# A Bioeconomic Model of a Multi-site Fishery with Nonlinear Demand Function: Number of Sites Optimizing the Total Catch

- 169 Downloads
- 2 Citations

## Abstract

We present a mathematical model of a fishery on several sites with a variable price. The model takes into account the evolution during the time of the resource, fish and boat movement between the different sites, fishing effort and price that varies with respect to supply and demand. We suppose that the movements of the boats and resource as well as the variation of the price go on at a fast time scale. We use methods of aggregation of variables in order to reduce the number of variables and we derive a reduced model governing two global variables, respectively the biomass of the resource and the fishing effort of the whole fishery. We look for the existence of equilibria of the aggregated model and perform local stability analysis. Two main cases can occur. The first one corresponds to over-exploitation leading to fish extinction. At extinction, the fishing effort tends to a positive value. The second case corresponds to a durable fishery equilibrium which is globally asymptotically stable. In the later case, we show that there exists a number of fishing sites that optimizes the total catch of the fishery.

## Keywords

Multi-site fishery Variable price Nonlinear demand function Aggregation of variables Optimum catch.## References

- Auger P, Mchich R, Raissi N, Kooi B (2010a) Effects of market price on the dynamics of a spatial fishery model: over-exploited fishery/traditional fishery. Ecol Complex 7:13–20Google Scholar
- Auger P, Lett C, Moussaoui A, Pioch S (2010b) Optimal number of sites in artificial pelagic multi-site fisheries. Can J Fish Aquat Sci 67:296–303CrossRefGoogle Scholar
- Auger P, Moussaoui A, Sallet G (2012) Basic reproduction ratio for a fishery model in a patchy environment. Acta Biotheor 60:167–188CrossRefGoogle Scholar
- Barbier EB, Strand I, Sathirathai S (2002) Do open access conditions affect the valuation of an externality? Estimating the welfare effects of mangrove-fishery linkages. Env Resour Econ 21:343–367CrossRefGoogle Scholar
- Clark CW (1990) Mathematical. Bioeconomics: the optimal management of renewable resources, 2nd edn. Wiley, New YorkGoogle Scholar
- de Lara M, Doyen L (2008) Sustainable management of renewable resources: mathematical models and methods. Springer, BerlinGoogle Scholar
- Dagorn L, Holland KN, Itano DG (2007) Behavior of yellowfin (
*Thunnus albacares*) and bigeye (*T. obesus*) tuna in a network of fish aggregating devices (FADs). Mar Biol 151(2):595–606Google Scholar - Fonteneau A, Ariz J, Gaertner D, Nordstrom T, Pallares P (2000) 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. Aquat Living Resour 13(4):253–257CrossRefGoogle Scholar
- Girard C, Benhamou S, Dagorn L (2004) FAD: fish aggregating device or fish attracting device? A new analysis of yellowfin tuna movements around floating objects. Anim Behav 67(2):319–326CrossRefGoogle Scholar
- Moreno G, Dagorn L, Sancho G, Itano D (2007) Fish behaviour from fisher’s knowledge: the case study of tropical tuna around drifting fish aggregating devices (DFADs). Can J Fish Aquat Sci 64(11):1517–1528CrossRefGoogle Scholar
- Moussaoui A, Auger P, Lett C (2011) Optimal number of sites in multi-site fisheries with fish stock dependent migrations. Math Biosci Eng 8:769–783CrossRefGoogle Scholar
- Ohta I, Kakuma S (2005) Periodic behavior and residence time of yellowfin and bigeye tuna associated with fish aggregating devices around Okinawa Islands, as identified with automated listening stations. Mar Biol 146(3):581–594CrossRefGoogle Scholar
- Schaefer MB (1957) Some considerations of population dynamics and economics in relation to the management of the commercial marine fisheries. J Fish Res Board Canada 14:669CrossRefGoogle Scholar
- Smith VL (1968) Economics of production from natural resources. Am Econ Rev 58(3):409–431Google Scholar
- Smith VL (1969) On models of commercial fishing. J Political Econ 77(2):181–198CrossRefGoogle Scholar
- Thiao D, Chaboud C, Samba A, Lalo F, Cury P (2012) Economic dimension of the collapse of the ‘false cod’.
*Epinephelus aeneus*in a context of ineffective management of the small-scale fisheries in Senegal. Afr J Mar Sci 34(3):305–311CrossRefGoogle Scholar