A Two-Species Model for a Fishing System with Marine Protected Areas

Authors: Felicia Magpantay, Kenzu Abdella

Abstract:

A model of a system concerning one species of demersal (inshore) fish and one of pelagic (offshore) fish undergoing fishing restricted by marine protected areas is proposed in this paper. This setup was based on the FISH-BE model applied to the Tabina fishery in Zamboanga del Sur, Philippines. The components of the model equations have been adapted from widely-accepted mechanisms in population dynamics. The model employs Gompertz-s law of growth and interaction on each type of protected and unprotected subpopulation. Exchange coefficients between protected and unprotected areas were assumed to be proportional to the relative area of the entry region. Fishing harvests were assumed to be proportional to both the number of fishers and the number of unprotected fish. An extra term was included for the pelagic population to allow for the exchange between the unprotected area and the outside environment. The systems were found to be bounded for all parameter values. The equations for the steady state were unsolvable analytically but the existence and uniqueness of non-zero steady states can be proven. Plots also show that an MPA size yielding the maximum steady state of the unprotected population can be found. All steady states were found to be globally asymptotically stable for the entire range of parameter values.

Keywords: fisheries modelling, marine protected areas, sustainablefisheries, Gompertz Law

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061070

References:

[1] White, A.T., Ali˜no, and Meneses, A.B.T., Creating and Managing Marine Protected Areas in the Philippines. Cebu City, Philippines: University of the Philippines Marine Science Institute, 2006
[2] Salm, R.V., and Clark, J.R., Marine and Coastal Protected Areas, A Guide for Planners and Managers. 3rd ed., Gland, Switzerland: International Union for Conservation of Nature and Natural Resources, 2000
[3] Roberts, C.M., McClean, C.J., Veron J.E.N., Hawkins, J.P., Allen, G.R., McAllister D.E., Mittermeier, C.G., Schueler, F.W., Spalding, M., Wells, F., Vynne, C., Werner, T.B., "Marine Biodiversity Hotspots and Conservation Priorities for Tropical Reefs," Science, V295, pp. 1280- 1284, 2002
[4] Licuanan, W.Y., Ali˜no, P.M., Campos, W.I., Castillo, G.B. and Juinio- Me˜nez, M.A., "A Decision Support Model for Determining Sizes of Marine Protected Areas: Biophysical Considerations," The Philippine Agricultural Scientist, V89(1), pp. 509-520, 2006
[5] Russ, G.R. and Alcala, A.C. "Do marine reserves export adult fish biomass? Evidence from Apo Island, central Philippines" Marine Ecology Progress Series, V132, pp. 1-9, 1996
[6] Alcala, A.C. and Luchavez, T. "Fish Yields of a Coral Reef Surrounding Apo Island, Central Visayas," Proceedings of the 4th International Coral Reef Symposium, V1, pp. 69-73, 1981
[7] Oracion, E.G., Miller, M.L. and Christie, P. "Marine Protected Areas for whom? Fisheries, Tourism, and Solidarity in a Philippine Community," Ocean and Coastal Management, V48, pp. 393-410, 2005
[8] Castillo, G.B., Licuanan, W.Y., Ali˜no, P.M., Campos, W. and Juinio- Me˜nez, A., "Fisheries Bio-Economic Model: A Decision Support Model for Local Fisheries Management," Philippine Environmental Governance Project,(Technical Report) Department of Environment and Natural Resources - United States Agency for International Development, 2004
[9] Baskett, M.L., Micheli, F. and Levin, S.A., "Designing Marine Reserves for Interacting Species: Insight from Theory," Biological Conservation, V137, pp. 163-179, 2007
[10] Dubey, B., Chandra, P. and Sinha, P. "A Model for Fishery Resource with Reserve Area" Nonlinear Analysis: Real World Applciations, V4, pp. 625-637, 2003
[11] Kar, T.K. and Misra, S. "Influence of Prey Reserve in a Prey-Predator Fishery" Nonlinear Analysis, V65, pp. 1725-1735, 2006
[12] Greenville, J. and MacAulay, T.G., "Protected Areas in Fisheries: a two-patch, two-species population model" The Australian Journal of Agricultural and Resource Economics, V50, pp. 207-226, 2006
[13] Kar, T.T. "Influence of Environmental Noises on the Gompertz Model of Two Species Fishery" Ecological Modelling, V173, pp. 283-293, 2004
[14] Pradhan, T. and Chaudhuri, K.S. "Bioeconomic Modeling of a Single- Species Fishery with Gompertz Law of Growth" Journal of Biological Systems, V6(4), pp. 393-409, 1998
[15] Shirai, Y. and Harada, Y., "Evaluation of the Advantage of Marine Protected Areas over the Reduction of the Fishing Mortality Coefficient," Fisheries Science, V69, pp. 465-472, 2003
[16] Armstrong, C.W. and Skonhoft, A. "Marine reserves: A bio-economic model with asymmetric density dependent migration" Ecological Economics, V57, pp. 466-476, 2006
[17] Roberts, C.M., Bohnsack, J.A., Gell, F., Hawkins, J.P. and Goodridge, R. "Effects of Marine Reserves on Adjacent Fisheries" Science, V294, pp. 1920-1923, 2001
[18] Pezzey, J.C.V., Reoberts, C.M. and Urdal, B.T. "A Simple Bioeconomic Model of a Marine Reserve" Ecological Economics, V33, pp. 77-91, 2000
[19] Nowlis, J.S., "Short-and Long-term Effects of Three Fishery- Management Tools on Depleted Fisheries" Bulletin of Marine Science, V66(3), pp. 651-662, 2000