A Mathematical Model for Optimal Management and Utilization of a Renewable Resource by Population

A dynamical model is proposed and analyzed to study the effect of the population on the resource biomass by taking into account the crowding effect. Biological and bionomical equilibria of the system are discussed. The global stability behavior of the positive equilibrium is studied via the output feedback control. An appropriate Hamiltonian function is formed for the discussion of optimal harvesting of resource which is utilized by the population using Pontryagin's Maximum Principal. A numerical simulation is performed on the model to analyze the theoretical results.

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Optimal management of a renewable resource utilized by a population with taxation as a control variable

Nonlinear Analysis Modelling and Control ◽

10.15388/na.18.1.14030 ◽

2013 ◽

Vol 18 (1) ◽

pp. 37-52 ◽

Cited By ~ 2

Author(s):

Balram Dubey ◽

Atasi Patra

Keyword(s):

Maximum Principle ◽

Control Variable ◽

Renewable Resource ◽

Dynamical Variable ◽

Dynamical Model ◽

Optimal Harvesting ◽

Optimal Management ◽

Crowding Effect ◽

Optimal Harvesting Policy ◽

Commercial Importance

A dynamical model is proposed and analyzed to discuss the effect of population on a resource biomass by taking into account the crowding effect. It is assumed that the resource biomass, which has commercial importance, is subjected to harvesting. The harvesting effort is assumed to be a dynamical variable and taxation is being used as a control variable. The optimal harvesting policy is discussed using the Pontryagin’s maximum principle.

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Theory of optimal harvesting for a size structured model of fish

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2019006 ◽

2020 ◽

Vol 15 ◽

pp. 1 ◽

Cited By ~ 1

Author(s):

Rong Liu ◽

Guirong Liu

Keyword(s):

Necessary Optimality Conditions ◽

Adjoint System ◽

Optimal Harvesting ◽

Optimal Management ◽

Negative Solution ◽

Structured Model ◽

The Maximum Principle ◽

Fish Resources ◽

Mazur’S Theorem ◽

Theoretical Results

This paper investigates the maximum principle for a nonlinear size structured model that describes the optimal management of the fish resources taking into account harvesting the fish and putting the fry. First, we show the existence of a unique non-negative solution of the system, and give a comparison principle. Next, we prove the existence of optimal policies by using maximizing sequence and Mazur’s theorem in convex analysis. Then, we obtain necessary optimality conditions by using tangent-normal cones and adjoint system techniques. Finally, some examples and numerical results demonstrate the effectiveness of the theoretical results in our paper.

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Globally Asymptotic Stability of Stochastic Nonlinear Systems with Time-Varying Delays via Output Feedback Control

Discrete Dynamics in Nature and Society ◽

10.1155/2016/7548156 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11

Author(s):

Mingzhu Song ◽

Wenwen Cheng ◽

Quanxin Zhu ◽

Hongwei Zhou ◽

Hui Wang

Keyword(s):

Nonlinear Systems ◽

Asymptotic Stability ◽

Output Feedback ◽

Output Feedback Control ◽

Lyapunov Theory ◽

Stochastic Nonlinear Systems ◽

Time Varying ◽

Globally Asymptotically Stable ◽

Globally Asymptotic Stability ◽

Theoretical Results

We address the problem of globally asymptotic stability for a class of stochastic nonlinear systems with time-varying delays. By the backstepping method and Lyapunov theory, we design a linear output feedback controller recursively based on the observable linearization for a class of stochastic nonlinear systems with time-varying delays to guarantee that the closed-loop system is globally asymptotically stable in probability. In particular, we extend the deterministic nonlinear system to stochastic nonlinear systems with time-varying delays. Finally, an example and its simulations are given to illustrate the theoretical results.

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