具有尺度结构和双加权的种群模型:稳定性与最优收获

Abstract

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

CLC Number:

O211.4

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.

Trendmd

References

[1] Ebenman B, Persson L. Size-Structured Populations:Ecology and Evolution. New York:Springer-Verlag, 1988
[2] Metz J A J, Diekmann O. The Dynamics of Physiologically Structured Populations. New York:Springer-Verlag, 1986
[3] Tuljapurkar S, Caswell H. Structured-Population Models in Marine, Terrestrial, and Freshwater Systems. New York:International Thomson Publishing, 1996
[4] Okubo A, Levin S A. Diffusion and Ecological Problems:Modern Perspectives (2ed). New York:Springer-Verlag, 2001
[5] Magal P, Ruan S. Structured Population Models in Biology and Epidemiology. Berlin:Springer-Verlag, 2008
[6] Tucker S L, Zimmerman S O. A nonlinear model of population dynamics containing an arbitrary number of structure variables. SIAM J Appl Math, 1988, 48(3):549-591
[7] Dercole F, Niklas K, Rand R. Self-thinning and community persistence in a simple size-structured dynamical model of plant growth. J Math Biol, 2005, 51:333-354
[8] Kohyama T. Size-structured multi-species model of rain forest trees. Functional Ecology, 1992, 6:206-212
[9] Botsford L W. Optimal fishery policy for size-specific density-dependent population models. J Math Biol, 1981, 12:265-293
[10] Kato N. Optimal harvesting in a two-species model of size-structured population//Boucekkine R, Hritonenko N, Yatsenko Y, et al. Optimal Control of Age-Structured Populations in Economy, Demography, and the Environment. London:Routledge Taylor & Francis Group, 2011:229-252
[11] He Z R, Liu Y. An optimal birth control problem for a dynamical population model with size-structure. Nonlinear Analysis:Real World Applications, 2012, 13:1369-1378
[12] Liu Y, He Z R. Stability results for a size-structured population model with resources-dependence and inflow. J Math Anal Appl, 2009, 360:665-675
[13] 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. 应用数学学报, 2014, 37(1):145-159He Z R, Liu R, Liu L L. Optimal harvesting of a size-structured population model in a periodic environment. Acta Math Appl Sin, 2014, 37(1):145-159
[14] 刘炎, 何泽荣. 具有Size结构的捕食种群系统的最优收获策略. 数学物理学报, 2012, 32A(1):90-102Liu Y, He Z R. Optimal harvesting of a size-structured predator-prey model. Acta Math Sci, 2012, 32A(1):90-102
[15] 何泽荣, 刘荣, 刘丽丽. 模拟周期环境和尺度结构的种群系统的最优收获率. 数学物理学报, 2014, 34A(3):684-690He Z R, Liu R, Liu L L. Optimal harvest rate for a population system modeling periodic environment and body size. Acta Math Sci, 2014, 34A(3):684-690
[16] 杨立志, 何泽荣, 邹世平. 具有尺度结构和加权总规模的种群模型的种群模型分析. 应用泛函分析学报, 2014, 16(3):212-219Yang L Z, He Z R, Zou S P. Analysis of a population model with size-structure and weighted size. Acta Anal Func Appl, 2014, 16(3):212-219
[17] Barbu V. Mathematical Methods in Optimization of Differential Systems. Dordrecht:Kluwer Academic Publishers, 1994
[18] Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47:324-353

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

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Lv Dongting. Globally Asymptotic Stability of 2-Species Reaction-Diffusion Systems of Spatially Inhomogeneous Models [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1171-1183.
[2]	Liang Qing. Existence, Uniqueness and Stability of the Global Solutions to Two Classes of Pantogragh Stochastic Functional Differential Equations with Markovian Switching [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1184-1205.
[3]	LI Yazhi, Liu Lili, Tian Yanni. Optimal Control Strategy for the SVEITR Pulmonary Tuberculosis Transmission Model Considering Diffusion [J]. Acta mathematica scientia,Series A, 2025, 45(3): 934-945.
[4]	Liu Yu, Chen Guanggan, Li Shuyong. Nonlinear Stability of Traveling Waves for Stochastic Kuramoto-Sivashinsky Equation [J]. Acta mathematica scientia,Series A, 2025, 45(3): 790-806.
[5]	Liu Jia, Bao Xiongxiong. Asymptotic Stability of Pyramidal Traveling Front for Nonlocal Delayed Diffusion Equation [J]. Acta mathematica scientia,Series A, 2025, 45(1): 44-53.
[6]	Xiao Jianglong, Song Yongli, Xia Yonghui. Spatiotemporal Dynamics Induced by the Interaction Between Fear and Schooling Behavior in a Diffusive Model [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1577-1594.
[7]	Tang Jun, Wu Ailong. Event-Triggered Control for a Class of Stochastic Time-Delay Nonlinear Systems [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1607-1616.
[8]	Zhang Yongxue, Jia Wensheng. Stability of Solutions for Controlled Systems of Generalized Multiobjective Multi-Leader-Follower Games Under Bounded Rationality [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1689-1702.
[9]	Yang Hong, Zhang Xiaoguang. A Stochastic SIS Epidemic Model on Simplex Complexes [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1392-1399.
[10]	Fan Tianjiao, Feng Lichao, Yang Yanmei. Generalization of Inequality and Its Application in Additive Time-Varying Delay Systems [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1335-1351.
[11]	Guo Yan, Xu Xiaochuan. Local Solvability and Stability of the Inverse Spectral Problems for the Discontinuous Sturm-Liouville Problem with the Mixed Given Data [J]. Acta mathematica scientia,Series A, 2024, 44(4): 859-870.
[12]	Pan Lijun, Lv Shun, Weng Shasha. Riemann Solution and Stability of Coupled Aw-Rascle-Zhang Model [J]. Acta mathematica scientia,Series A, 2024, 44(4): 885-895.
[13]	Zhang Chunguo, Sun Baonan, Fu Yuzhi, Yu Xin. Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping [J]. Acta mathematica scientia,Series A, 2024, 44(4): 946-959.
[14]	Gao Caixia, Zhao Dongxia. The Delayed Control and Input-to-State Stability of ARZ Traffic Flow Model with Disturbances [J]. Acta mathematica scientia,Series A, 2024, 44(4): 960-977.
[15]	Bin Maojun, Shi Cuiyun. Time Optimal Control for Semilinear Riemann-Liouville Fractional Evolution Feedback Control Systems [J]. Acta mathematica scientia,Series A, 2024, 44(3): 687-698.