Considering Economic Efficiency in Ecosystem-Based Management: The Case of Horseshoe Crabs in Delaware Bay

Abstract

The welfare gains from incorporating ecosystem considerations into fisheries management are unclear and can vary widely between systems. Additionally, welfare gains depend on how ecosystem considerations are adopted. This paper uses an empirically parameterized bioeconomic model to explore the welfare implications of two definitions of ecosystem-based fisheries management (EBFM). We first define EBFM as a fishery management plan that maximizes the net present value of ecosystem services. We then explore an alternative definition that adds ecosystem considerations to a fishery managed with regulated open access. Our biological model reflects horseshoe crabs in Delaware Bay, which are harvested in a commercial fishery and are ecologically linked to migrating shorebirds populations, e.g. the endangered red knot. We find that introducing ecosystem considerations to a regulated open access fishery generates welfare gains on par with gains from addressing the commons problem even when fishery rents are completely dissipated. Additionally, solving the commons problem within an EBFM approach can provide substantial welfare gains above those from solving the commons problem in a single-species management framework.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 1.

    3 C.F.R. 227 (2010), reprinted in 33 U.S.C. §857–19 (2015).

  2. 2.

    As a point of clarification Smith (2007b) considered a “quasi-optimized” system rather than a fully optimized system where fishing effort was fixed at a constant level over time to maximize the net present value of the system rather than allowed to vary over time.

  3. 3.

    79 Fed. Reg. 73,705 (11 Dec 2014).

  4. 4.

    For example, in 2008 the Public Broadcasting System (PBS) released a documentary “Crash: a Tale of Two Species” detailing the importance of horseshoe crabs for the survival of the red knot.

  5. 5.

    There were limited existing restrictions in New Jersey, Delaware, and Maryland (see ASMFC 1998b for more detail).

  6. 6.

    The red knot threshold was originally 45,000 birds and was then adjusted to 81,900 birds in 2013 to reflect the change in the method of monitoring the red knot population. See ASMFC (2013b) for the detail on the adjustment.

  7. 7.

    In the ASMFC’s model the fishery manager weighs harvest of female and male horseshoe crabs differently and takes into account the operational sex ratio.

  8. 8.

    Empirical evidence supports assuming well-defined populations. First, tagging and genetic evidence shows limited exchange between the Delaware Bay horseshoe crab population and the Chesapeake Bay horseshoe crab population or the Raritan Bay horseshoe crab population (Swan 2005; Pierce et al. 2000). Second, the red knot population that winters along the Argentinian coast from Tierra del Fuego to Río Negro comprises the majority of red knots that stopover at Delaware Bay (Niles et al. 2008) although there are at least two other smaller populations identified feeding on horseshoe crab eggs in Delaware Bay (Atkinson et al. 2005).

  9. 9.

    We ignore the sex composition and assume 1:1 sex ratios for both populations.

  10. 10.

    Because red knots are believed to breed in their second year (Harrington and Morrison 1980), their recruitment delay is relatively short compared to that of horseshoe crabs. Therefore, to simplify the model we assume instantaneous recruitment for red knots.

  11. 11.

    See also USFWS (2014, pp. 28–33) for a review of the literature on this matter.

  12. 12.

    This is motivated by McGowan et al. (2011b), who modeled the probability of red knots transitioning from light-weight (departure weight, i.e. weight upon departing Delaware Bay, \(< 180 \mathrm{g}\)) to heavy-weight (departure weight \(\ge 180 \mathrm{g}\)) during the stopover as a logistic function of the abundance of spawning female horseshoe crabs.

  13. 13.

    Smith (2007a) found a slightly sigmoidal relationship between the number of eggs disturbed by subsequent spawning and density of spawning female horseshoe crabs through simulation. Sweka et al. (2007) modeled the number of horseshoe crab eggs available to shorebirds as a convex function of the number of spawning females. Both studies modeled after horseshoe crabs in the Delaware Bay area.

  14. 14.

