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
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.
1 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
2
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.
3 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
4 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.
5 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.
6 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 – a 2x – Mtuna ) – h tuna (1)
= y (b 1 – b 2y – Mmack ψ) – h mack (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)
7
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
ζ M& mack = (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
M& tuna = η M& mack (5)
8
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
∞ 2 2 max ∫ [( ptuna htuna – c tuna htuna ) + (p mack hmack – c mack hmack ] dt (6) 0
subject to
−γ mtuna ptuna = tuna e (7)
−γ mmack pmack = mack e (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
9 λ λ same rate for a given increase in mercury content. The Current Value Hamiltonian, where 1, 2,
λ3, λ4, and λ5 denote the co-state variables for x, y, m, M tuna , and M mack , respectively, is
2 2 = [(p tuna htuna – c tuna htuna ) + (p mack hmack – c mack hmack )]
+ 1[x(a 1 – a 2x – M tuna ) – h tuna ]
+ 2[y(b 1 –b2y – M mack ψ) – h mack ]
+ 3[F – φ m]
+ 4 [η M& mack ]
+ 5 [ζ ] (9)
If we assume that there are only two controls, h tuna and h mack , and that the parameter for the flow
of mercury into the environment, F, is constant, then the First Order Conditions for this system
are
~ ∂H = P tuna – 2C tuna htuna – 1 = 0 (10)
∂htuna
~ ∂H = P mack – 2C mack hmack – 2 = 0 (11)
∂hmack
~ ∂H − = 1 – 1 = – 1[a 1 – 2a 2x – M tuna ω] = 0 (12) ∂x
~ ∂H 2 − = 2 – 2 = – [b 1 – 2b 2y – M mack ψ] = 0 (13) ∂y
~ ∂H 3 − = 3 – 3 = φ = 0 (14) ∂m
10 ~ −γ mtuna – ∂H = 4 – 4 = γ tuna e = 0 (15) ∂M tuna
~ γ −γ mmack – ∂H = 5 – 5 = mack e = 0 (16) ∂M mack
= x[a 1 – a 2x – Mtuna ω] – h tuna (17)
= y(b 1 – b 2y – Mmack ψ) – h mack (18)
= F – φ m (3)
ζ M& mack = (4)
M& tuna = η M& mack (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[a 1 – a 2x – M tuna ω] – h tuna = 0 (1’)
= y(b 1 – b 2y – Mmack ψ) – h mack = 0 (2’)
= F – φm = 0 (3’)
ζ M& mack = = 0 (4’)
M& tuna = η M& mack = 0 (5’)
11 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 = δλ 1 – λ1[a 1 – 2a 2x – Mtuna ω] = 0 (12’)
2 = 2 – 2[b 1 – 2b 2y – Mmack ψ] = 0 (13’)
3 = 3 + 3φ = 0 (14’)
−γ mtuna 4 = 4 + γ tuna e = 0 (15’)
γ −γ mmack 5 = 5 + mack e = 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
F mss = (19) ϕ
Because also appears in the function for M& mack , we can substitute into (4) to obtain
0 M& mack = (21)
Integrating from 0 to τ and solving yields
12
ss 0 Mmack = ζ (F/ φ – m ) (23)
Similarly, when substituting this expression into (5), integrating and solving yields
ss 0 0 Mtuna = η [ ζ (F/ φ – m ) – M mack ] (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 = P tuna – 2C tuna htuna (25)
λ2 = P mack – 2C mack hmack (26)
Substituting into (12) and (13) yields
1 – (P tuna – 2C tuna htuna )δ =
– (Ptuna – 2C tuna htuna ) (a1 – 2a 2x – Mtuna ω) = 0 (27)
2 – (P mack – 2C mack hmack )δ =
– (P mack – 2C mack hmack ) (b 1 – 2b 2y – Mmack ψ) = 0 (28)
13 This provides, in conjunction with (1’) and (2’), two systems of equations that can be solved for the steady-state values of x and h tuna , and y and h mack . Solving for h tuna and hmack in (1’) and (2’) results in
htuna = x[a 1 – a 2x – M tuna ω] (29)
hmack = y[b 1 – b 2y – M mack ψ] (30)
By substituting into (27) and (28) we obtain
0 = (2C tuna (x (a 1 – a2x –Mtuna ω)) – P tuna ) (a 1 – 2a 2x – M tuna ω – δ) (31)
0 = (2C mack (y (b 1 – b 2y –Mmack ψ)) – P mack ) (b 1 – 2b 2y – M mack ψ – δ) (32)
Solving these cubics for x and y yields expressions defining the steady-state stocks of tuna and mackerel.
