Optimal harvesting for a periodic $ n $-dimensional food chain model with size structure in a polluted environment

Tainian Zhang; Zhixue Luo; Hao Zhang; Tainian Zhang; Zhixue Luo; Hao Zhang

doi:10.3934/mbe.2022352

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 7481-7503. doi: 10.3934/mbe.2022352

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Research article

Optimal harvesting for a periodic $ n $-dimensional food chain model with size structure in a polluted environment

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

Academic Editor: Hermann J. Eberl

Received: 12 March 2022 Revised: 19 April 2022 Accepted: 02 May 2022 Published: 19 May 2022

This study examines an optimal harvesting problem for a periodic $ n $-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the $ i $th population, the concentration of toxicants in the $ i $th population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. 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. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting $ n $ species.
- food chain model,
- size structure,
- optimal harvesting,
- pollution,
- finite difference method
Citation: Tainian Zhang, Zhixue Luo, Hao Zhang. Optimal harvesting for a periodic $ n $-dimensional food chain model with size structure in a polluted environment[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7481-7503. doi: 10.3934/mbe.2022352

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Abstract

This study examines an optimal harvesting problem for a periodic $ n $-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the $ i $th population, the concentration of toxicants in the $ i $th population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. 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. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting $ n $ species.

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