    Several studies have revealed that increased egg density has diminishing returns on red knots’ egg-intake rate. See Niles et al. (2008, pp. 36–39).

  15. 15.

    Our calculation with landings data shows that the inflation-adjusted ex-vessel price of horseshoe crabs was relatively low and stable from the late 1970s through the early 1990s. Although the price has risen considerably since the late 1990s, it exhibited much less variation than landings did in some years when landings fluctuated dramatically. See the online supplementary material.

  16. 16.

    We note that optimal management may conflict with limits to incidental take defined by the Endangered Species Act. As ours is a conceptual analysis conducted for the purposes of exploring the welfare gains from various frameworks for ecosystem-based management, we do not incorporate any constraints imposed by the Endangered Species Act.

  17. 17.

    While the total economic value of horseshoe crabs should also include any non-use values for both horseshoe crabs and red knots, these values have not been estimated in the literature.

  18. 18.

    This valuation function is motivated by Kellner et al. (2011), who modeled the non-fishing value proportional to the square root of fish density.

  19. 19.

    Additionally, we check that applying our valuation function to the lowest population count data on red knot does not yield an infinite marginal value of red knots. In fact, the highest marginal value calculated with observed data was $118.5 (2009 dollars).

  20. 20.

    We implicitly assume that the manager assigns equal weights to the rents from horseshoe crabs and the economic value from red knots. We conduct a sensitivity analysis on our red knot value function later by varying w, which is equivalent to varying the relative weight.

  21. 21.

    16 U.S.C. §1851(a)(1) (2015); see also 50 C.F.R. §600.310 (2015).

  22. 22.

    To determine \(F_\text {MSY }\) in our model, we first solve Eq. 1 for sustainable harvest, which gives \(h = g_c C \exp (- C / K_c^*) - \eta _c C\). Maximizing the preceding equation with respect to C gives the MSY harvest rate, \(h_\text {MSY }\), and the stock level that delivers it, \(C_\text {MSY }\). Then the upper bound on fishery mortality is \(F_\text {MSY } = h_\text {MSY } / C_\text {MSY }\).

  23. 23.

    State-level fisheries are managed with gear-specific permit restrictions and quotas.

  24. 24.

    Rents earned depended on the elasticity of substitution between restricted and unrestricted inputs.

  25. 25.

    MATLAB code is available from the authors upon request.

  26. 26.

    The non-negativity constraint on effort level \(E_t\) binds when \(\lambda _t > p\), at which point effort level is no longer determined by Eq. 16 but constrained to zero. On the other hand, the non-negativity constraints on \(C_t\) and \(R_t\) turn out to be non-binding in our numerical solutions. Proper treatment of these non-negativity constraints is included in the “Appendix”.

  27. 27.

    Full set of necessary optimality conditions is included in the “Appendix”.

  28. 28.

    The technique of historical decomposition in vector autoregression models was pioneered by Sims (1980) and subsequently developed by Burbidge and Harrison (1985).

  29. 29.

    To our best knowledge, the turnpike property for finite-horizon delayed optimal control problems with discount criterion has not yet been formally established in the literature, although we speculate that it is true. We observe that, within the management horizon, the effort level, the harvest rate, and the horseshoe crab and the red knot populations approach certain stationary levels. The levels that are sustained for the majority of the optimization horizon should be very close to the respective turnpikes.

  30. 30.

    Such extremely low population sizes are no surprise and are direct consequences of matching the predictions of our model with real-world trends in populations. For instance, the Delaware 30-foot trawl survey found the lowest horseshoe crab abundance index in 2004, which was only 1.1% of the index in 1990. Refer to Fig. 2a.

  31. 31.

    By “stabilize” or “sustain,” we mean that subsequent changes in the population or harvest level are by less than 1% (except towards the terminal periods).

  32. 32.

    Strictly speaking, effort level has a very small lead in time. We do not imply causality here, however.

  33. 33.

    Our function is calibrated to fit through two data points based on the literature and our assumptions; see the online supplementary material for more detail.

  34. 34.

    This statement holds when the stock size is larger than \(R_\text {m } + 1\), which is always true in our case.

