Bulletin of Mathematical Biology

, Volume 76, Issue 9, pp 2122–2143 | Cite as

Modelling the Impact of Marine Reserves on a Population with Depensatory Dynamics

  • Matthew H. Chan
  • Peter S. Kim
Original Article


In this study, we use a spatially implicit, stage-structured model to evaluate marine reserve effectiveness for a fish population exhibiting depensatory (strong Allee) effects in its dynamics. We examine the stability and sensitivity of the equilibria of the modelled system with regards to key system parameters and find that for a reasonable set of parameters, populations can be protected from a collapse if a small percentage of the total area is set aside in reserves. Furthermore, the overall abundance of the population is predicted to achieve a maximum at a certain ratio \(A\) of reserve area to fished area, which depends heavily on the other system parameters such as the net export rate of fish from the marine reserves to the fished areas. This finding runs contrary to the contested “equivalence at best” result when comparing fishery management through traditional catch or effort control and management through marine reserves. Lastly, we analyse the problem from a bioeconomics perspective by computing the optimal harvesting policy using Pontryagin’s Maximum Principle, which suggests that the value for \(A\) which maximizes the optimal equilibrium fishery yield also maximizes population abundance when the cost per unit harvest is constant, but can increase substantially when the cost per unit harvest increases with the area being harvested.


Population dynamics Stage structure Depensation  Strong Allee effect Marine reserve 



The work of MHC was supported by the Australian Postgraduate Award. PSK was supported by the ARC Discovery Early Career Research Award (DE120101113).


