductive markers as sperm
motility, fertilization rates, and larval viability (Latif et al., 1999; Devlin, 1992; Khan and Weis,
1987; Chyb et al., 2001). However, the strength of detrimental effects of mercury varies widely
among different species and many of the species studied in the literature are freshwater species.
Consequently, we simulate several scenarios using different values for ω and ψ. These values
range from 0.02 to 0.06 which is sufficient to differentiate these simulation results from the base
case and from each other. A starting value of 0.02 corresponds roughly to approximately a 2%
decrease in hatching success in walleye when moving from an environment without mercury to
one in which concentrations are approximately 2ng/L (Latif et al., 2001), which is at the high end
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of values reported for concentrations of mercury in the ocean (Gill and Fitzgerald, 1988; Gill and
Fitzgerald, 1987; Gill and Fitzgerald, 1985), but accurate enough for the scope of our model.
The next step is to parameterize the expression for harvest: p*h = (rents)*p*h + c*h2
Empirical values for price and harvest might be considered even more unreliable than population
statistics because, while more empirical data exists, price varies greatly depending on the
particular market, and a substantial portion of harvest goes unreported. Therefore, it best serves
our purpose to establish parameter values that aid our inquiry into steady state dynamics.
Pintassilgo and Costa Duarte calculated Bluefin prices between $5/kg and $25/kg, depending on
the gear type (2002). NMFS lists the average price for Bluefin sales at $14.35/kg (2006). For
convenience, we use an average price of $15/kg or rather, because the model is designed for
tons, $15,000/mt. ICCAT (2008) indicates that reported harvest is 34,030 tons, but suggests that
total harvest approaches 61,100 tons. For finding a value for the cost parameter, we use a
harvest value of 60,000 tons. Rents in rationalized fisheries will be a fraction of total revenues.
They may plausibly range between 25% and 60% depending on prices, cost structure of the
fishery, and biological productivity of the stock (Smith, 2009). This large span allows some
leeway to find a value that satisfies the base case conditions described above, in this case, 50%
of the rents from the fishery. So the equation becomes (15000)(60000) = (.5)(15000)(60000) + c
(60000)2 and when solved: c = 0.000000121171.
Estimates for the population size and harvest values for mackerel combined in such a way
as to result in larger logistic growth and lower harvest than fits properly into the model. To
compensate, we assume rents for mackerel to be an unusually high 65%. This, combined with a
low value for intrinsic growth, was necessary to ensure that the profit maximization level
remained above the logistic growth curve. Prices for mackerel have been volatile in recent years,
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falling from about $1500/mt to $750/mt in 2006 (fishupdate.com, 2006) and averaged out at over
$1000/mt in 2007 (NMFS, 2007). For simplicity, we use a price of $1000/mt. The Northeast
Fisheries Science Center mackerel Stock Assessment Review Committee used a 2005 projected
catch of 95,000mt for deterministic projections but believes that long term MSY is closer to
89,000 (2006); we will use 90,000mt in our model to find the cost parameter. The same
expression for harvest from above specified for mackerel is: (1000)(90000) = (.5)(1000)(90000)
+ c (90000)2 and when solved: c = 0.0055556.
The final piece of the model is simulating the flow of mercury through the system,
beginning with emissions. As noted by Mason et al., (2002), the relationship between mercury
emissions and fish concentrations is likely not a simple linear equation due to the intricacies of
the global mercury cycle. Incorporating all the possible fluxes into the system, many of which
are not understood, is beyond the scope of this study and it should simply be noted that the
model attempts to approximate this relationship. The state equation for mercury requires a value
for φ to enable outputs. Assuming a deposition to the marine environment of 2000mt/yr
(Lamborg et al., 2002) and the volume of the global ocean to be about 1.347 billion cubic
kilometers (Gleick, 1996), the base flow, F, into the marine environment is 0.0014847ng/L. For
m, we will use the slightly high end estimate from Gill and Fitzgerald of 0.02ng/L (1985, 1987,
1988). In order to derive a value for φ, it is necessary to assume that the system is in steady state
such that (3’) is true. Substitution into this equation and solving yields φ = 0.000742391. This
allows us to examine the change in environmental mercury concentration for a given change in
F. To see how this influences the concentrations of mercury in mackerel and tuna, it is necessary
to derive values for the parameters ζ and η which can be done by applying empirical data to
equations (4) and (5). As a general rule, concentrations of mercury bioaccumulate by
21
approximately a factor of 10 with each increase in trophic level (Watras and Bloom, 1992;
Lindqvist et al., 1991). Because these experiments take place in freshwater with fish other than
the study species and because they operate on a short time scale, they must be regarded with
some suspicion, but these numbers remain a good approximation nevertheless. Several studies
demonstrate that the primary influx of mercury into the food web occurs at the transfer between
water and fish at a factor of about 106 (Watras and Bloom, 1992; Lindqvist et al., 1991).