  35. 35.

    We mean the harvest rate maximizing the instantaneous profit. Due to the delay in recruitment of horseshoe crabs and discounting, the sustained harvest rate under ECON-EBFM does not maximize the instantaneous profit.

  36. 36.

    We ignore the \(R < R_\text {m }\) branch of the red knot value function Eq. 7 since the infinite derivative at \(R = R_\text {m }\) would keep the optimal trajectory of \(R_t\) away from \(R_\text {m }\). It immediately renders the nonnegativity constraint on \(R_t\) and the subsequent introduction of the multiplier \(\zeta ^\text {p }_{r,t}\) redundant. Yet we keep \(\zeta ^\text {p }_{r,t}\) in the Hamiltonian for the sake of completeness. Alternatively, we could have set up the optimal control problem with the constraint \(R_t \ge R_\text {m }\), \(t \in [0, T]\), or \(R_t \ge R_\text {m } + \epsilon \), \(t \in [0, T]\), where \(\epsilon \) is a sufficiently small positive number. Additionally, it can be easily verified that the rank condition (Göllmann et al. 2009, Eq. 10) for the nonnegativity constraints Eq. 11 is satisfied.

References

  1. ACCSP (Atlantic Coastal Cooperative Statistics Program) (2016) Non-confidential commercial landings database. http://www.accsp.org/data-warehouse. Cited 21 July 2016

  2. ASMFC (Atlantic States Marine Fisheries Commission) (1998) Interstate fishery management plan for horseshoe crab. Fishery Management Report No. 32, Atlantic States Marine Fisheries Commission, Washington, DC

  3. ASMFC (Atlantic States Marine Fisheries Commission) (2004) 2003 review of the fishery management plan for horseshoe crab (Limulus polyphemus). http://www.asmfc.org/uploads/file/fmpreview2004.pdf

  4. ASMFC (Atlantic States Marine Fisheries Commission) (2009a) A framework for adaptive management of horseshoe crab harvest in the Delaware Bay constrained by red knot conservation. Stock Assessment Report No. 09-02 (Supplement B), Atlantic States Marine Fisheries Commission, Washington, DC

  5. ASMFC (Atlantic States Marine Fisheries Commission) (2009b) Horseshoe crab stock assessment for peer review. Stock Assessment Report No. 09-02 (Supplement A), Atlantic States Marine Fisheries Commission, Washington, DC

  6. ASMFC (Atlantic States Marine Fisheries Commission) (2013a) 2013 horseshoe crab stock assessment update. http://www.asmfc.org/uploads/file//52a88db82013HSC_StockAssessmentUpdate.pdf

  7. ASMFC (Atlantic States Marine Fisheries Commission) (2013b) Horseshoe crab Delaware Bay Ecosystem Technical Committee meeting summary, Arlington, VA, September 24, 2013. http://www.asmfc.org/uploads/file/DBETCMeetingSummary_Sept2013.pdf

  8. ASMFC (Atlantic States Marine Fisheries Commission) (2015) 2015 review of the Atlantic States Marine Fisheries Commission fishery management plan for horseshoe crab (Limulus polyphemus), 2014 fishing year. http://www.asmfc.org/uploads/file/56d76a40hscFMPReview2015.pdf

  9. Atkinson PW, Baker AJ, Bevan RM, Clark NA, Cole KB, Gonzalez PM, Newton J, Niles LJ, Robinson RA (2005) Unravelling the migration and moult strategies of a long-distance migrant using stable isotopes: Red Knot Calidris canutus movements in the Americas. Ibis 147(4):738–749

    Article  Google Scholar 

  10. Baker AJ, González PM, Piersma T, Niles LJ, de Lima Serrano do Nascimento I, Atkinson PW, Clark NA, Minton CDT, Peck MK, Aarts G, (2004) Rapid population decline in red knots: fitness consequences of decreased refuelling rates and late arrival in Delaware Bay. Proc R Soc Lond B Biol Sci 271(1541):875–882

