Acta Biotheoretica

, Volume 62, Issue 3, pp 339–353 | Cite as

Effect of Small Versus Large Clusters of Fish School on the Yield of a Purse-Seine Small Pelagic Fishery Including a Marine Protected Area

  • Nguyen Trong Hieu
  • Timothée Brochier
  • Nguyen-Huu Tri
  • Pierre Auger
  • Patrice Brehmer
Regular Article

Abstract

We consider a fishery model with two sites: (1) a marine protected area (MPA) where fishing is prohibited and (2) an area where the fish population is harvested. We assume that fish can migrate from MPA to fishing area at a very fast time scale and fish spatial organisation can change from small to large clusters of school at a fast time scale. The growth of the fish population and the catch are assumed to occur at a slow time scale. The complete model is a system of five ordinary differential equations with three time scales. We take advantage of the time scales using aggregation of variables methods to derive a reduced model governing the total fish density and fishing effort at the slow time scale. We analyze this aggregated model and show that under some conditions, there exists an equilibrium corresponding to a sustainable fishery. Our results suggest that in small pelagic fisheries the yield is maximum for a fish population distributed among both small and large clusters of school.

Keywords

Optimal yield Small pelagic fish Fish school Clusters Marine protected area Aggregation of variables  Three level system 

References

  1. Arreguín-Sánchez F (1996) Catchability: a key parameter for fish stock assessment. Rev Fish Biol Fish 6:221–242CrossRefGoogle Scholar
  2. Auger P, Bravo de la Parra R, Poggiale JC, Sanchez E, Nguyen Huu T (2008a) Aggregation of variables and applications to population dynamics. In: Magal P, Ruan S (eds) Structured population models in biology and epidemiology. Lecture Notes in Mathematics, 1936, Mathematical Biosciences Subseries. Springer, Berlin, pp 209–263Google Scholar
  3. Auger P, Bravo De La Parra R, Poggiale JC, Sanchez E, Sanz L (2008b) Aggregation methods in dynamical systems variables and applications in population and community dynamics. Phys Life Rev 5:79–105CrossRefGoogle Scholar
  4. Auger P, Poggiale J-C, Sanchez E (2012) A review on spatial aggregation methods involving several time scales. Ecol Complex 10:12–25 P.CrossRefGoogle Scholar
  5. Bazykin AD (1998) Nonlinear dynamics of interacting populations. World Scientific series on nonlinear science. World Scientific, SingaporeGoogle Scholar
  6. Brehmer P, Do TC, Laugier T, Galgani F, Laloë F, Darnaude AM, Fiandrino A, Caballero PI, Mouillot D (2011) Field investigations and multi-indicators for management and conservation of shallow water lagoons: practices and perspectives. Aquat Conserv Mar Freshw Ecosyst 21(7):728–742CrossRefGoogle Scholar
  7. Brehmer P, Georgakarakos S, Josse E, Trygonis V, Dalen J (2008) Adaptation of fisheries sonar for monitoring large pelagic fish school: dependence of schooling behaviour on fish finding efficiency. Aquat Living Resour 20:377–384CrossRefGoogle Scholar
  8. Brehmer P, Gerlotto F, Laurent C, Cotel P, Achury A, Samb B (2007) Schooling behaviour of small pelagic fish: phenotypic expression of independent stimuli. Mar Ecol Prog Ser 334:263–272CrossRefGoogle Scholar
  9. Brehmer P, Lafont T, Georgakarakos S, Josse E, Gerlotto F, Collet C (2006) Omnidirectional multibeam sonar monitoring: applications in fisheries science. Fish Fish 7(3):165–179CrossRefGoogle Scholar
  10. Brochier T, Echevin V, Tam J, Chaigneau A, Goubanova K, Bertrand A (2013) Climate change scenarios experiments predict a future reduction in small pelagic fish recruitment in the humboldt current system. Glob Change Biol 19:1841–1853CrossRefGoogle Scholar
  11. Brochier T, Ecoutin JM, de Morais LT, Kaplan DM, Lae R (2013) A multi-agent ecosystem model for studying changes in a tropical estuarine fish assemblage within a marine protected area. Aquat Living Resour 26:147–158CrossRefGoogle Scholar
  12. Clark CW (1990) Mathematical bioeconomics: the optimal management of renewable resources, 2nd edn. Wiley, New YorkGoogle Scholar