Assuming biomagnification by a factor of 10, η = 10. Plugging these parameter values into (3),
(4), and (5) allows us to apply changes to the system and yield evaluable outputs.
Values of 1.0 ppm and 0.05 ppm for Mmack and Mtuna used below correspond roughly to
studies in the toxicology literature (Nakagawa, 1997; Storelli, 2001; Srebocan, 2007; Licata,
2004; FDA, 2006).
VISUALIZING MODEL DYNAMICS
Below are a series of phase-space diagrams showing the relationship between stocks and
harvests under various model simulations. We first examine base cases in which the terms Mtuna
and Mmack are set to zero to allow examination of the system without considering the effects of
mercury. Subsequent simulations examine first the effects of mercury on the price of the fish,
and second the effects of mercury on fish population growth. These effects are then combined to
examine the countervailing forces between them. Lastly, we simulate how an increased flow of
mercury into the environment affects price degradation and growth inhibition.
For the base cases, in addition to setting mercury concentrations to zero, we assume that
ψ = 0, ω = 0, and γ = 0. This corresponds to a situation in which the market price of the fish does
not degrade as a result of high mercury concentrations in fish and, likewise, the fish do not suffer
22
biologically as a result of high mercury concentrations. Figures 1 and 2 display the base cases.
The curve is the logistic growth curve which describes the rate of growth of the fish for a
particular population size and represents the isocline at which harvest and growth counterbalance
each other such that no changes in population occur over time. The vertical and horizontal lines
are solutions to the equations (37) and (38). The intersections of solution lines and the logistic
growth curve are optimal steady state solutions such that the system can indefinitely maintain the
harvest and population sizes at the point of intersection. The horizontal lines also represent the
harvest levels which maximize profits. Harvesting never occurs above these horizontal lines, as
doing so would be economically unbeneficial. Black arrows detail the directions that harvest and
stock size will shift when the system is in flux at a particular location on the diagram. When
harvesting occurs above the logistic growth curve, population growth is insufficient to maintain
stocks, and population decreases. Correspondingly, when harvesting below the logistic growth
curve, the fish population grows despite harvesting, and stocks increase over time. Colored
arrows indicate example trajectories towards particular outcomes. The red and green arrows
show paths to an optimal steady state result. The light purple arrows show paths to extinction.
23
Figure 1 – Tuna Base Case (ω = 0; γ = 0)
0
20000
40000
60000
80000
100000
120000
140000
1
61
121
181
241
301
361
421
481
541
601
661
721
781
841
901
961
1021
1081
1141
1201
1261
1321
1381
1441
1501
1561
1621
1681
1741
y (thousands of tons)
hm
ack
(to
ns)
Figure 2 – Mackerel Base Case (ψ = 0; γ = 0)
24
Because the profit maximizing harvest level is above the logistic growth curve, only one
optimal steady state solution exists in these situations. The fact that profit maximizing harvest is
above the growth curve also suggests what is likely the current state of the fisheries in which
fishers could achieve greatest profit by driving the fish to extinction.
Figures 3 – 8 are representative cases (using tuna) that depict how the stock and harvest
change over time for the three trajectory paths demarcated by the colored arrows. Figures 3 and
4 illustrate the case where the harvest at t = 0 is along the profit maximizing harvest level that
decreases stocks and eventually drops to attain the optimal steady state solution at the
intersection of the logistic growth curve and the vertical solution line. Figures 5 and 6 consider
the same outcome with different initial harvest and population sizes wherein the harvest at t = 0
is below the isocline, thus leading eventually to the steady state optimal solution. Figures 7 and
8 illustrate the feasible, but non-optimal case where the harvest at t = 0 is above the isocline and
initial population is low, thus leading to extinction.
