×

zbMATH — the first resource for mathematics

A model for fishery resource with reserve area. (English) Zbl 1011.92049
Summary: We propose and analyse a mathematical model to study the dynamics of a fishery resource system in an aquatic environment that consists of two zones: a free fishing zone and a reserve zone where fishing is strictly prohibited. Biological and bionomic equilibria of the system are obtained, and criteria for local stability, instability and global stability of the system are derived. It is shown that even if fishery is exploited continuously in the unreserved zone, fish populations can be maintained at an appropriate equilibrium level in the habitat. An optimal harvesting policy is also discussed using Pontryagin’s maximum principle.

MSC:
92D40 Ecology
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bhattacharya, D.K.; Begum, S., Bionomic equilibrium of two species system, Math. biosci., 135, 2, 111-127, (1996) · Zbl 0856.92018
[2] Birkhoff, G.; Rota, G.S., Ordinary differential equations, (1882), Ginn Boston
[3] Burges, D.N.; Graham, A., Introduction to control theory including optimal control, (1980), Wiley New York · Zbl 0428.93001
[4] Chattopadhyay, J.; Mukhopadhyay, A.; Tapaswi, P.K., A resourced based competitive system in three species fishery, Nonlinear stud., 3, 73-83, (1996) · Zbl 0853.92021
[5] Clark, C.W., Profit maximization and the extinction of animal species, Journal of political economy, 81, 950-961, (1973)
[6] Clark, C.W., Mathematical bioeconomics: the optimal management of renewable resources, (1976), Wiley New York · Zbl 0364.90002
[7] Clark, C.W., Mathematical models in the economics of renewable resources, SIAM rev., 21, 81-99, (1979) · Zbl 0401.90028
[8] Clark, C.W., Bioeconomic modelling and fisheries management, (1985), Wiley Interscience New York
[9] Clark, C.W., Mathematical bioeconomics: the optimal management of renewable resources, (1990), Wiley New York · Zbl 0712.90018
[10] Clark, C.W.; Clarke, F.H.; Mundro, G.R., The optimal exploitation of renewable resource stocksproblems of irreversible investment, Econometrica, 47, 1, 25-47, (1979) · Zbl 0396.90026
[11] Clark, C.W.; De Pree, J.D., A simple linear model for the optimal exploitation of renewable resources, Appl. math. optim., 5, 181-196, (1979) · Zbl 0443.49002
[12] Chaudhuri, K.S., A bioeconomic model of harvesting a multispecies fishery, Ecol. model., 32, 267-279, (1986)
[13] Chaudhuri, K.S., Dynamic optimisation of combined harvesting of a two species fishery, Ecol. model., 41, 17-25, (1988)
[14] Dubey, B.; Chandra, P.; Sinha, P., A resource dependent fishery model with optimal harvesting policy, J. biol. systems, 10, 1-13, (2002) · Zbl 1109.92313
[15] Fan, M.; Wang, K., Optimal harvesting policy for single population with periodic coefficients, Math. biosci., 152, 165-177, (1998) · Zbl 0940.92030
[16] Ganguli, S.; Chaudhuri, K.S., Regulation of a single-species fishery by taxation, Ecol. model., 82, 51-60, (1995)
[17] Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Amsterdam
[18] Hanson, F.B.; Ryan, D., Optimal harvesting with both population and price dynamics, Math. biosci., 148, 129-146, (1998) · Zbl 0938.92032
[19] Kitabatake, Y., A dynamic predator – prey model for fishery resourcesa case of lake kasumigaura, Environ. planning A, 14, 225-235, (1982)
[20] La Salle, J.; Lefschetz, S., Stability by Liapunov’s direct method with applications, (1961), Academic Press New York, London · Zbl 0098.06102
[21] Leung, A.; Wang, A., Analysis of models for commercial fishingmathematical and economical aspects, Econometrica, 44, 2, 295-303, (1976) · Zbl 0319.90017
[22] Mesterton-Gibbons, M., On the optimal policy for combined harvesting of independent species, Nat. res. model., 2, 109-134, (1987) · Zbl 0850.92071
[23] Mesterton-Gibbons, M., On the optimal policy for combined harvesting of predator-prey, Nat. res. model., 3, 63-90, (1988)
[24] Mesterton-Gibbons, M., A technique for finding optimal two species harvesting policies, Ecol. model., 92, 235-244, (1996)
[25] Mukhopadhyay, A.; Chattopadhyay, J.; Tapaswi, P.K., Selective harvesting in a two species fishery model, Ecol. model., 94, 243-253, (1997)
[26] Pradhan, T.; Chaudhuri, K.S., Bioeconomic modelling of a single-species fishery with Gompertz law of growth, J. biol. systems, 6, 4, 393-409, (1998)
[27] Pradhan, T.; Chaudhuri, K.S., A dynamic reaction model of a two species fishery with taxation as a control instrumenta capital theoretic analysis, Ecol. model., 121, 1-16, (1999)
[28] Pradhan, T.; Chaudhuri, K.S., Bioeconomic harvesting of a schooling fish speciesa dynamic reaction model, Korean J. comput. appl. math., 6, 1, 127-141, (1999) · Zbl 0914.92023
[29] Pradhan, T.; Chaudhuri, K.S., Bioeconomic modelling of selective harvesting in an inshore – offshore fishery, Differential equations dyn. systems, 7, 3, 305-320, (1999) · Zbl 0973.92036
[30] Ragozin, D.L.; Brown, G., Harvest policies and nonmarket valuation in a predator – prey system, J. environ. econom. manage., 12, 155-168, (1985)
[31] Song, X.; Chen, L., Optimal harvesting and stability for a two-species competitive system with stage structure, Math. biosci., 170, 2, 173-186, (2001) · Zbl 1028.34049
[32] Zhang, X.; Chen, L.; Neumann, A.U., The stage structured predator – prey model and optimal harvesting policy, Math. biosci., 168, 2, 201-210, (2000) · Zbl 0961.92037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.