  13. Dao DK, Auger P, Nguyen-Huu T (2008) Predator density dependent prey dispersal in a patchy environment with a refuge for the prey. S Afr J Sci 104(5–6):180–184Google Scholar
  14. de Lara M, Doyen L (2008) Sustainable management of renewable resources: mathematical models and methods. Springer, BerlinGoogle Scholar
  15. Fréon P, Werner F, Chavez FP (2009) Conjectures on future climate effects on marine ecosystems dominated by small pelagic fish. Climate change and small pelagic fish. Cambridge University Press, Cambridge, pp 312–343Google Scholar
  16. Fulton EA, Link JS, Kaplan IC, Savina-Rolland M, Johnson P, Ainsworth C, Horne P, Gorton R, Gamble RJ, Smith ADM, Smith DC (2011) Lessons in modelling and management of marine ecosystems: the atlantis experience. Fish Fish 12:171–188CrossRefGoogle Scholar
  17. Gonzalez EO, Ramos RJ (2003) Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol Model 166(1–2):135–146CrossRefGoogle Scholar
  18. Iwasa Y, Andreasen V, Levin SA (1987) Aggregation in model ecosystems. I. Perfect aggregation. Ecol Model 37:287–302CrossRefGoogle Scholar
  19. Iwasa Y, Levin SA, Andreasen V (1989) Aggregation in model ecosystems. II. Approximate aggregation. IMA J Math Appl Med Biol 6:1–23CrossRefGoogle Scholar
  20. Krivan V (2011) On the gause predator–prey model with a refuge: a fresh look at the history. J Theor Biol 274(1):67–73CrossRefGoogle Scholar
  21. Leah EK (2005) Mathematical models in biology. SIAMs classics in applied mathematics, vol 46. Random House, New YorkGoogle Scholar
  22. MacLennan DN, Simmonds EJ (2005) Fisheries acoustics: theory and practice, 2nd edn. Blackwell, LondonGoogle Scholar
  23. Maury O (2010) An overview of apecosm, a spatialized mass balanced “apex predators ecosystem model” to study physiologically structured tuna population dynamics in their ecosystem. Progr Oceanogr 84:113–117CrossRefGoogle Scholar
  24. Mchich R, Charouki N, Auger P, Raissi N, Ettahihi O (2006) Optimal spatial distribution of the fishing effort in a multi fishing zone model. Ecol Model 197(3/4):274–280CrossRefGoogle Scholar
  25. Nguyen ND, Nguyen-Huu T, Auger P (2012) Effects of refuges and density dependent dispersal on interspecific competition dynamics. Int J Bifurc Chaos 22(2):1–10Google Scholar
  26. Petitgas P, Levenez JJ (1996) Spatial organization of pelagic fish: echogram structure, spatio-temporal condition, and biomass in senegalese waters. ICES J Mar Sci 53:147–153CrossRefGoogle Scholar
  27. Pinsky ML, Jensen OP, Ricard D, Palumbi SR (2011) Unexpected patterns of fisheries collapse in the worlds oceans. Proc Natl Acad Sci USA 108:8317–8322CrossRefGoogle Scholar
  28. Schaefer MB (1957) Some considerations of population dynamics and economics in relation to the management of the commercial marine fisheries. J Fish Res Board Can 14:669–681CrossRefGoogle Scholar
  29. Smith VL (1968) Economics of production from natural resources. Am Econ Rev 58(3):409–431Google Scholar
  30. Smith VL (1969) On models of commercial fishing. J Polit Econ 77(2):181–198CrossRefGoogle Scholar
  31. Tacon AGJ (2004) Use of fish meal and fish oil in aquaculture: a global perspective. Aquat Resour Cult Dev 1:3–14Google Scholar
  32. Yemane D, Shin Y-J, Field JG (2009) Exploring the effect of marine protected areas on the dynamics of fish communities in the southern Benguela: an individual-based modelling approach. ICES J Mar Sci 66:378–387CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Nguyen Trong Hieu
    • 1
    • 2
    • 3
  • Timothée Brochier
    • 4
    • 5
  • Nguyen-Huu Tri
    • 1
    • 6
  • Pierre Auger
    • 1
    • 7
  • Patrice Brehmer
    • 4
    • 5
  1. 1.IRD UMI 209 UMMISCOBondy CedexFrance
  2. 2.École doctorale Pierre Louis de santé publiqueUniversité Pierre et Marie CurieParisFrance
  3. 3.Faculty of Mathematics, Informatics and MechanicsVietnam National UniversityHanoiVietnam
  4. 4.IRD UMR195 LemarDakarSenegal
  5. 5.ISRA, CRODTPole de recherche de HannDakarSenegal
  6. 6.IXXIENS LyonFrance
  7. 7.University Cheikh-Anta-DiopDakarSenegal

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