Figure 3 & 4 – Tuna Base Case: htuna0 at profit maximization (time vs. x; time vs. htuna)
25
Figures 5 & 6 – Tuna Base Case: htuna0 Below Logistic Growth Curve Isocline (time vs. x; time vs. htuna)
Figures 7 & 8 – Tuna Base Case: htuna0 above isocline, left of vertical line solution (Time vs. x; time vs. htuna)
Figures 9 and 10 consider the case in which fish-specific mercury concentrations drive
consumer decision-making, thus leading to reduced demand and subsequent price decay that
results in lower profit maximizing harvest levels. The fish populations remain unaffected by
direct toxicity from mercury, but because of a declining profit maximizing harvest level for tuna,
the horizontal line creates new intersections with the logistic growth curve isocline, indicating
26
new optimal steady state solutions. The black arrows depict the shift in the profit maximizing
harvest level for varying levels of consumer concern (represented by changes in γ) and the blue
arrows show the corresponding shift in the optimal steady state solutions from the base case
intersection to the new intersections. Note that although two viable points of intersection exist
for each of these simulations, the intersection points at lower population values are not optimal
and should the system reach those intersection points, the optimal path will travel along the profit
maximization level to the solution at a higher fish population. In terms of management, the
important thing to recognize is that under these price decay conditions, the optimal steady state
results in a much higher population of tuna than when this effect is not considered.
0
10000
20000
30000
40000
50000
60000
70000
1
74
147
220
293
366
439
512
585
658
731
804
877
950
1023
1096
1169
1242
1315
x (thousands of tons)
htu
na
(to
ns)
Figure 9 – Tuna Price Decay A (ω = 0; γ = 0. 2, γ = 0. 35, γ = 0. 5)
27
0
20000
40000
60000
80000
100000
120000
140000
1
73
145
217
289
361
433
505
577
649
721
793
865
937
1009
1081
1153
1225
1297
1369
1441
1513
1585
1657
1729
y (thousands of tons)
hm
ack
(to
ns)
Figure 10 – Mackerel Price Decay A (ψ = 0; γ = 0. 2, 0. 35, 0. 5)
Note also that both price and harvest change relatively little for mackerel compared to tuna. This
is due both to relatively low mercury concentration and low price for mackerel.
Figures 11 and 12 show the other effect due to mercury considered in this model, namely,
population growth inhibition. In these simulations, the price decay effect is considered
insignificant and ignored for the sake of focusing on the negative effects of mercury on fish
reproduction. The diagrams show different scenarios corresponding to different values of the
reproductive damage variables ω and ψ (colored) compared to the base cases (black). The
important aspect of this situation is that despite continued demand for fish at the same harvest
level, declines in the growth potential result in intersections between the vertical harvest solution
and the growth curve at harvest levels less than those of the base cases. This means that while
people would prefer to consume more fish, high concentrations of mercury limit fish population
size enough to reduce the level of harvest which can be sustained in perpetuity. As with the
previous case, tuna is affected to a much greater degree than mackerel.
28
0
10000
20000
30000
40000
50000
60000
70000
1
53
105
157
209
261
313
365
417
469
521
573
625
677
729
781
833
885
937
989
1041
1093
1145
1197
1249
1301
1353
x (thousands of tons)
htu
na (
ton
s)
Figure 11 – Tuna Inhibition (ω = 0.02, ω = 0.04, ω = 0.06; γ = 0; Mtuna = 1)
0
20000
40000
60000
80000
100000
120000
140000
1
83
165
247
329
411
493
575
657
739
821
903
985
1067
1149
1231
1313
1395
1477
1559
1641
1723
y (thousands of tons)
hm
ack
(to
ns)
Figure 12 – Mackerel Inhibition (ψ = 0.02, ψ = 0.04, ψ = 0.06; γ = 0; Mmack = 0.05)
Some studies have shown that higher trophic level predators such as marine mammals can
tolerate very high levels of mercury by transforming methylmercury into an inorganic form
29
which is less toxic (Beckvar, 1996). It could be the case that bluefin tuna possess such resistance
and can tolerate higher mercury concentrations than Atlantic mackerel, thus justifying lower
values for ω relative to ψ, but this has not been studied for the study species.
