Afrika Statistika

An application of optimal control to the effective utilization of a renewable resource

Mahmud Ibrahim and Francis Benyah

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Abstract

The study explores the optimal harvesting of renewable resources like fisheries. The fish biomass dynamics is described by a nonlinear growth model that maximizes the total net revenue whilst taking into consideration the sustainable and effective utilization of the resource. In addition, stability dynamics of the model is assessed through bifurcation analysis. Pontryagin's maximum principle is used to derive the optimality system and characterize the optimal control. A numeric iterative method employing the fourth order Runge-Kutta scheme facilitates the solution of the optimality system. The simulation results obtained are then discussed. The results show that the sum of the maximum allowable harvest and the final biomass level must not exceed the maximum sustainable yield (MSY).

Résumé

L'étude explore la récolte optimale des ressources renouvelables comme la pêche. La dynamique de la biomasse des poissons est décrite par un modèle de croissance non linéaire qui maximise le revenu net total tout en tenant compte de l'utilisation effective et durable de la ressource. En outre, la dynamique de la stabilité du modèle est évaluée grâce une analyse de bifurcation. Le principe maximal de Pontryagin est utilisé pour dériver le système d'optimalité et caractériser le contrôle optimal. Une méthode itérative numérique employant le schéma Runge-Kutta de quatrième ordre facilite la solution du système d'optimalité. Les résultats de simulation obtenus sont ensuite discutés. Ils montrent que la somme de la récolte maximale admissible et du niveau de biomasse finale ne doit pas dépasser le rendement maximal durable (MSY).

Article information

Source
Afr. Stat., Volume 12, Number 2 (2017), 1313-1331.

Dates
Received: 4 July 2017
Accepted: 10 August 2017
First available in Project Euclid: 14 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.as/1505354426

Digital Object Identifier
doi:10.16929/as/2017.1313.106

Mathematical Reviews number (MathSciNet)
MR3703442

Zentralblatt MATH identifier
1376.49025

Subjects
Primary: 49J15: Optimal control problems involving ordinary differential equations 49N05: Linear optimal control problems [See also 93C05] 92B05: General biology and biomathematics

Keywords
Optimal control Effective utilization rate Fish biomass Logistic growth model Bifurcation analysis Shadow price Maximum sustainable yield (MSY)

Citation

Ibrahim, Mahmud; Benyah, Francis. An application of optimal control to the effective utilization of a renewable resource. Afr. Stat. 12 (2017), no. 2, 1313--1331. doi:10.16929/as/2017.1313.106. https://projecteuclid.org/euclid.as/1505354426


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