  11. Bertram C, Quaas MF (2017) Biodiversity and optimal multi-species ecosystem management. Environ Resour Econ 67(2):321–350

    Article  Google Scholar 

  12. Botsford LW, Castilla JC, Peterson CH (1997) The management of fisheries and marine ecosystems. Science 277(5325):509–515

    Article  Google Scholar 

  13. Botton ML, Loveland RE (2003) Abundance and dispersal potential of horseshoe crab (Limulus polyphemus) larvae in the Delaware estuary. Estuar Coasts 26(6):1472–1479

    Article  Google Scholar 

  14. Brodziak J, Link J (2002) Ecosystem-based fishery management: What is it and how can we do it? Bull Mar Sci 70(2):589–611

    Google Scholar 

  15. Burbidge J, Harrison A (1985) An historical decomposition of the great depression to determine the role of money. J Monet Econ 16(1):45–54

    Article  Google Scholar 

  16. Clark CW (1990) Mathematical bioeconomics: the optimal management of renewable resources, 2nd edn. Wiley, New York

    Google Scholar 

  17. Deacon RT, Finnoff D, Tschirhart J (2011) Restricted capacity and rent dissipation in a regulated open access fishery. Resour Energy Econ 33(2):366–380

    Article  Google Scholar 

  18. Edwards PET, Parsons GR, Myers KH (2011) The economic value of viewing migratory shorebirds on the Delaware Bay: an application of the single site travel cost model using on-site data. Hum Dimens Wildl 16(6):435–444

    Article  Google Scholar 

  19. Essington TE (2001) The precautionary approach in fisheries management: the devil is in the details. Trends Ecol Evol 16(3):121–122

    Article  Google Scholar 

  20. Eubanks TL Jr., Stoll JR, Kerlinger P (2000) The economic impact of tourism based on the horseshoe crab–shorebird migration in New Jersey. Report to New Jersey Division of Fish and Wildlife, Fermata, Inc

  21. Falk-Petersen J, Armstrong CW (2013) To have one’s cake and eat it too: managing the alien invasive red king crab. Mar Resour Econ 28(1):65–81

    Article  Google Scholar 

  22. Gerrodette T, Dayton PK, Macinko S, Fogarty MJ (2002) Precautionary management of marine fisheries: moving beyond burden of proof. Bull Mar Sci 70(2):657–668

    Google Scholar 

  23. Göllmann L, Kern D, Maurer H (2009) Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optim Control Appl Methods 30(4):341–365

    Article  Google Scholar 

  24. Gurney WSC, Blythe SP, Nisbet RM (1980) Nicholson’s blowflies revisited. Nature 287:17–21

    Article  Google Scholar 

  25. Guttormsen AG, Kristofersson D, Nævdal E (2008) Optimal management of renewable resources with Darwinian selection induced by harvesting. J Environ Econ Manag 56(2):167–179

    Article  Google Scholar 

  26. Hannesson R (1983) Optimal harvesting of ecologically interdependent fish species. J Environ Econ Manag 10(4):329–345

    Article  Google Scholar 

  27. Haramis GM, Link WA, Osenton PC, Carter DB, Weber RG, Clark NA, Teece MA, Mizrahi DS (2007) Stable isotope and pen feeding trial studies confirm the value of horseshoe crab Limulus polyphemus eggs to spring migrant shorebirds in Delaware Bay. J Avian Biol 38(3):367–376

    Article  Google Scholar 

  28. Harrington BA, Morrison RIG (1980) An investigation of wintering areas of red knots (Calidris canutus) and hudsonian godwits (Limosa haemastica) in Argentina. Report to World Wildlife Federation. Washington D.C. and Toronto, ON, Canada

    Google Scholar 

  29. Hilborn R (2004) Ecosystem-based fisheries management: The carrot or the stick? In: Perspectives on ecosystem-based approaches to the management of marine resources. Mar Ecol Prog Ser 274:275–278