The combination of the price decay effect and the reproductive inhibition effect is shown
(in black) in Figures 13 and 16 in contrast to the base cases (in gray). The simulation is
particularly interesting for tuna due to the presence of three intersections (solutions). Only two,
however, are optimal steady state solutions, A and C, since arrival at B is not optimal compared
to C. Should initial harvest and population occur below the logistic growth curve, the system
will move upwards and to the next solution to the right. Initial values above the isolcine near A
and B will move towards A while initial values above the isocline to the right of C will move
towards C. Five such trajectories are described in Table 1.
0
10000
20000
30000
40000
50000
60000
70000
1
64
127
190
253
316
379
442
505
568
631
694
757
820
883
946
1009
1072
1135
1198
1261
1324
x (thousands of tons)
htu
na (
ton
s)
Figure 13 – Tuna Mercury Effects Combination (ω = 0.02; γ = .2; Mtuna = 1)
A
B C
30
Table 1 - Tuna Combination (ω = 0.02; γ = .2; Mtuna = 1)
Now suppose that the effects of growth inhibition occur first and that the price decay
effect follows afterwards such that the horizontal line (profit maximizing harvest level) remains
at 60,000mt until after the growth curve and vertical solution line have shifted (similar to the
case illustrated in Figure 11). Perhaps consumers do not appreciate the dangers of mercury until
they learn the damage it has done to the fish population. From the previous optimal steady state,
harvest increases instantaneously and then declines over time to a new steady state. Additional
intersections (optimal solutions) occur as the profit maximizing harvest level drops and
eventually intersects the logistic growth curve. Harvest remains static until the horizontal line
passes below the intersection of the vertical line with the logistic growth curve, at which point
harvesting declines along the path of the isocline with increasing γ (see Figure 14).
The population size also shifts rapidly, though not instantaneously as fish need time to
reproduce as opposed to harvesting which is without limits on rapidity of change. Figure 15
displays the change in population size as γ increases. With a change in harvesting solutions
along the profit maximizing harvesting level, the tuna population is allowed to increase
substantially and continues to do so as γ continues to increase just as it does in Figure 9.
Initial harvest & population Outcome
Below isocline, left of vertical solution line Approach A from below the isocline
Below isocline, right of vertical solution line, left of B
Approach B from below the isocline, continue to C along static profit maximizing harvest line
Above isocline, left of B Approach A from above the isocline
Below isocline, right of B Approach C from below the isocline
Above isocline, right of C Approach C from above isocline along static profit maximizing harvest line
31
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time
htuna (tons)
Figure 14 – Tuna combination: Inhibition occurs before γ increases (Tuna: Time vs. Htuna)
Yet again, the price decay and population growth inhibition effects alter the mackerel
fishery very little. Indeed, the base case is barely differentiable by eye from the combination of
effects in Figure 15.
0
20000
40000
60000
80000
100000
120000
140000
160000
1
83
165
247
329
411
493
575
657
739
821
903
985
1067
1149
1231
1313
1395
1477
1559
1641
1723
y (thousands of tons)
hm
ack (to
ns)
Figure 15 – Mackerel Mercury Effects Combination (ψ = .5; γ = 0.3; Mmack = 0.05)
32
The last scenarios we consider deal with increasing mercury pollution, in our model
defined as F, the flow of mercury into the marine environment. Increases in F in turn increase
Mmack and Mtuna and these increased concentrations modify the price decay and population
growth inhibition effects accordingly.
