2011, 8(3): 769-783. doi: 10.3934/mbe.2011.8.769

Optimal number of sites in multi-site fisheries with fish stock dependent migrations

1. 

Université Aboubekr Belkaid, Faculté Des Sciences, Département de Mathématiques, 13000, Tlemcen,, Algeria

2. 

IRD, UMI 209, UMMISCO, IRD France Nord, F-93143, Bondy, France

3. 

UMI IRD 209 UMMISCO, Centre de Recherche Halieutique Méditerranéenne et Tropicale, Avenue Jean Monnet, BP 171, 34203 Séte Cedex, France

Received  July 2010 Revised  March 2011 Published  June 2011

We present a stock-effort dynamical model of a fishery subdivided into fishing zones. The stock corresponds to a fish population moving between these zones, on which they are harvested by fishing fleets. We consider a linear chain of identical fishing zones. Fish movements between the zones, as well as vessels displacements, are assumed to take place at a faster time scale than the variation of the stock and the change of the fleet size. The vessels movements between the fishing areas are assumed to be stock dependent, i.e. the larger the stock density is in a zone the more vessels tends to remain in it. We take advantage of these two time scales to derive a reduced model governing the dynamics of the total harvested stock and the total fishing effort. Under some assumption, we obtain either a stable equilibrium or a stable limit cycle which involves large cyclic variations of the total fish stock and fishing effort. We show that there exists an optimal number of fishing zones that maximizes the total catch at equilibrium. We discuss the results in relation to fish aggregating devices (FADs) fisheries.
Citation: Ali Moussaoui, Pierre Auger, Christophe Lett. Optimal number of sites in multi-site fisheries with fish stock dependent migrations. Mathematical Biosciences & Engineering, 2011, 8 (3) : 769-783. doi: 10.3934/mbe.2011.8.769
References:
[1]

D. K. Arrowsmith and C. M. Place, "Dynamical Systems,", in, (1992).   Google Scholar

[2]

P. Auger, C. Lett, A. Moussaoui and S. Pioch, Optimal number of sites in artificial pelagic multi-site fisheries,, Canadian Journal of Fisheries and Aquatic Sciences, 67 (2010), 296.  doi: 10.1139/F09-188.  Google Scholar

[3]

P. Auger and J.-C. Poggiale, Emergence of population growth models: Fast migration and slow growth,, J. Theor. Biol., 182 (1996), 99.  doi: 10.1006/jtbi.1996.0145.  Google Scholar

[4]

P. Auger and J.-C. Poggiale, Aggregation and emergence in systems of ordinary differential equations,, Math. Comput. Model., 27 (1998), 1.  doi: 10.1016/S0895-7177(98)00002-8.  Google Scholar

[5]

P. Auger and R. Roussarie, Complex ecological models with simple dynamics: From individuals to population,, Acta Biotheor., 42 (1994), 111.  doi: 10.1007/BF00709485.  Google Scholar

[6]

P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics,, in, 1936 (2008), 209.   Google Scholar

[7]

P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and L. Sanz, Aggregation methods in dynamical systems variables and applications in population and community dynamics,, Physics of Life Reviews, 5 (2008), 79.  doi: 10.1016/j.plrev.2008.02.001.  Google Scholar

[8]

H. Belvéze, "Biologie et Dynamique des Populations de Sardine (Sardina Pilchardus Walbaum) Peuplant les CU tes Atlantiques Marocaines et Propositions pour un Aménagement des PIcheries,", Ph.D thesis, (1984).   Google Scholar

[9]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, 35 (1981).   Google Scholar

[10]

D. K. Dao, P. Auger and H. T. Nguyen, Predator density dependent prey dispersal in a patchy environment with a refuge for the prey,, South African Journal of Science, 104 (2008), 180.   Google Scholar

[11]

T. Dempster and M. Taquet, Fish aggregation device (FAD) research: Gaps in current knowledge and future directions for ecological studies,, Reviews in Fish Biology and Fisheries, 14 (2004), 21.  doi: 10.1007/s11160-004-3151-x.  Google Scholar

[12]

A. El Abdllaoui, P. Auger, R. Bravo de la Parra, B. Kooi and R. Mchich, Effects of density-dependent migrations on stability of a two-patch predator-prey model,, Mathematical Biosciences, 210 (2007), 335.  doi: 10.1016/j.mbs.2007.03.002.  Google Scholar

[13]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, Indiana University Mathematical Journal, 21 (): 193.   Google Scholar

