《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (11): 18-25.

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### 依赖尺度结构的竞争种群的最优出生率控制

1. 兰州交通大学数理学院, 甘肃 兰州 730070
• 发布日期:2020-11-17
• 作者简介:张萍(1996— ),女,硕士研究生,研究方向为生物数学及最优控制理论. E-mail:1361049574@qq.com*通信作者简介:雒志学(1963— ),男,博士,教授,博士生导师,研究方向为生物数学及最优控制理论. E-mail:luozhix@263.net
• 基金资助:
国家自然科学基金资助项目(11561041);甘肃省自然科学基金资助项目(1506RJZA071)

### Optimal birth rate control for competing populations dependent on scale structure

ZHANG Ping, LUO Zhi-xue*

1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
• Published:2020-11-17

Abstract: A competitive population model with scale structure-dependent is established, and the optimal birth-control problem of the model is discussed. Firstly, the existence and uniqueness of the system solution are proved by using the characteristic line method and the Banach fixed point theorem, and the continuous dependence of the system solution on the control variables is also proved by the comparison principle. At the end of the paper, the necessity condition of the existence of optimal control is proved by using the definition of normal cone.

• O715.1
 [1] 刘汉武, 周立, 刘伟, 等. 利用不育技术防治高原鼠兔的理论模型[J]. 生态学杂志,2008,27(7):1238-1243. LIU Hanwu, ZHOU Li, LIU Wei, et al. Theoretical model of ochotona curzoniae control via contraception[J]. Chinese Journal of Ecology, 2008, 27(7):1238-1243.[2] 雒志学, 王绵森. 具有年龄结构的线性周期种群动力系统的最优收获控制问题[J]. 数学物理学报,2005,25A(6):905-912. LUO Zhixue, WANG Miansen. Optimal harvesting control for linear periodic age-dependent population dynamic system[J]. Acta Mathematica Scientia, 2005, 25A(6):905-912.[3] 何泽荣,郑敏,周娟. 带有性别比和尺度结构的非线性种群的全局演化行为[J]. 系统科学与数学,2013,33(12):1480-1490. HE Zengrong, ZHENG Min, ZHOU Juan. Global evolutionary behavior of nonlinear populations with sex ratios and scale structures[J]. Journal of Systems Science and Mathematical Sciences, 2013, 33(12):1480-1490.[4] 张美明, 张凤琴, 吕江, 等. 具有性别结构的免疫不育控制模型[J]. 信阳师范学院学报(自然科学版),2011,24(4):425-429. ZHANG Meiming, ZHANG Fengqin, LV Jiang, et al. Immunocontraception control models with sex-structure[J]. Journal of Xinyang Normal University(Natural Science Edition), 2011, 24(4):425-429.[5] 何泽荣, 刘荣, 刘丽丽. 周期环境中基于个体尺度的种群模型的最优收获策略[J]. 应用数学学报,2014,37(1):145-159. HE Zerong, LIU Rong, LIU Lili. Optimal harvesting of a size-structured population model in a periodic environment[J]. Acta Mathematicae Applicatae Sinica, 2014, 37(1):145-159.[6] HE Z, LIU Y. An optimal birth control problem for a dynamical population model with size structure[J]. Nonlinear Analysis: Real World Applications, 2012, 13(3):1369-1318.[7] 刘炎, 何泽荣. 具有size结构的捕食种群系统的最优收获策略[J]. 数学物理学报,2012,32A(1):90-102. LIU Yan, HE Zerong. Optimal harvesting of a size-structured predator-prey model[J]. Acta Mathematica Scientia, 2012, 32A(1):90-102.[8] 何泽荣, 刘荣, 刘丽丽. 依赖个体尺度结构的种群资源开发模型理论分析[J]. 系统科学与数学,2012,32(9):1109-1120. HE Zerong, LIU Rong, LIU Lili. Theoretical analysis for a nonlinear size-structured population resources model[J]. Journal of Systems Science and Mathematical Sciences, 2012, 32(9):1109-1120.[9] LIU Yan, CHENG Xiaoliang, HE Zerong. On the optimal harvesting of size-structured population dynamics[J]. Applied Mathematics: Journal of Chinese Universities(Series B), 2013, 28(2):173-186.[10] 李秋英, 张凤琴, 刘汉武. 不育控制下的具有性别结构的单种群模型[J]. 数学的实践与认识,2009,39(19):145-151. LI Qiuying, ZHANG Fengqin, LIU Hanwu. Model of single-species population with sex-structure under contraception control[J]. Journal of Mathematics in Practice and Theory, 2009, 39(19):145-151.[11] 吕江, 张凤琴, 刘汉武, 等. 具有竞争性繁殖干扰的不育控制害鼠种群模型[J]. 工程数学学报,2013,30(2):263-270. LV Jiang, ZHANG Fengqin, Liu Hanwu, et al. Dynamic model of bandicoot population with competitive reproductive interference under virus-vectored contraception control[J]. Chinese Journal of Engineering Mathematics, 2013, 30(2):263-270.[12] 刘汉武. 两性具有不同出生率和死亡率的种群动态[J]. 生态学杂志,2003,22(1):63-65. LIU Hanwu. Population dynamics of sex with different birth and death rates[J]. Chinese Journal of Ecology, 2003, 22(1):63-65.
 [1] 郑瑞瑞,孙同军. 一类捕食与被捕食模型最优控制问题的有限元方法的先验误差估计[J]. 《山东大学学报(理学版)》, 2020, 55(1): 23-32. [2] 梁丽宇,雒志学. 周期环境中具有尺度结构的两种群系统的最优控制[J]. 《山东大学学报(理学版)》, 2019, 54(9): 69-75. [3] 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. [4] 刘兵兵,郝庆一. 一类悲观二层规划问题的一阶必要最优性条件[J]. 山东大学学报（理学版）, 2016, 51(3): 44-50. [5] 余丽. 集值映射的ε-强次微分及应用[J]. J4, 2013, 48(3): 99-105. [6] 宿 洁, . 值型线性双层规划的共轭对偶及最优性条件[J]. J4, 2007, 42(10): 13-17 .
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