  30. Holland D, Schnier KE (2006) Individual habitat quotas for fisheries. J Environ Econ Manag 51(1):72–92

    Article  Google Scholar 

  31. Homans FR, Wilen JE (1997) A model of regulated open access resource use. J Environ Econ Manag 32(1):1–21

    Article  Google Scholar 

  32. Homans FR, Wilen JE (2005) Markets and rent dissipation in regulated open access fisheries. J Environ Econ Manag 49(2):381–404

    Article  Google Scholar 

  33. Jardine SL, Sanchirico JN (2015) Fishermen, markets, and population diversity. J Environ Econ Manag 10:125

    Google Scholar 

  34. Kamien MI, Schwartz NL (1991) Dynamic optimization: the calculus of variations and optimal control in economics and management, 2nd edn. Elsevier, Amsterdam

    Google Scholar 

  35. Kellner JB, Sanchirico JN, Hastings A, Mumby PJ (2011) Optimizing for multiple species and multiple values: tradeoffs inherent in ecosystem-based fisheries management. Conserv Lett 4(1):21–30

    Article  Google Scholar 

  36. Levin PS, Fogarty MJ, Murawski SA, Fluharty D (2009) Integrated ecosystem assessments: developing the scientific basis for ecosystem-based management of the ocean. PLoS Biol 7(1):e1000,014

  37. Link JS (2002) What does ecosystem-based fisheries management mean. Fisheries 27(4):18–21

    Article  Google Scholar 

  38. Lubchenco J, Sutley N (2010) Proposed U.S. policy for ocean, coast, and great lakes stewardship. Science 328(5985):1485–1486

    Article  Google Scholar 

  39. McGowan CP, Hines JE, Nichols JD, Lyons JE, Smith DR, Kalasz KS, Niles LJ, Dey AD, Clark NA, Atkinson PW, Minton CDT, Kendall W (2011a) Demographic consequences of migratory stopover: linking red knot survival to horseshoe crab spawning abundance. Ecosphere 2(6):1–22

    Article  Google Scholar 

  40. McGowan CP, Smith DR, Sweka JA, Martin J, Nichols JD, Wong R, Lyons JE, Niles LJ, Kalasz K, Brust J, Klopfer M, Spear B (2011b) Multispecies modeling for adaptive management of horseshoe crabs and red knots in the Delaware Bay. Nat Resour Model 24(1):117–156

    Article  Google Scholar 

  41. Myers JP, Morrison RIG, Antas PZ, Harrington BA, Lovejoy TE, Sallaberry M, Senner SE, Tarak A (1987) Conservation strategy for migratory species. Am Sci 75:18–26

    Google Scholar 

  42. Myers KH, Parsons GR, Edwards PET (2010) Measuring the recreational use value of migratory shorebirds on the Delaware Bay. Mar Resour Econ 25(3):247–264

    Article  Google Scholar 

  43. Niles LJ, Sitters HP, Dey AD, Atkinson PW, Baker AJ, Bennett KA, Carmona R, Clark KE, Clark NA, Espoz C, González PM, Harrington BA, Hernández DE, Kalasz KS, Lathrop RG, Matus RN, Minton CDT, Morrison RIG, Peck MK, William P, Robinson RA, Serrano IL (2008) Status of the Red Knot (Calidris canutus rufa) in the Western Hemisphere. No. 36 in Studies in Avian Biology, Cooper Ornithological Society

  44. Niles LJ, Bart J, Sitters HP, Dey AD, Clark KE, Atkinson PW, Baker AJ, Bennett KA, Kalasz KS, Clark NA, Clark J, Gillings S, Gates AS, González PM, Hernandez DE, Minton CDT, Morrison RIG, Porter RR, Ross RK, Veitch CR (2009) Effects of horseshoe crab harvest in Delaware Bay on red knots: Are harvest restrictions working? BioScience 59(2):153–164

    Article  Google Scholar 

  45. Pierce JC, Tan G, Gaffney PM (2000) Delaware bay and Chesapeake Bay populations of the horseshoe crab (Limulus polyphemus) are genetically distinct. Estuaries 23(5):690–698