[14]

A. Fonteneau, J. Ariz, D. Gaertner, T. Nordstrom and P. Pallares, Observed changes in the species composition of tuna schools in the Gulf of Guinea between 1981 and 1999, in relation with the Fish Aggregrating Device fishery,, Aquatic Living Resources, 13 (2000), 253.  doi: 10.1016/S0990-7440(00)01054-8.  Google Scholar

[15]

C. Girard, S. Benhamou and S. L. Dagorn, FAD: Fish Aggregating Device or Fish Attracting Device? A new analysis of yellowfin tuna movements around floating objects,, Animal Behaviour, 67 (2004), 319.  doi: 10.1016/j.anbehav.2003.07.007.  Google Scholar

[16]

R. Hilborn and P. Medley, Tuna purse-seine fishing with Fish-Aggregating Devices (FAD)- Models of tuna FAD interactions,, Canadian Journal of Fisheries and Aquatic Sciences, 46 (1989), 28.  doi: 10.1139/f89-004.  Google Scholar

[17]

R. Hilborn, F. Micheli and G. A. De Leo, Integrating marine protected areas with catch regulation,, Canadian Journal of Fisheries and Aquatic Sciences, 63 (2006), 642.  doi: 10.1139/f05-243.  Google Scholar

[18]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds,, Bull. Am. Math. Soc., 76 (1970), 1015.  doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[19]

Y. Iwasa, V. Andreasen and S. A. Levin, Aggregation in model ecosystems I. Perfect aggregation,, Ecological Modelling, 37 (1987), 287.  doi: 10.1016/0304-3800(87)90030-5.  Google Scholar

[20]

Y. Iwasa, S. A. Levin and V. Andreasen, Aggregation in model ecosystems. II. Approximate aggregation,, IMA Journal of Mathematics Applied in Medicine and Biology, 6 (1989), 1.  doi: 10.1093/imammb/6.1.1-a.  Google Scholar

[21]

H. Kakimoto, Artificial fishing reef studies and effects,, Japanese Institute of Technology on Fishing Ports, II (2004), 150.   Google Scholar

[22]

C. H. Lan and C. Y. Hsui, The deployment of artificial reef ecosystem: Modelling, simulation and application,, Simulation Modelling Practice and Theory, 14 (2006), 673.  doi: 10.1016/j.simpat.2005.10.011.  Google Scholar

[23]

R. Mchich, P. Auger and J.-C. Poggiale, Effect of predator density dependent dispersal of prey on stability of a predator-prey system,, Mathematical Biosciences, 206 (2007), 343.  doi: 10.1016/j.mbs.2005.11.005.  Google Scholar

[24]

R. Mchich, P. Auger, R. Bravo de la Parra and N. Raïssi, Dynamics of a fishery on two fishing zones with fish stock dependent migrations: Aggregation and control,, Ecol. Model., 158 (2002), 51.  doi: 10.1016/S0304-3800(02)00237-5.  Google Scholar

[25]

J. Michalski, J.-C. Poggiale, R. Arditi and P. Auger, Macroscopic dynamic effects of migrations in patchy predator-prey systems,, J. Theor. Biol., 185 (1997), 459.  doi: 10.1006/jtbi.1996.0327.  Google Scholar

[26]

G. Moreno, L. Dagorn, G. Sancho and D. Itano, Fish behaviour from fishers' knowledge: The case study of tropical tuna around Drifting Fish Aggregating Devices (DFADs),, Canadian Journal of Fisheries and Aquatic Sciences, 64 (2007), 1517.  doi: 10.1139/f07-113.  Google Scholar

[27]

A. Moussaoui, Effect of a toxicant on the dynamics of a spatial fishery,, African Diaspora Journal of Mathematics, 10 (2010), 122.   Google Scholar

[28]

P. A. Nelson, Marine fish assemblages associated with Fish Aggregating Devices (FADs): Effects of fish removal, FAD size, fouling communities, and prior recruits,, Fishery Bulletin, 101 (2003), 835.   Google Scholar

[29]

S. Levin and S. Pacala, Theories of simplification and scaling of spatially distributed processes,, in, (1997), 204.   Google Scholar

[30]

S. Pioch, "Les 'Habitats Artificiels': Élément de Stratégie pour une Gestion Intégrée des Zones Côtières? Essai de Méthodologie d'Aménagement en 325 Ré cifs Artificiels Adapté à la Pêche Artisanale Côtière,", Ph.D. thesis, (2008).   Google Scholar

