污染环境下具有尺度结构的捕食种群系统的最优控制

Optimal control for predator-prey system with size structure in a polluted environment

摘要

研究一类污染环境下依赖个体尺度的捕食种群系统的最优控制问题, 其控制变量为生育率与毒素的输入量。 运用切锥法锥理论、Dubovitskii-Milyutin 定理和共轭系统技巧分别给出固定时间区间上的端点自由问题、无穷时间问题、状态约束问题的最优性条件,为治理环境污染、保护生物多样性、科学开发生物资源等方面提供理论支撑。

Abstract

We investigate the optimal control problem for a predator-prey system that depends on individual size in a polluted environment. The control variables include fertility and the input rate of exogenous toxicants. The optimality conditions for various problems-free terminal, infinite horizon, and constrained endpoint problem on fixed horizon-are derived using the theory of tangent-normal cones, the Dubovitskii-Milyutin theorem, and the adjoint system technique. These results offer theoretical underpinnings for controlling environmental pollution, protecting biodiversity, and scientifically exploiting biological resources.

关键词

环境污染 / 尺度结构 / 最优控制 / 共轭系统

Key words

environmental pollution / size structure / optimal control / adjoint system

参考文献

[1] Hallam T G , Clark C E , Lassiter R R . Effects of toxicants on population: a qualitative approach Ⅰ. Equilibrium environmental exposure[J]. Ecological Modelling, 1983, 18: 291-304.

[2] Hallam T G , Clark C E , Jordan G S . Effects of toxicants on populations: a qualitative approach Ⅱ. First order kinetics[J]. Journal of Mathematical Biology, 1983, 18: 25-37.

[3] Hallam T G, DeLuna J T. Effects of toxicants on populations: a qualitative approach Ⅲ. Environmental and food chain pathways[J]. Journal of Theoretical Biology, 1984, 109: 411-429.

[4] Kato N. Linear size—structured population models with spacial diffusion and optimal harvesting problems[J]. Mathematical Modelling of Natural Phenomena, 2014, 9(4): 122-130.

[5] Liu R, Liu G R. Optimal birth control problems for a nonlinear vermin population model with size—structure[J]. Journal of Mathematical Analysis Applications, 2017, 449: 265-291.

[6] Li Y J, Zhang Z H, Lv Y F, et al. Optimal harvesting for a size—stage—structured population model[J]. Nonlinear Analysis: Real World Applications, 2018, 44: 616-630.

[7] Liu J, Wang X S. Numerical optimal control of a size—structured PDE model for metastatic cancer treatment[J]. Mathematical Biosciences, 2019, 314: 28-42.

[8] Liu R, Liu G R. Optimal contraception control for a nonlinear vermin population model with size—structure[J]. Applied Mathematics and Optimization, 2019, 79: 231-256.

[9] 郑秀娟, 雒志学, 张昊. 基于尺度结构的非线性竞争种群的最优控制[J]. 山东大学学报 (理学版), 2021, 56(11): 11-20.

[10] Zheng Xiujuan, Luo Zhixue, Zhang Hao. Optimal control of nonlinear competing populations based on the size—structure[J]. Journal of Shandong University (Natural Science), 2021, 56(11): 11-20.

[11] He Z R, Han M J. Theoretical results of optimal harvesting in a hierarchical size—structured population system with delay[J]. International Journal of Biomathematics, 2021, 14(7): 2150054.

[12] 张昊, 雒志学, 郑秀娟. 基于尺度结构的周期三种群系统的最优收获[J]. 山东大学学报 (理学版), 2022, 57(1): 1-12.

[13] Zhang Hao, Luo Zhixue, Zheng Xiujuan. Optimal harvesting for three species system with size—structures in periodic environments[J]. Journal of Shandong University (Natural Science), 2022, 57(1): 1-12.

[14] 刘荣, 何泽荣. 周期变化环境中一类尺度结构种群系统的最优控制[J]. 系统科学与数学, 2022, 42(8): 1973-1989.

[15] Liu Rong, He Zerong. Optimal contraception control for a nonlinear vermin model with size—structure in a periodic environment[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(8): 1973-1989.

[16] 何泽荣, 窦艺萌, 韩梦杰. 具有尺度等级和时滞的种群系统的最优边界控制[J]. 数学物理学报, 2022, 42A(3): 867-880.

[17] He Zerong, Dou Yimeng, Han Mengjie. Optimal boundary control for a hierarchical size—structured population model with delay[J]. Acta Mathematica Scientia, 2022, 42A(3): 867-880.

[18] Ainseba B, Louison L, Omrane A. A population harvesting model with time and size competition dependence function[J]. Journal of Optimization Theory and Applications, 2022, 195: 647-665.

[19] Liu R, Zhang F Q, Chen Y M. Optimal contraception control problems in a nonlinear size—strucured vermin model[J]. Journal of Optimization Theory and Applications, 2023, 199: 1188-1221.

[20] 何泽荣, 江晓东, 杨立志. 个体尺度差异下的竞争种群模型的最优控制问题[J]. 数学进展, 2015, 44(3): 449-458.

[21] He Zerong, Jiang Xiaodong, Yang Lizhi. Optimal control problems of a size—structured competitive population system[J]. Advances in Mathematics (China), 2015, 44(3): 449-458.

[22] 于景元, 郭宝珠, 朱广田. 人口分布参数系统控制理论[M]. 武汉:华中理工大学出版社, 1999: 92-123.

[23] Yu Jingyuan, Guo Baozhu, Zhu Guangtian. Control theory of population distributional parameter systems[M]. Wuhan: Press of Central China University of Science and Technology, 1999: 92-123.

[24] Girsanov I V. Lectures on mathematical theory of extremun problems[M]//Beckmann M, Künzi H P. Lecture Notes in Economics and Mathematical Systems. New York: Springer—Verlag, 1972: 67.

基金资助

甘肃省自然科学基金项目(23JRRG0006)

河西学院校长基金青年科研项目(QN2024004)

河西学院校长基金创新团队项目(CXTD2024002)