0
10000
20000
30000
40000
50000
60000
70000
80000
1
64
127
190
253
316
379
442
505
568
631
694
757
820
883
946
1009
1072
1135
1198
1261
1324
x (thousands of tons)
htu
na (
ton
s)
Figure 16 – Flow of Hg into Environment Increases by 1% (F = 1.49963*10-9; ω = 0.02; γ = .2; Mtuna = 1.2)
A second interesting aspect of this particular simulation evolves from considering further
increases in mercury concentrations in the tuna due to continuing mercury pollution (Mtuna
increasing) when consumers no longer react to increases in mercury concentrations by changing
their purchasing behavior (γ drops to zero after a certain threshold, perhaps because all the
people willing to reduce their consumption have done so already) such that the profit maximizing
harvest level remains in place. Fish continue to suffer reproductive inhibition as a result of
increasing mercury concentrations, thus shrinking the logistic growth curve. Eventually, the
curve falls low enough that it no longer intersects the profit maximizing harvest level and the
33
previous optimal harvesting solution is lost. The result is an instantaneous shift leftwards on the
diagram to the vertical solution line in order to maintain indefinite harvesting. A leftward shift
means a reduction in tuna population, shown in Figure 17.
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Mmack (ppm)
hm
ac
k (
ton
s)
Figure 17 – F increases: Tuna (Mmack vs. hmack)
0
100000
200000
300000
400000
500000
600000
700000
1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2
Mmack (ppm)
x (
ton
s)
Figure 18 – F increases Tuna: (Mmack vs. x)
34
This result is the opposite of the price decay effect described in Figure 14. Both
outcomes result in lower harvests, but price decay leads to higher stocks, while growth inhibition
leads to lower stocks relative to the base case. When considering both effects together, the
ultimate steady-state outcome depends on which variables exert more control over the system.
Because this model is highly theoretical without a large body of literature to rely on, postulating
reasonable values for parameters is a significant challenge, thus making any conjectures on the
relative importance of particular variables highly uncertain.
0
20000
40000
60000
80000
100000
120000
140000
160000
1
83
165
247
329
411
493
575
657
739
821
903
985
1067
1149
1231
1313
1395
1477
1559
1641
1723
y (thousands of tons)
hm
ack
(to
ns)
Figure 19 – Flow of Hg into Environment Increases by 1% (F = 1.49963*10-9; ω = 0.02; γ = .2; Mtuna = 0.07)
The resistance of the mackerel fishery to changes resulting from mercury is obvious
throughout this section. Even simulating a 10% increase in mercury concentration in mackerel
resulted in very little change in either harvesting or population growth potential. We must
conclude that under the assumptions of this particular model that mackerel is a resilient fishery.
35
It is possible that using other models which incorporate detailed aspects of the system which do
not appear in the contaminated fisheries model such as population interactions between species
could change the above results.
INTERPRETATION OF CONTAMINATED FISHERIES MODEL
The various scenarios above demonstrate a variety of outcomes for harvests and stocks
depending on the values of the variables and parameters. We explore many different options
because very little information exists on the values of most of the input values, resulting in a
wide range of possible scenarios that cannot be narrowed without further research. These values
are gleaned from the literature to the greatest extent possible, but many must be estimated and
may not be accurate. Nonetheless, for the purposes of qualitative modeling, using approximate
numbers allows for a theoretical analysis that can direct further research to develop empirical
results. Current management lacks substantive information in this area and a range of
predictions provides tools for managers to more effectively oversee marine resources and their
consumption for many possible future outcomes.
The economic interpretation for most of these situations in steady-state is relatively
straightforward. The difficulty lies in understanding how the countervailing effects of mercury
contamination alter harvesting and stock sizes. The struggle between the strength of price decay
and negative reproductive effects that hamper stocks causes this system to operate in some
interesting ways. Should the rate of price decay spike without a corresponding reduction in the
intrinsic growth rate of tuna stocks, harvest will decrease in profitability and stocks will
eventually reach a new steady-state equilibrium commensurate with the reproductive damage of
mercury. Alternately, if consumers remain relatively indifferent to the dangers of mercury, but
36
fish begin to suffer biologically as a result of high mercury concentrations, then we experience a
cases in which both harvesting and mercury toxicity contribute to the decline of stocks.
A major lesson for management is that with higher concentrations of mercury, both the
market price decay of mackerel and the reproductive damage to mackerel are likely to be less
than for bluefin tuna. Depending on the parameters, this will likely result in greater decreases in
harvest of bluefin relative to mackerel. One might speculate that as global demand for seafood
continues to increase, this will put greater upward pressure on seafood prices—unless a
particular species is contaminated. High levels of contamination combined with scarce seafood
could further drive up prices and harvests of remaining, uncontaminated species such as Atlantic
mackerel. This would suggest a further disparity in relative harvesting of bluefin and mackerel
compared to the present. It is unfortunate that time and the complexity of initial research on a
predator-prey model necessitated dispensing with the original intention to incorporate consumer
choices into the population interaction dynamics of the system in this particular study, but we
discuss the initial stages of a more detailed model below as a precursor to continuing research on
the subject.