    Article  Google Scholar 

  46. Pikitch EK, Santora C, Babcock EA, Bakun A, Bonfil R, Conover DO, Dayton P, Doukakis P, Fluharty D, Heneman B, Houde ED, Link J, Livingston PA, Mangel M, McAllister MK, Pope J, Sainsbury KJ (2004) Ecosystem-based fishery management. Science 305(5682):346–347

    Article  Google Scholar 

  47. Schaefer MB (1954) Some aspects of the dynamics of populations important to the management of the commercial marine fisheries. Inter Am Trop Tuna Comm Bull 1(2):23–56

    Google Scholar 

  48. Shuster CN Jr, Sekiguchi K (2003) Growing up takes about ten years and eighteen stages. In: Shuster CN Jr, Brockmann HJ, Barlow RB (eds) The American Horseshoe Crab. Harvard University Press, Cambridge, pp 103–132

    Google Scholar 

  49. Sims CA (1980) Macroeconomics and reality. Econometrica 48(1):1–48

    Article  Google Scholar 

  50. Singh R, Weninger Q (2009) Bioeconomies of scope and the discard problem in multiple-species fisheries. J Environ Econ Manag 58(1):72–92

    Article  Google Scholar 

  51. Smith DR (2007a) Effect of horseshoe crab spawning density on nest disturbance and exhumation of eggs: a simulation study. Estuar Coasts 30(2):287–295

    Article  Google Scholar 

  52. Smith DR, Millard MJ, Carmichael RH (2009) Comparative status and assessment of Limulus polyphemus with emphasis on the New England and Delaware Bay populations. In: Tanacredi JT, Botton ML, Smith DR (eds) Biology and Conservation of Horseshoe Crabs. Springer, Berlin, pp 361–386

    Google Scholar 

  53. Smith MD (2007b) Generating value in habitat-dependent fisheries: the importance of fishery management institutions. Land Econ 83(1):59–73

    Article  Google Scholar 

  54. Smith VL (1968) Economics of production from natural resources. Am Econ Rev 58(3):409–431

    Google Scholar 

  55. Swan BL (2005) Migrations of adult horseshoe crabs, Limulus polyphemus, in the Middle Atlantic Bight: a 17-year tagging study. Estuaries 28(1):28–40

    Article  Google Scholar 

  56. Sweka JA, Smith DR, Millard MJ (2007) An age-structured population model for horseshoe crabs in the Delaware Bay area to assess harvest and egg availability for shorebirds. Estuar Coasts 30(2):277–286

    Article  Google Scholar 

  57. USFWS (United States Fish and Wildlife Service) (2014) Rufa red knot ecology and abundance, supplement to Endangered and Threatened Wildlife and Plants; Proposed Threatened status for the rufa red knot (Calidris canutus rufa)

Download references

Acknowledgements

We thank Lee Anderson for his support. We also thank Amanda Dey at the New Jersey Department of Environmental Protection, James Lyons at the USGS Patuxent Wildlife Research Center, Stewart Michels at the Delaware Department of Natural Resources and Environmental Control (DNREC), and Kevin Kalasz formerly at DNREC for help with the data. This research was supported by NOAA Sea Grant #NA14OAR4170087 to S.L.J.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yue Tan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

This research was supported by NOAA Sea Grant #NA14OAR4170087 to S.L.J. The majority of work was completed when Yue Tan attended the Ph.D. program at the Department of Economics at the University of Delaware.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 765 KB)

Appendix

Appendix

In this appendix we derive the first-order necessary optimality conditions for the fishery manager’s delayed optimal control problem under ECON-EBFM defined in Eqs. 812. We assume all conditions required by Theorem 4.2 in Göllmann et al. (2009) are satisfied so that the theorem applies.Footnote 36 We first work with the present-value Hamiltonian and then turn to the current-value Hamiltonian.

