具有尺度结构和双加权的种群模型:稳定性与最优收获

数学物理学报 ›› 2016, Vol. 36 ›› Issue (3): 584-600.

• 论文 • 上一篇

具有尺度结构和双加权的种群模型:稳定性与最优收获

何泽荣, 杨立志

杭州电子科技大学运筹与控制研究所杭州 310018

收稿日期:2015-10-11 修回日期:2016-03-23 出版日期:2016-06-26 发布日期:2016-06-26
作者简介:何泽荣,zrhe@hdu.edu.cn
基金资助:
国家自然科学基金（10771048，11271104）资助

A Weighted Population Model with Size-Structure: Stability and Optimal Harvesting

He Zerong, Yang Lizhi

Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou 310018

Received:2015-10-11 Revised:2016-03-23 Online:2016-06-26 Published:2016-06-26
Supported by:
Supported by the NSFC (10771048, 11271104)

摘要/Abstract

摘要：

研究一类带有新生个体调控的非线性尺度结构种群模型，其中密度制约对繁殖率和死亡率的影响不同. 应用压缩映像原理证明了平衡态的存在唯一性，给出了平衡态的表达式. 导出了平衡态的特征方程，由此给出平衡态稳定性的判定条件. 对于最优控制问题，借助凸分析范畴的切锥法锥理论获得了最优反馈策略；再用Ekeland变分原理确立了最优控制器的存在唯一性. 此外还用迎风差分法对模型离散化，并通过两个算例展示种群系统的演化历程.

关键词: 个体尺度, 种群模型, 稳定性, 最优控制, 法锥, Ekeland原理

Abstract:

This paper is concerned with the stability and optimal harvesting for a size-structured population model with control of newborns, where fertility and mortality depend the density in different ways. A formal equilibrium is derived and existence of unique steady state is shown via a contraction mapping. Some conditions for asymptotical stability and instability are presented by means of characteristic equation. As for the optimal harvesting problem, we cite the tangent-normal cones to establish an optimal feedback policy, and employ the Ekeland's variational principle to prove the existence and uniqueness of optimal strategies. Two examples demonstrate the evolution of the species.

Key words: Size-structure, Population model, Stability, Optimal control, Normal cone, Ekeland's variational principle

中图分类号:

O211.4

何泽荣, 杨立志. 具有尺度结构和双加权的种群模型:稳定性与最优收获[J]. 数学物理学报, 2016, 36(3): 584-600.

He Zerong, Yang Lizhi. A Weighted Population Model with Size-Structure: Stability and Optimal Harvesting[J]. Acta mathematica scientia,Series A, 2016, 36(3): 584-600.

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具有尺度结构和双加权的种群模型:稳定性与最优收获

A Weighted Population Model with Size-Structure: Stability and Optimal Harvesting

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