− 2a C M ω + − 8a C P + 2( a C − 2C M ω)2 xss = − 1 tuna tuna 2 tuna tuna 1 tuna tuna tuna (33.1) 4a2Ctuna
− 2a C M ω − − 8a C P + 2( a C − 2C M ω)2 xss = − 1 tuna tuna 2 tuna tuna 1 tuna tuna tuna (33.2) 4a2Ctuna
a − M ω xss = 1 tuna (33.3) 2a2
− 2b C M ψ − − 8b C P + 2( b C − 2C M ζ )2 y ss = − 1 mack mack 2 mack mack 1 mack mack mack (34.1) 4b2Cmack
14 − 2b C M ψ + − 8b C P + 2( b C − 2C M ζ )2 y ss = − 1 mack mack 2 mack mack 1 mack mack mack (34.2) 4b2Cmack
b − M ψ y ss = 1 mack (34.3) 2b2
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
P&tuna − λ&1 h&tuna = (35) 2Ctuna
P&mack − λ&2 h&mack = (36) 2Cmack
In turn, we can then rearrange (27) and (28) and substitute for 1 and 2 yielding
(Ptuna − 2Ctuna htuna )( a1 − a2 x − Mtuna ω −δ ) + P&tuna h&tuna = = 0 (37) 2Ctuna
(Pmack − 2Cmack hmack )( b1 − b2 y − M mack ζ −δ ) + P&mack h&mack = = 0 (38) 2Cmack
15 where P&tuna and P&mack necessarily equal zero in the steady-state. Graphing these expressions in x
versus h tuna and y versus h mack 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 a 2 = a 1 / k tuna , 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 a 1. 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
16 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 k tuna = 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 k tuna and a 2 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
-7 1,375,740 tons and a value for a 2 of 1.21171 * 10 . These are suitable for our purposes.
The same logistic growth equation is used for mackerel with growth parameters labeled
differently: b 2 = b 1 / 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 b 1 =
17 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 k mack = 1,624,000mt. Applying this to the logistic equation and solving
-7 yields b 2 = 1.84729×10 .
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
18 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*h 2
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,
19 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
20 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 10 6 (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 M mack and M tuna 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 M tuna and M mack 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
21 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.
22
Figure 1 – Tuna Base Case ( ω = 0; γ = 0)
140000
120000
100000
80000
60000 hmack (tons) hmack 40000
20000
0 1 61 121 181 241 301 361 421 481 541 601 661 721 781 841 901 961 y (thousands of tons)1021 1081 1141 1201 1261 1321 1381 1441 1501 1561 1621 1681 1741
Figure 2 – Mackerel Base Case ( ψ = 0; γ = 0)
23 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.
0 Figure 3 & 4 – Tuna Base Case: h tuna at profit maximization (time vs. x; time vs. h tuna )
24
0 Figures 5 & 6 – Tuna Base Case: h tuna Below Logistic Growth Curve Isocline (time vs. x; time vs. h tuna )
0 Figures 7 & 8 – Tuna Base Case: h tuna above isocline, left of vertical line solution (Time vs. x; time vs. h tuna )
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
25 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.
70000
60000
50000 40000
30000 20000 htuna (tons) 10000 0 1 7 4 1 4 7 2 2 0 2 9 3 3 6 6 4 3 9 5 1 2 5 8 5 6 5 8 7 3 1 8 0 4 8 7 7 9 5 0 1 0 2 3 1 0 9 6 1 1 6 9 1 2 4 2 1 3 1 5 x (thousands of tons)
Figure 9 – Tuna Price Decay A ( ω = 0; γ = 0. 2, γ = 0. 35, γ = 0. 5)
26 140000
120000
100000
80000 (tons) 60000 mack h 40000
20000
0 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)
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.
27 70000
60000
50000
40000 (tons) 30000 tuna h 20000
10000
0 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)
Figure 11 – Tuna Inhibition (ω = 0.02, ω = 0.04, ω = 0.06; γ = 0; M tuna = 1)
140000
120000
100000
80000 (tons) 60000 mack h
40000
20000
0 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)
Figure 12 – Mackerel Inhibition (ψ = 0.02, ψ = 0.04, ψ = 0.06; γ = 0; M mack = 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
28 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.