ALTERNATIVE MODEL INCLUDING STOCK EFFECTS
The approach described above models a situation in which the size of the stocks of tuna
and mackerel fail to change the outcome of the problem from the perspective of a harvester.
This is a great simplification as the cost of catching fish can change dramatically depending on
the stock size. With high stocks, less fishing is necessary to reach the optimal harvest because
fish occur in more locations and are therefore easier to catch. When stocks are low, fishers must
often increase their effort and costs in order to catch the optimal harvest of fish. A simple way to
37
address this issue is to include additional stock terms in the state equations influencing the cost
such that the objective function becomes
max∫∞
0
[(ptunahtuna – (Ctuna/x)htuna2) + (pmackhmack – (Cmack/y)hmack
2] dt (6’’)
This addition modifies the First Order Conditions (10), (11), (12), and (13):
tunah
H
∂
∂~
= Ptuna – (2Ctuna/x)htuna – 1 = 0 (10’’)
mackh
H
∂
∂~
= Pmack – (2Cmack/y)hmack – 2 = 0 (11’’)
x
H
∂
∂−
~= 1 – 1 = (Ctuna/x
2)htuna2 – 1[a1 – 2a2x – Mtunaω] = 0 (12’’)
y
H
∂
∂−
~ = 2 – 2 = (Cmack/y
2)hmack2 – 2[b1 – 2b2y – Mmackψ] = 0 (13’’)
This in turn affects the expressions for the co-state variables when solving for the steady-state
λ1 = Ptuna – (2Ctuna/x)htuna (25’’)
λ2 = Pmack – (2Cmack/y)hmack (26’’)
Which after substituting into (12’’) and (13’’) yields
0 = (Ctuna/x2)htuna
2 – [Ptuna – 2(Ctuna/x)htuna](a1 – 2a2x – Mtunaω – δ) (31’’)
0 = (Cmack/y2)hmack
2 – [Pmack – 2(Cmack/y)hmack](b1 – 2b2y – Mmackψ – δ) (32’’)
38
These results can be applied in a similar manner as with the first model. Expressions for the
change in harvest over time are
tuna
tuna
tunaC
Pxh
21λ&&
&&−
= = 0 (33’’)
mack
mack
mackC
Pyh
22λ&&
&&−
= = 0 (34’’)
Substituting in (1’), (2’), (31’’), and (32’’), we obtain
( )
tuna
tunatunatuna
tunatunatuna
tuna
tunatunaC
Mxaahx
cPh
x
cP
Mxaaxh2
2)(2()()(
212
2
21
δω
ω
−−−
−−−
−−=
&
&
(35’’)
( )
mack
mackmackmack
mackmackmack
mack
mackmackC
Mybbhy
cPh
y
cP
Mybbyh2
2)(2()(
)(21
2
2
21
δψ
ψ
−−−
−−−
−−=
&
&
(36’’)
Solving (1’) and (2’) for htuna and hmack and then substituting into (35’’) and (36’’) results in two
fourth order polynomials. Solving for x and y numerically yields three non-negative solutions
for each. Parameterizing this model and then evaluating these solutions as with the previous
model will yield different results that can describe the system in greater detail and complexity.
39
Unfortunately, further examination of this particular model is beyond the scope of this study, but
remains the likely next step for further investigation.
DISCUSSION
The contaminated fisheries model posed several difficulties. In standard predator-prey
interactions, the predator feeds on more species than the prey under consideration and the prey is
consumed by other predators not considered. This is particularly pertinent for the current
analysis because tuna are opportunistic foragers and accumulate a great deal of mercury from
sources other than mackerel. We assume in this model for the sake of simplicity that
tunaM& depends only on the change in concentration of mercury in mackerel. A more detailed
approach would include constructing a more representative equation for the accumulation of
mercury in tuna by including other species, whether by replacing the role of mackerel in the
model with an aggregate of bluefin prey or by incorporating additional individual species state
equations. An alternate approach to addressing this problem is changing the study species. This
model is not species specific and applying it to other species may prove more appropriate.