Construct the present-value Hamiltonian as

$$\begin{aligned}&\mathcal {H}^\text {p }(t, C_t, C_{t-\tau }, R_t, E_t, \lambda ^\text {p }_t, \xi ^\text {p }_t, \zeta ^\text {p }_{e,t}, \zeta ^\text {p }_{c,t}, \zeta ^\text {p }_{r,t}) \\&\quad = {} e ^ {- \rho t} \left[ p q C_t E_t - \delta E_t^2 + w (R_t - R_\text {m }) ^ \alpha \right] \\&\qquad {} + \lambda ^\text {p }_t \left[ g_c C_{t-\tau } \exp (- C_{t-\tau } / K_c^*) - \eta _c C_t - q C_t E_t \right] \\&\qquad {} + \xi ^\text {p }_t g_r R_t \left\{ 1 - R_t / K_r / a \cdot [1 + \exp (b_0 + b_1 C_t)] \right\} \\&\qquad {} + \zeta ^\text {p }_{e,t} E_t + \zeta ^\text {p }_{c,t} C_t + \zeta ^\text {p }_{r,t} R_t, \end{aligned}$$

where a superscript p indicates association with the present-value Hamiltonian, \(\lambda ^\text {p }_t\) and \(\xi ^\text {p }_t\) are the two costate variables associated respectively with Eqs. 9 and 10, and \(\zeta ^\text {p }_{e,t}\), \(\zeta ^\text {p }_{c,t}\), and \(\zeta ^\text {p }_{r,t}\) are multipliers associated with the nonnegativity constraints Eq. 11.

The first-order necessary optimality conditions are then given by

$$\begin{aligned} 0 = \frac{\partial \mathcal {H}^\text {p }}{\partial E_t} = e ^ {- \rho t} (p q C_t - 2 \delta E_t) - \lambda ^\text {p }_t q C_t + \zeta ^\text {p }_{e,t}, \quad 0 \le t \le T, \end{aligned}$$
(23)
$$\begin{aligned} \begin{aligned} - \dot{\lambda }^\text {p }_t&= \frac{\partial \mathcal {H}^\text {p }}{\partial C_t} + \left. \frac{\partial \mathcal {H}^\text {p }}{\partial C_{t-\tau }} \right| _{t+\tau } = e ^ {- \rho t} p q E_t - \lambda ^\text {p }_t (\eta _c + q E_t) \\&\quad - \xi ^\text {p }_t g_r R_t ^ 2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) + \zeta ^\text {p }_{c,t} \\&\quad + \lambda ^\text {p }_{t+\tau } g_c (1 - C_t / K_c^*) \exp (- C_t / K_c^*), \quad 0 \le t < T - \tau , \end{aligned} \end{aligned}$$
(24)
$$\begin{aligned} \begin{aligned} - \dot{\lambda }^\text {p }_t = \frac{\partial \mathcal {H}^\text {p }}{\partial C_t}&= e ^ {- \rho t} p q E_t - \lambda ^\text {p }_t (\eta _c + q E_t) - \xi ^\text {p }_t g_r R_t ^ 2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) \\&\quad + \zeta ^\text {p }_{c,t}, \quad T - \tau \le t \le T, \end{aligned} \end{aligned}$$
(25)

and

$$\begin{aligned} \begin{aligned} - \dot{\xi }^\text {p }_t = \frac{\partial \mathcal {H}^\text {p }}{\partial R_t}&= e ^ {- \rho t} w \alpha (R_t - R_\text {m }) ^ {\alpha - 1} \\&\quad + \xi ^\text {p }_t g_r \left\{ 1 - 2 R_t / K_r / a \cdot [1 + \exp (b_0 + b_1 C_t)] \right\} \\&\quad + \zeta ^\text {p }_{r,t}, \quad 0 \le t \le T. \end{aligned} \end{aligned}$$
(26)

Also, the optimal solution should maximize the Hamiltonian among all admissible control and state trajectories that satisfy the nonnegativity constraints Eq. 11. The transversality condition is simply

$$\begin{aligned} \lambda ^\text {p }_T = 0. \end{aligned}$$
(27)