  1. Abesamis RA, Russ GR (2005) Density-dependent spillover from a marine reserve: long-term evidence. Ecol Appl 15(5):1798–1812CrossRefGoogle Scholar
  2. Ban NC, Klein CJ (2009) Spatial socioeconomic data as a cost in systematic marine conservation planning. Conserv Lett 2(5):206–215CrossRefGoogle Scholar
  3. Botsford LW, Kaplan DM, Hastings A (2004) Sustainability and yield in marine reserve policy. Am Fish Soc Symp 42:75–86Google Scholar
  4. Botsford LW, Micheli F, Hastings A (2003) Principles for the design of marine reserves. Ecol Appl 13:25–31CrossRefGoogle Scholar
  5. Bundy A, Fanning LP (2005) Can Atlantic cod (Gadus morhua) recover? Exploring trophic explanations for the non-recovery of the cod stock on the eastern Scotian Shelf, Canada. Can J Fish Aquat Sci 62(7):1474–1489CrossRefGoogle Scholar
  6. Clark CW (1976) Pure and applied mathematics., Mathematical bioeconomics: the optimal management of renewable resourcesWiley-Interscience, New YorkGoogle Scholar
  7. Costello C, Ovando D, Hilborn R, Gaines SD, Deschenes O, Lester SE (2012) Status and solutions for the worlds unassessed fisheries. Science 338(6106):517–520CrossRefGoogle Scholar
  8. Côté IM, Mosqueira I, Reynolds JD (2001) Effects of marine reserve characteristics on the protection of fish populations: a meta-analysis. J Fish Biol 59:178–189CrossRefGoogle Scholar
  9. Courchamp F, Clutton-Brock T, Grenfell BT (1999) Inverse density dependence and the Allee effect. Trends Ecol Evol 14(10):405–410CrossRefGoogle Scholar
  10. Dubey B, Chandra P, Sinha P (2003) A model for fishery resource with reserve area. Nonlinear Anal Real 4(4):625–637MathSciNetCrossRefMATHGoogle Scholar
  11. Frank KT, Brickman D (2000) Allee effects and compensatory population dynamics within a stock complex. Can J Fish Aquat Sci 57(3):513–517CrossRefGoogle Scholar
  12. Gaylord B, Gaines SD, Siegel DA, Carr MH (2005) Marine reserves exploit population structure and life history in potentially improving fisheries yields. Ecol Appl 15(6):2180–2191CrossRefGoogle Scholar
  13. Gell FR, Roberts CM (2003) Benefits beyond boundaries: the fishery effects of marine reserves. Trends Ecol Evol 18(9):448–455CrossRefGoogle Scholar
  14. Gerber LR, Andelman SJ, Botsford LW, Gaines SD, Hastings A, Palumbi SR, Possingham HP (2003) Population models for marine reserve design: a retrospective and prospective synthesis. Ecol Appl 13:47–S64CrossRefGoogle Scholar
  15. Grafton QR, Kompas T, Pham VH (2006) The economic payoffs from marine reserves: resource rents in a stochastic environment. Econ Rec 82:469–480CrossRefGoogle Scholar
  16. Grafton RQ, Kompas T, Lindenmayer D (2005) Marine reserves with ecological uncertainty. Bull Math Biol 67(5):957–971MathSciNetCrossRefGoogle Scholar
  17. Grüss A, Kaplan D, Guenette S, Roberts CM, Botsford LW (2011) Consequences of adult and juvenile movement for marine protected areas. Biol Cons 144:692–702CrossRefGoogle Scholar
  18. Halpern B, Warner R (2002) Marine reserves have rapid and lasting effects. Ecol Lett 5:361–366CrossRefGoogle Scholar
  19. Halpern BS (2003) The impact of marine reserves: do reserves work and does reserve size matter? Ecol Appl 13:117–137CrossRefGoogle Scholar
  20. Hannesson R (1998) Marine reserves: what would they accomplish? Mar Resour Econ 13:159–170Google Scholar
  21. Hannesson R (2008) The economics of marine reserves. Nat Resour Model 15:273–290CrossRefGoogle Scholar
  22. Harrison HB, Williamson DH, Evans RD, Almany GR, Thorrold SR, Russ GR, Feldheim KA, van Herwerden L, Planes S, Srinivasan M, Berumen ML, Jones GP (2012) Larval export from marine reserves and the recruitment benefit for fish and fisheries. Curr Biol 22(11):1023–1028CrossRefGoogle Scholar
  23. Hart, DR, Sissenwine MP (2009) Marine reserve effects on fishery profits: a comment on white et al. (2008) Ecol Lett 12(3):E9–E11Google Scholar
  24. Hastings A, Botsford LW (1999) Equivalence in yield from marine reserves and traditional fisheries management. Science 284(5419):1537–1538CrossRefGoogle Scholar
  25. Holland DS, Brazee RJ (1996) Marine reserves for fisheries management. Mar Resour Econ 11:157–172Google Scholar
  26. Horwood JW, Nichols JH, Milligan S (1998) Evaluation of closed areas for fish stock conservation. J Appl Ecol 35(6):893–903CrossRefGoogle Scholar
  27. Huo HF, Jiang HM, Meng XY (2012) A dynamic model for fishery resource with reserve area and taxation. J Appl Math. doi: 10.1155/2012/794719
  28. Hutchings JA (2013) Renaissance of a caveat: Allee effects in marine fish. ICES J Mar Sci. doi: 10.1093/icesjms/fst179
  29. Hutchings JA, Myers RA (1994) What can be learned from the collapse of a renewable resource? Atlantic cod, Gadus morhua, of Newfoundland and Labrador. Can J Fish Aquat Sci 51(9):2126–2146CrossRefGoogle Scholar
  30. Lauck T, Clark CW, Mangel M, Munro GR (1998) Implementing the precautionary principle in fisheries management through marine reserves. Ecol Appl 8:S72–S78CrossRefGoogle Scholar
  31. Liermann M, Hilborn R (2001) Depensation: evidence, models and implications. Fish Fish 2(1):33–58CrossRefGoogle Scholar
