A bioeconomic model of a fishery with saturated catch and variable price: Stabilizing effect of marine reserves on fishery dynamics |
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| Institution: | 1. Laboratoire d’Analyse Non linéaire et Mathématiques Appliquées, Université de Tlemcen, Faculté des Sciences, Département de Mathématiques, Algérie;2. UMI IRD 209 UMMISCO Centre IRD de lIle de France 32 Avenue Henri Varagnat, 93143 Bondy Cedex, France;3. Sorbonne Universités, UPMC Univ Paris 06, France;1. Department of Mathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi campus, Ghana;2. Department of Mathematics, Federal University of Technology, Akure, Nigeria;3. Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India;4. Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India;1. Université Catholique de l’Afrique de l’Ouest/ UUZ, Daker, Senegal;2. Laboratory Decision Mathematics and Numerical Analysis (LMDAN / UCAD), Dakar, Senegal;3. Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, Dakar Fann BP 5683, Senegal;1. Department of Mathematics, Sultan Qaboos University, Al-Khoudh 123, Oman;2. Center of Preparatory Studies, Sultan Qaboos University, Al-Khoudh 123, Oman;3. School of IT Information and Control Engineering, Kunsan National University, Gunsan 54150, South Korea |
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| Abstract: | In this paper, we study a mathematical bio-economic model of a fishery with varying price. The three dimensional model describes the time evolution of the resource, the fishing effort and the price. The model is original because it considers a nonlinear harvesting function assumed to depend upon stock size and fishing effort with a saturation effect with respect to the resource as well as a price equation depending on demand and supply which is in addition proportional to price. Assuming that the price varies at a fast time scale, we are able to use ”aggregation of variables methods” in order to reduce the model in a two dimensional model at a slow time scale. This aggregated (reduced) model is analyzed. Several numerical simulations of the model are performed to substantiate our analytical findings. The existence of nonlinear harvesting makes the dynamics of the model more complicated, including multiple equilibria, bi-stability and limit cycle. Such large amplitude cycle variations are not desirable because they generate periods of overfishing at periods of very low activity. We then study the effects of marine reserves on the dynamics of the fishery, showing that for an adequate number of small reserves, limit cycle oscillations are switched off. |
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