Nonnegativity of multipliers and the complementarity condition guarantee

$$\begin{aligned} \zeta ^\text {p }_{e,t}, \zeta ^\text {p }_{c,t}, \zeta ^\text {p }_{r,t} \ge 0 \quad \text {and} \quad \zeta ^\text {p }_{e,t} E_t = \zeta ^\text {p }_{c,t} C_t = \zeta ^\text {p }_{r,t} R_t = 0. \end{aligned}$$
(28)

We now turn to the current-value Hamiltonian, defined simply as \(\mathcal {H} = e ^ {\rho t} \mathcal {H}^\text {p }\). The current-value costate variables and current-value multipliers are defined accordingly as

$$\begin{aligned} \lambda _t = e ^ {\rho t} \lambda ^\text {p }_t, \quad \xi _t = e ^ {\rho t} \xi ^\text {p }_t, \end{aligned}$$
(29)

and

$$\begin{aligned} \zeta _{e,t} = e ^ {\rho t} \zeta ^\text {p }_{e,t}, \quad \zeta _{c,t} = e ^ {\rho t} \zeta ^\text {p }_{c,t}, \quad \zeta _{r,t} = e ^ {\rho t} \zeta ^\text {p }_{r,t}. \end{aligned}$$
(30)

Differentiation with respect to time in Eq. 29 yields

$$\begin{aligned} \dot{\lambda }_t = \rho \lambda _t + e ^ {\rho t} \dot{\lambda }^\text {p }_t \quad \text {and} \quad \dot{\xi }_t = \rho \xi _t + e ^ {\rho t} \dot{\xi }^\text {p }_t. \end{aligned}$$
(31)

Substitute Eqs. 2931 into Eqs. 2328 and then we obtain the conditions in current-value terms. Eqs. 2326 become

$$\begin{aligned}\begin{gathered} E_t = [q C_t (p - \lambda _t) + \zeta _{e,t}] / (2 \delta ), \quad 0 \le t \le T, \\ \begin{aligned} - \dot{\lambda }_t + \rho \lambda _t&= p q E_t - \lambda _t (\eta _c + q E_t) - \xi _t g_r R_t^2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) + \zeta _{c,t} \\*&\quad + e ^ {- \rho \tau } \lambda _{t+\tau } g_c (1 - C_t / K_c^*) \exp (- C_t / K_c^*), \quad 0 \le t < T - \tau , \end{aligned} \\ \begin{aligned} - \dot{\lambda }_t + \rho \lambda _t&= p q E_t - \lambda _t (\eta _c + q E_t) - \xi _t g_r R_t^2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) \\*&\quad + \zeta _{c,t}, \quad T - \tau \le t \le T, \quad \text {and} \end{aligned} \\ \begin{aligned} - \dot{\xi }_t + \rho \xi _t&= w \alpha (R_t - R_\text {m }) ^ {\alpha - 1} \\*&\quad + \xi _t g_r \left\{ 1 - 2 R_t / K_r / a \cdot [1 + \exp (b_0 + b_1 C_t)] \right\} + \zeta _{r,t}, \quad 0 \le t \le T. \end{aligned} \end{gathered}\end{aligned}$$

Equation 27 becomes \(\lambda _T = 0\). Equation 28 becomes

$$\begin{aligned} \zeta _{e,t}, \zeta _{c,t}, \zeta _{r,t} \ge 0 \quad \text {and} \quad \zeta _{e,t} E_t = \zeta _{c,t} C_t = \zeta _{r,t} R_t = 0. \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tan, Y., Jardine, S.L. Considering Economic Efficiency in Ecosystem-Based Management: The Case of Horseshoe Crabs in Delaware Bay. Environ Resource Econ 72, 511–538 (2019). https://doi.org/10.1007/s10640-017-0204-x

Download citation

Keywords

  • Bioeconomics
  • Delayed optimal control
  • Ecosystem-based fisheries management
  • Horseshoe crab (Limulus polyphemus)
  • Non-fishing values
  • Open access
  • Red knot (Calidris canutus rufa)

JEL Classification

  • Q22
  • Q57