A systemic problem in the contaminated fisheries model remains a heavy reliance on
non-empirical assumptions. Scientists have yet to uncover values for many of the parameters
required by the model. Some, particularly γ, represent any number of collective inputs
interacting simultaneously. A strong hedonic pricing push could uncover some of the consumer
choices that result in a market price for fish, but ultimately, the degree to which preferences
offset each other may defy comprehension. When considering bluefin tuna, consumers may
balance mercury content and sustainability of fishing on one side with the high levels of omega-3
fatty acids and a highly desirable taste on the other.
40
From an environmental chemistry perspective, the model oversimplifies a number of
processes in the global mercury cycle that affect the relationships between emissions,
environmental, and biological concentrations of mercury. The effects of fluxes between the
atmosphere, the oceans, and the terrestrial environment the localized nature of some mercury
pollution increasingly complicates the system, but may have significant effects on the outcomes
discussed above. Additionally, some research describes the build up of tolerance in some fish
species to toxics in the environment (Khan and Weis, 1987), a fact that requires consideration for
any future work concerning reductions in fish growth potential as a result of high mercury
concentrations.
The decline of Bluefin tuna stocks compounds the problems posed by mercury. Because
stocks remain depleted and overfishing continues to occur (International Commission for the
Conservation of Atlantic Tuna, 2008), the initial value of the stock is likely close to the origin in
a phase-space diagram, suggesting that under current conditions, steady-state equilibriums will
tend to be at harvests less than Maximum Sustainable Yield (MSY).
Because bluefin tuna stocks are in global decline, this problem has significant
implications for fisheries management when examined from the viewpoint of conservationists.
Should mercury drive consumer choices as we assume in this model, demand for tuna could drop
enough to result in a profit maximizing level of harvest low enough to allow tuna stocks to
recover. Alternately, should mercury reduce population growth in tuna, stocks could decline
even further. Determining which of these effects is more sensitive to increases in environmental
mercury concentrations is critical for future management of this species. Conservation groups
focusing effort on bluefin may find that educating consumers about the dangers of mercury is
more effective in reducing overfishing and increasing stocks than traditional methods. This
41
strategy may appear an indirect way of achieving tuna conservation, but remains a possible
outreach tool to direct at those consumers more concerned with health than with conservation of
marine resources. This model is not limited to bluefin-mackerel interactions and could be
applied to any other situations in which a contaminated predator feeds upon a seafood resource
which we also consume.
A major avenue of further study if given more time and resources would be to
incorporate additional empirical data into the model. More accurate numbers would enable us to
construct the optimal approach path to the steady state and gain more understanding of γ by
manipulating price through changing mercury concentrations. One broader benefit of such work
could be to help develop estimates for the social costs of mercury in a manner entirely different
than that used by Trasande and the EPA (Griffiths, 2007). The difficulties of this approach lie in
the assumptions about information dissemination to the public and other factors not considered
in the model such as the health benefits of omega-3 fatty acids found in both fishes.
Given the long residence time of mercury, both in the natural environment and in human
beings, it might make sense to examine the issues of long run discounting. Constant exponential
discounting is standard practice, but strongly tilts extraction towards the present. Given Booth’s
(2005) findings that global warming will exacerbate mercury bioaccumulation, and the
unlikelihood of reversing such a trend, prices of bluefin could fall relative to prices of mackerel
in the far distant future. Such a result is undesirable from a conservation or ecology point of
view because this relative price change encourages fishers to further overexploit bluefin in the
present. Avoiding this situation requires explicitly accounting for non-market values of bluefin.
The framework constructed in this paper attempts to model the optimal management of
two harvested fish species where one becomes contaminated as a result of preying upon the
42
other. The framework incorporates both the concerns consumers have about the dangers of
mercury toxicity from consuming these fish as reflected in the price of the fish and the decline in
maximum growth rates of the fish resulting from mercury-reduced fecundity. With a properly
working model, the price could tell us how consumers value the damages due to a given increase
in mercury. Managers, policymakers, and scientists alike can benefit from understanding these
complex biological, chemical, and economic interactions and how they will affect the future.
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