[31]

J.-C. Poggiale, "Applications des Variétés Invariantes a la Modélisation de l'Hétèrogén éité en Dynamique des Populations,", Ph.D. thesis, (1994).   Google Scholar

[32]

J.-C. Poggiale and P. Auger, Impact of spatial heterogeneity on a predator-prey system dynamics,, Comptes Rendus Biologies, 327 (2004), 1058.  doi: 10.1016/j.crvi.2004.06.006.  Google Scholar

[33]

R. A. Rountree, Association of fishes with fish aggregating devices: Effects of structure size on fish abundance,, Bulletin of Marine Science, 44 (1989), 960.   Google Scholar

[34]

K. Sakamoto, Invariant manifolds in singular perturbations problems for ordinary differential equations,, Proc. Roy. Soc. Ed. Sect. A, 116 (1990), 45.   Google Scholar

show all references

References:
[1]

D. K. Arrowsmith and C. M. Place, "Dynamical Systems,", in, (1992).   Google Scholar

[2]

P. Auger, C. Lett, A. Moussaoui and S. Pioch, Optimal number of sites in artificial pelagic multi-site fisheries,, Canadian Journal of Fisheries and Aquatic Sciences, 67 (2010), 296.  doi: 10.1139/F09-188.  Google Scholar

[3]

P. Auger and J.-C. Poggiale, Emergence of population growth models: Fast migration and slow growth,, J. Theor. Biol., 182 (1996), 99.  doi: 10.1006/jtbi.1996.0145.  Google Scholar

[4]

P. Auger and J.-C. Poggiale, Aggregation and emergence in systems of ordinary differential equations,, Math. Comput. Model., 27 (1998), 1.  doi: 10.1016/S0895-7177(98)00002-8.  Google Scholar

[5]

P. Auger and R. Roussarie, Complex ecological models with simple dynamics: From individuals to population,, Acta Biotheor., 42 (1994), 111.  doi: 10.1007/BF00709485.  Google Scholar

[6]

P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics,, in, 1936 (2008), 209.   Google Scholar

[7]

P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and L. Sanz, Aggregation methods in dynamical systems variables and applications in population and community dynamics,, Physics of Life Reviews, 5 (2008), 79.  doi: 10.1016/j.plrev.2008.02.001.  Google Scholar

[8]

H. Belvéze, "Biologie et Dynamique des Populations de Sardine (Sardina Pilchardus Walbaum) Peuplant les CU tes Atlantiques Marocaines et Propositions pour un Aménagement des PIcheries,", Ph.D thesis, (1984).   Google Scholar

[9]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, 35 (1981).   Google Scholar

[10]

D. K. Dao, P. Auger and H. T. Nguyen, Predator density dependent prey dispersal in a patchy environment with a refuge for the prey,, South African Journal of Science, 104 (2008), 180.   Google Scholar

[11]

T. Dempster and M. Taquet, Fish aggregation device (FAD) research: Gaps in current knowledge and future directions for ecological studies,, Reviews in Fish Biology and Fisheries, 14 (2004), 21.  doi: 10.1007/s11160-004-3151-x.  Google Scholar

[12]

A. El Abdllaoui, P. Auger, R. Bravo de la Parra, B. Kooi and R. Mchich, Effects of density-dependent migrations on stability of a two-patch predator-prey model,, Mathematical Biosciences, 210 (2007), 335.  doi: 10.1016/j.mbs.2007.03.002.  Google Scholar

[13]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, Indiana University Mathematical Journal, 21 (): 193.   Google Scholar

[14]

A. Fonteneau, J. Ariz, D. Gaertner, T. Nordstrom and P. Pallares, Observed changes in the species composition of tuna schools in the Gulf of Guinea between 1981 and 1999, in relation with the Fish Aggregrating Device fishery,, Aquatic Living Resources, 13 (2000), 253.  doi: 10.1016/S0990-7440(00)01054-8.  Google Scholar

[15]

C. Girard, S. Benhamou and S. L. Dagorn, FAD: Fish Aggregating Device or Fish Attracting Device? A new analysis of yellowfin tuna movements around floating objects,, Animal Behaviour, 67 (2004), 319.  doi: 10.1016/j.anbehav.2003.07.007.  Google Scholar

[16]

R. Hilborn and P. Medley, Tuna purse-seine fishing with Fish-Aggregating Devices (FAD)- Models of tuna FAD interactions,, Canadian Journal of Fisheries and Aquatic Sciences, 46 (1989), 28.  doi: 10.1139/f89-004.  Google Scholar