  32. Myers RA, Barrowman NJ, Hutchings JA, Rosenberg A (1995) Population dynamics of exploited fish stocks at low population levels. Science 269(5227):1106–1108CrossRefGoogle Scholar
  33. Naidoo R, Balmford A, Ferraro PJ, Polasky S, Ricketts TH, Rouget M (2006) Integrating economic costs into conservation planning. Trends Ecol Evol 21(12):681–687CrossRefGoogle Scholar
  34. Nehlsen W, Williams JE, Lichatowich JA (1991) Pacific salmon at the crossroads: stocks at risk from California, Oregon, Idaho, and Washington. Fisheries 16(2):4–21CrossRefGoogle Scholar
  35. Neubert MG (2003) Marine reserves and optimal harvesting. Ecol Lett 6(9):843–849CrossRefGoogle Scholar
  36. Parrish R (1999) Marine reserves for fisheries management: why not? Calif Coop Ocean Fish 40:77–86Google Scholar
  37. Price MHH, Darimont CT, Temple NF, MacDuffee SM (2008) Ghost runs: management and status assessment of Pacific salmon (Oncorhynchus spp.) returning to British Columbia’s central and north coasts. Can J Fish Aquat Sci 65(12):2712–2718CrossRefGoogle Scholar
  38. Ralston S, OFarrell MR (2008) Spatial variation in fishing intensity and its effect on yield. Can J Fish Aquat Sci 65(4):588–599CrossRefGoogle Scholar
  39. Ricard D, Minto C, Jensen OP, Baum JK (2012) Examining the knowledge base and status of commercially exploited marine species with the ram legacy stock assessment database. Fish Fish 13(4):380–398CrossRefGoogle Scholar
  40. Roberts CM, Hawkins JP, Gell FR (2005) The role of marine reserves in achieving sustainable fisheries. Phil Trans R Soc B 360(1453):123–132CrossRefGoogle Scholar
  41. Rowe S, Hutchings JA, Bekkevold D, Rakitin A (2004) Depensation, probability of fertilization, and the mating system of atlantic cod (Gadus morhua L.). ICES J Mar Sci 61(7):1144–1150CrossRefGoogle Scholar
  42. Russ GR (2002) Yet another review of marine reserve as reef fishery management tools. In: Sale PF (ed) Coral reef fishes: dynamics and diversity in a complex ecosystem. Academic Press, San DiegoGoogle Scholar
  43. Russ GR, Alcala AC (2011) Enhanced biodiversity beyond marine reserve boundaries: the cup spillith over. Ecol Appl 21:241–250Google Scholar
  44. Shelton PA, Healey BP (1999) Should depensation be dismissed as a possible explanation for the lack of recovery of the northern cod (Gadus morhua) stock? Can J Fish Aquat Sci 56(9):1521–1524CrossRefGoogle Scholar
  45. Sinclair AF (2001) Natural mortality of cod (Gadus morhua) in the Southern Gulf of st Lawrence. ICES J Mar Sci J du Conseil 58(1):1–10CrossRefGoogle Scholar
  46. Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the allee effect? Oikos 87(1):185–190CrossRefGoogle Scholar
  47. Stoner AW (1997) The status of queen conch, strombus gigas, research in the caribbean. Mar Fish Rev 59(3):14–33Google Scholar
  48. Stoner AW, Davis MH, Booker CJ (2012) Negative consequences of allee effect are compounded by fishing pressure: comparison of queen conch reproduction in fishing grounds and a marine protected area. Bull Mar Sci 88:89–104CrossRefGoogle Scholar
  49. Stoner AW, Lipcius RN, Marshall LS, Bardales AT (1988) Synchronous emergence and mass migration in juvenile queen conch. Mar Ecol Prog Ser 49(1–2):51–55CrossRefGoogle Scholar
  50. Sumaila UR (1998) Protected marine reserves as fisheries management tools: a bioeconomic analysis. CMI working papers. Chr. Michelsen InstituteGoogle Scholar
  51. Taylor CM, Hastings A (2005) Allee effects in biological invasions. Ecol Lett 8(8):895–908CrossRefGoogle Scholar
  52. Tetreault I, Ambrose RF (2007) Temperate marine reserves enhance targeted but not untargeted fishes in multiple no-take mpas. Ecol Appl 17(8):2251–2267CrossRefGoogle Scholar
  53. Walters CJ, Kitchell JF (2001) Cultivation/depensation effects on juvenile survival and recruitment: implications for the theory of fishing. Can J Fish Aquat Sci 58(1):39–50CrossRefGoogle Scholar
  54. White C, Kendall BE (2007) A reassessment of equivalence in yield from marine reserves and traditional fisheries management. Oikos 116(12):2039–2043CrossRefGoogle Scholar
  55. White C, Kendall BE, Gaines SD, Siegel DA, Costello CJ (2008) Marine reserve effects on fishery profit. Ecol Lett 11:370–379CrossRefGoogle Scholar
  56. Wicklund RI, Hepp LA, Wenz GA (1991) Preliminary studies on the early life history of the queen conch (strombus gigas) in the exuma cays, bahamas. In: Proceedings of the Gulf and Caribbean Fisheries Institute, vol 40. Gulf and Caribbean Fisheries Institute, pp 283–298Google Scholar
  57. Willis TJ, Millar RB, Babcock RC, Tolimieri N (2003) Burdens of evidence and the benefits of marine reserves: putting descartes before des horse? Environ Conserv 30(2):97–103CrossRefGoogle Scholar
  58. Worm B, Hilborn R, Baum JK, Branch TA, Collie JS, Costello C, Fogarty MJ, Fulton EA, Hutchings JA, Jennings S, Jensen OP, Lotze HK, Mace PM, McClanahan TR, Minto C, Palumbi SR, Parma AM, Ricard D, Rosenberg AA, Watson R, Zeller D (2009) Rebuilding global fisheries. Science 325(5940):578–585CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

Personalised recommendations