Optimal harvesting for a periodic competing system with size structure in a polluted environment

Tainian Zhang; Zhixue Luo; Tainian Zhang; Zhixue Luo

doi:10.3934/math.2022808

AIMS Mathematics

2022, Volume 7, Issue 8: 14696-14717. doi: 10.3934/math.2022808

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Research article

Optimal harvesting for a periodic competing system with size structure in a polluted environment

Tainian Zhang ^{1
,
,},
Zhixue Luo ²

1.
School of Environmental and Municipal Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
2.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China

Received: 28 March 2022 Revised: 25 May 2022 Accepted: 27 May 2022 Published: 08 June 2022
MSC : 49J20, 92B05

As a renewable resource, biological population not only has direct economic value to people's lives, but also has important ecological and environmental value. This study examines an optimal harvesting problem for a periodic, competing hybrid system of three species that is dependent on size structure in a polluted environment. The existence and uniqueness of the nonnegative solution are proved via an operator theory and fixed point theorem. The necessary optimality conditions are derived by constructing an adjoint system and using the tangent-normal cone technique. The existence of unique optimal control pair is verified by means of the Ekeland variational principle and a feedback form of the optimal policy is presented. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. The objective functional represents the total profit obtained from harvesting three species. The results obtained in this work can be extended to a wide variety of fields.
- size structure,
- optimal harvesting,
- competing system,
- pollution,
- finite difference method
Citation: Tainian Zhang, Zhixue Luo. Optimal harvesting for a periodic competing system with size structure in a polluted environment[J]. AIMS Mathematics, 2022, 7(8): 14696-14717. doi: 10.3934/math.2022808

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Abstract

As a renewable resource, biological population not only has direct economic value to people's lives, but also has important ecological and environmental value. This study examines an optimal harvesting problem for a periodic, competing hybrid system of three species that is dependent on size structure in a polluted environment. The existence and uniqueness of the nonnegative solution are proved via an operator theory and fixed point theorem. The necessary optimality conditions are derived by constructing an adjoint system and using the tangent-normal cone technique. The existence of unique optimal control pair is verified by means of the Ekeland variational principle and a feedback form of the optimal policy is presented. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. The objective functional represents the total profit obtained from harvesting three species. The results obtained in this work can be extended to a wide variety of fields.

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