考虑污染和种内关系的种群扩散最优控制模型

doi:10.12005/orms.2023.0009

运筹与管理 ›› 2023, Vol. 32 ›› Issue (1): 54-59.DOI: 10.12005/orms.2023.0009

考虑污染和种内关系的种群扩散最优控制模型

杨露, 高伟

西安财经大学统计学院,陕西西安 710100

收稿日期:2021-01-12 出版日期:2023-01-25 发布日期:2023-03-01
通讯作者: 高伟(1978-),女,陕西西安人,博士,教授,硕士生导师,研究方向:时间序列分析。
作者简介:杨露(1989-),女,陕西西安人,博士,副教授,硕士生导师,研究方向:随机控制。
基金资助:
陕西省自然科学基金项目(2022JQ-042);全国统计科学研究重点项目(2021LZ28);西安财经大学青年英才发展支持计划;西安财经大学雁塔学者支持项目;国家社会科学基金项目(20CTJ008)

Optimal Control Model of Population Diffusion System Affected by Pollution and Intraspecific Relationship

YANG Lu, GAO Wei

Department of Statistics, Xi'an University of Finance and Economics, Xi'an 710100, China

Received:2021-01-12 Online:2023-01-25 Published:2023-03-01

摘要/Abstract

摘要： 针对污染和种内关系均影响细菌种群扩散这一管理生态学问题,本文建立了基于非线性拟抛物方程的最优控制模型,将外界环境向细菌种群输入的毒素率作为控制变量,运用控制理论和方法探讨污染和种内关系双重影响下种群扩散系统的最优控制问题。利用Schauder不动点定理证明了该种群扩散系统的适定性;同时,通过建立新的Carleman型估计,给出了容许控制和最优控制的存在性。最后,通过数值算例分析了理论推导的结果,在算例中都找到一对时间最优控制,验证了种群扩散系统最优控制模型的有效性。该研究结果对现代传染病预防具有借鉴意义,也为有效控制瘟疫的爆发和流行提供理论参考。

关键词: 种群扩散系统, 最优控制, 适定性, Carleman型估计

Abstract: In order to solve the problem in management ecology that both pollution and inter-individual interactions (intraspecific relationships) seriously restrict the diffusion of bacteria population, an optimal control model based on nonlinear quasi-parabolic equation is established in this paper, in which the toxin rate imported into the population by the outside environment is regarded as the control variable. Meanwhile, the optimal control problem of population diffusion system under the influence of pollution and intraspecific relations is studied by using the control theory and method. The well-posedness of the population diffusion system is proved by the Schauder Fixed-point Theorem, and the existence of the admissible control and the optimal control is obtained by establishing a new Carleman-type estimate. Finally, a numerical example is given to analyze the results of theoretical derivation, and a pair of time optimal control is found in various cases, which verifies the effectiveness of the optimal control model of population diffusion system. The results can be used for reference in the prevention of modern infectious diseases and provide theoretical reference for the effective control of the outbreak and epidemic of plague.

Key words: population diffusion system, optimal control, well-posedness, carleman inequality

中图分类号:

O231.1

杨露, 高伟. 考虑污染和种内关系的种群扩散最优控制模型[J]. 运筹与管理, 2023, 32(1): 54-59.

YANG Lu, GAO Wei. Optimal Control Model of Population Diffusion System Affected by Pollution and Intraspecific Relationship[J]. Operations Research and Management Science, 2023, 32(1): 54-59.

参考文献

[1] 张大庆.人类与瘟疫的不懈斗争[J].求是,2020,12(6):70-76.
[2] 李俊杰,张红.地方政府间治理空气污染行为的演化博弈与仿真研究[J].运筹与管理,2019,28(8):27-34.
[3] 谭德庆,徐浩.基于前景理论的环境污染群体事件演化均衡分析[J].运筹与管理,2020, 29(5):161-170.
[4] Skellam J. Random dispersal in theoretical population[J]. Biometrika, 1951, 38(1): 196-218.
[5] 何泽荣,王晶晶.离散等级种群系统的能控性与最优控制[J].高校应用数学学报A辑,2020,35(4):57-65.
[6] Srinivasu P D N. Control of environmental pollution to conserve a population[J]. Nonlinear Analysis, 2002, 3(3): 397-411.
[7] Liu M, Wang K. Survival analysis of stochastic single-species population models in polluted environments[J]. Ecological Modelling, 2009, 220(9): 1347-1357.
[8] 燕雪飞,殷红,曹莹.污染环境中单种群生存分析[J].生物数学学报,2009,24(1):87-92.
[9] 何泽荣,倪冬冬,郑敏.具有尺度结构和时滞的种群系统遍历性与最优控制[J].系统科学与数学,2018,38(1):1-15.
[10] 贠晓菊,王战平.污染环境中具有年龄结构的非线性时变种群扩散系统的最优控制[J].应用数学,2018,31(3):621-630.
[11] 申柳肖,赵春.基于尺度结构的竞争种群系统的最优输入率控制[J].山东大学学报(理学版),2018,53(7):21-29.
[12] 贠晓菊,王战平.带环境污染的与年龄相关的非线性种群动力系统的最优控制[J].数学的实践与认识,2017,47(5):208-218.
[13] 雒志学,李沐春.具有扩散和年龄结构种群动力系统的最优控制[J].工程数学学报,2006,23(4):641-646.
[14] 毛子昆,郝占庆,原作强.物种聚集分布与种间关系的多度不对称性[J].中国科学,2020,50(4):381-390.
[15] 杨露,高伟.PM_2.5污染扩散模型及适定性分析[J].系统工程学报,2020,35(3):301-314.
[16] Jia C H, Feng D X. The time optimal control of a nonlinear parabolic equation with a nonlocal lower order term[J]. Acta Analysis Functional Applicatiton, 2008, 10(2): 123-138.
[17] Wang G S. The existence of time optimal control of semilinear parabolic equations[J]. Systems and Control Letters, 2004, 53(3): 171-175.
[18] Wang L J, Wang G S. The optimal time control of a phase-field system[J]. SIAM Journal on Control and Optimization, 2003, 42(4): 1483-1508.
[19] Vancostenoble J. Improved Hardy-poincare inequalities and sharp carleman estimates for degenerate/singular parabolic problems[J]. Discrete and Continuous Dynamical Systems, 2010, 4(3): 761-790.
[20] Hassi E, Khodja F A, Hajjaj A. Carleman estimates and null controllability of coupled degenerate systems[J]. Evolution Equations and Control Theory, 2017, 2(3): 441-459.
[21] Fu X, Liu X. A weighted identity for stochastic partial differential operators and its applications[J]. Journal of Differential Equations, 2017, 262(6): 3551-3582.
[22] Francini E, Vessella S. Carleman estimates for the parabolic transmission problem and holder propagation of smallness across an interface[J]. Journal of Differential Equations, 2018, 265(6): 2375-2430.
[23] Louis-Rose C. A null controllability problem with a finite number of constraints on the normal derivative for the semilinear heat equation[J]. Electronic Journal of Qualitative Theory of Differential Equations, 2018, 55(95): 1-34.

考虑污染和种内关系的种群扩散最优控制模型

Optimal Control Model of Population Diffusion System Affected by Pollution and Intraspecific Relationship

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