Maximum principle for the optimal harvesting problem of a size-stage-structured population model

Miaomiao Chen; Rong Yuan

doi:10.3934/dcdsb.2021245

Article Contents

2022, Volume 27, Issue 8: 4619-4648. Doi: 10.3934/dcdsb.2021245

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Maximum principle for the optimal harvesting problem of a size-stage-structured population model

Miaomiao Chen^1, , and
Rong Yuan^2,

1.
College of science, Qingdao University of Technology, Qingdao, 266033, China
2.
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

^*Corresponding author: Miaomiao Chen
^*Corresponding author: Miaomiao Chen

Received: February 28, 2021

Revised: July 31, 2021

Early access: October 2021

Published: August 2022

This work is supported by the National Natural Science Foundation of China (Nos. 11771044, 11871007 and 12171039)

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Abstract

The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.

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Mathematics Subject Classification: Primary: 34C60; Secondary: 35Q92, 35Q93, 34H05.

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Figure 1. Time-series of (a) optimal harvesting effort, (b) adult population

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Figure 2. Time-series of (a) growth function, (b) total juvenile population, and juvenile population distributions (c) without harvesting, (d) with optimal harvesting

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Figure 3. (a) Time-series of the optimal harvesting effort, adult population, and total juvenile population, (b) contours of the juvenile population distribution

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Figure 4. Time-series of (a) adult population and growth function, (b) total juvenile population and harvested adult

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