70000
60000 50000 B C 40000 A
(tons) 30000 tuna
h 20000 10000 0 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)
Figure 13 – Tuna Mercury Effects Combination ( ω = 0.02; γ = .2; M tuna = 1)
29 Initial harvest & population Outcome
Below isocline, left of vertical solution line Approach A from below the isocline
Below isocline, right of vertical solution line, Approach B from below the isocline, continue left of B 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
Table 1 - Tuna Combination ( ω = 0.02; γ = .2; M tuna = 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.
30 50000
45000
40000
35000
30000
25000
20000
15000 htuna (tons) htuna
10000
5000
0 1 2 3 4 5 6 7 8 9 1011121314151617181920 Time
Figure 14 – Tuna combination: Inhibition occurs before γ increases (Tuna: Time vs. H tuna )
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.
160000
140000
120000
100000
(tons) 80000
mack 60000 h 40000 20000
0 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)
Figure 15 – Mackerel Mercury Effects Combination (ψ = .5; γ = 0.3; M mack = 0.05)
31 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 M tuna and these increased concentrations modify the price decay and population
growth inhibition effects accordingly.
80000 70000 60000 50000 40000 (tons) 30000 tuna h 20000 10000 0 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)
-9 Figure 16 – Flow of Hg into Environment Increases by 1% (F = 1.49963*10 ; ω = 0.02; γ = .2; M tuna = 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 (M tuna 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
32 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.
45000
40000
35000
30000
25000 (tons) 20000 mack
h 15000
10000
5000
0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Mmack (ppm)
Figure 17 – F increases: Tuna (M mack vs. h mack )
700000
600000
500000
400000
300000 x (tons)
200000
100000
0 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)
Figure 18 – F increases Tuna: (Mmack vs. x)
33 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.
160000 140000 120000 100000 80000 (tons) 60000 mack
h 40000 20000 0 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)
-9 Figure 19 – Flow of Hg into Environment Increases by 1% (F = 1.49963*10 ; ω = 0.02; γ = .2; M tuna = 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.
34 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
35 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
36 address this issue is to include additional stock terms in the state equations influencing the cost such that the objective function becomes
∞ 2 2 max ∫ [(ptuna htuna – (C tuna /x)h tuna ) + (p mack hmack – (C mack /y)h mack ] dt (6’’) 0
This addition modifies the First Order Conditions (10), (11), (12), and (13):
~ ∂H = P tuna – (2C tuna /x)htuna – 1 = 0 (10’’)
∂htuna
~ ∂H = P mack – (2C mack /y)hmack – 2 = 0 (11’’)
∂hmack
~ ∂H 2 2 − = 1 – 1 = (Ctuna /x )htuna – 1[a 1 – 2a 2x – Mtuna ω] = 0 (12’’) ∂x
~ 2 2 ∂H 2 − = 2 – 2 = (C mack /y )h mack – [b 1 – 2b 2y – Mmack ψ] = 0 (13’’) ∂y
This in turn affects the expressions for the co-state variables when solving for the steady-state
λ1 = P tuna – (2C tuna /x)h tuna (25’’)
λ2 = P mack – (2C mack /y)h mack (26’’)
Which after substituting into (12’’) and (13’’) yields
2 2 0 = (Ctuna /x )htuna – [P tuna – 2(C tuna /x)h tuna ](a1 – 2a 2x – Mtuna ω – δ) (31’’)
2 2 0 = (Cmack /y )h mack – [P mack – 2(C mack /y)h mack ](b 1 – 2b 2y – M mack ψ – δ) (32’’)
37
These results can be applied in a similar manner as with the first model. Expressions for the change in harvest over time are
P&tuna − λ&1 h&tuna = x& = 0 (33’’) 2Ctuna
P&mack − λ&2 h&mack = y& = 0 (34’’) 2Cmack
Substituting in (1’), (2’), (31’’), and (32’’), we obtain
ctuna 2 ctuna P&tuna − ( 2 )htuna − (Ptuna − (2 )htuna ()a1 − 2a2 x − M tuna ω − δ x x h& = x(a − a x − M ω) tuna 1 2 tuna 2C tuna
(35’’)
cmack 2 cmack P&mack − ( )hmack − (Pmack − (2 )hmack ()b1 − 2b2 y − Mmack ψ −δ y2 y h& = y(b − b y − M ψ ) mack 1 2 mack 2C mack
(36’’)
Solving (1’) and (2’) for h tuna and h mack 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.
38 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
M& tuna 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.
39 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
40 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
41 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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