[17]

R. Hilborn, F. Micheli and G. A. De Leo, Integrating marine protected areas with catch regulation,, Canadian Journal of Fisheries and Aquatic Sciences, 63 (2006), 642.  doi: 10.1139/f05-243.  Google Scholar

[18]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds,, Bull. Am. Math. Soc., 76 (1970), 1015.  doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[19]

Y. Iwasa, V. Andreasen and S. A. Levin, Aggregation in model ecosystems I. Perfect aggregation,, Ecological Modelling, 37 (1987), 287.  doi: 10.1016/0304-3800(87)90030-5.  Google Scholar

[20]

Y. Iwasa, S. A. Levin and V. Andreasen, Aggregation in model ecosystems. II. Approximate aggregation,, IMA Journal of Mathematics Applied in Medicine and Biology, 6 (1989), 1.  doi: 10.1093/imammb/6.1.1-a.  Google Scholar

[21]

H. Kakimoto, Artificial fishing reef studies and effects,, Japanese Institute of Technology on Fishing Ports, II (2004), 150.   Google Scholar

[22]

C. H. Lan and C. Y. Hsui, The deployment of artificial reef ecosystem: Modelling, simulation and application,, Simulation Modelling Practice and Theory, 14 (2006), 673.  doi: 10.1016/j.simpat.2005.10.011.  Google Scholar

[23]

R. Mchich, P. Auger and J.-C. Poggiale, Effect of predator density dependent dispersal of prey on stability of a predator-prey system,, Mathematical Biosciences, 206 (2007), 343.  doi: 10.1016/j.mbs.2005.11.005.  Google Scholar

[24]

R. Mchich, P. Auger, R. Bravo de la Parra and N. Raïssi, Dynamics of a fishery on two fishing zones with fish stock dependent migrations: Aggregation and control,, Ecol. Model., 158 (2002), 51.  doi: 10.1016/S0304-3800(02)00237-5.  Google Scholar

[25]

J. Michalski, J.-C. Poggiale, R. Arditi and P. Auger, Macroscopic dynamic effects of migrations in patchy predator-prey systems,, J. Theor. Biol., 185 (1997), 459.  doi: 10.1006/jtbi.1996.0327.  Google Scholar

[26]

G. Moreno, L. Dagorn, G. Sancho and D. Itano, Fish behaviour from fishers' knowledge: The case study of tropical tuna around Drifting Fish Aggregating Devices (DFADs),, Canadian Journal of Fisheries and Aquatic Sciences, 64 (2007), 1517.  doi: 10.1139/f07-113.  Google Scholar

[27]

A. Moussaoui, Effect of a toxicant on the dynamics of a spatial fishery,, African Diaspora Journal of Mathematics, 10 (2010), 122.   Google Scholar

[28]

P. A. Nelson, Marine fish assemblages associated with Fish Aggregating Devices (FADs): Effects of fish removal, FAD size, fouling communities, and prior recruits,, Fishery Bulletin, 101 (2003), 835.   Google Scholar

[29]

S. Levin and S. Pacala, Theories of simplification and scaling of spatially distributed processes,, in, (1997), 204.   Google Scholar

[30]

S. Pioch, "Les 'Habitats Artificiels': Élément de Stratégie pour une Gestion Intégrée des Zones Côtières? Essai de Méthodologie d'Aménagement en 325 Ré cifs Artificiels Adapté à la Pêche Artisanale Côtière,", Ph.D. thesis, (2008).   Google Scholar

[31]

J.-C. Poggiale, "Applications des Variétés Invariantes a la Modélisation de l'Hétèrogén éité en Dynamique des Populations,", Ph.D. thesis, (1994).   Google Scholar

[32]

J.-C. Poggiale and P. Auger, Impact of spatial heterogeneity on a predator-prey system dynamics,, Comptes Rendus Biologies, 327 (2004), 1058.  doi: 10.1016/j.crvi.2004.06.006.  Google Scholar

[33]

R. A. Rountree, Association of fishes with fish aggregating devices: Effects of structure size on fish abundance,, Bulletin of Marine Science, 44 (1989), 960.   Google Scholar

[34]

K. Sakamoto, Invariant manifolds in singular perturbations problems for ordinary differential equations,, Proc. Roy. Soc. Ed. Sect. A, 116 (1990), 45.   Google Scholar

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