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]	Gu Haibo, Gao Caixia. Razumikhin-Type Theorems on P-th Moment and Almost Sure Stability of Neutral Stochastic Switched Nonlinear Systems [J]. Acta mathematica scientia,Series A, 2018, 38(5): 970-983.
[2]	Zheng Lifei, Guo Jie, Wu Meihua, Wang Xiaorui, Wan Aying. The Research on a Class of the Predator-Prey-Mutualist System with Delays [J]. Acta mathematica scientia,Series A, 2018, 38(5): 1001-1013.
[3]	Pan Quyu, Tang Yinghui. Analysis of M/G/1 Repairable Queueing System and Optimal Control Policy with a Replaceable Repair Facility Under Delay Min(N,D)-Policy [J]. Acta Mathematica Scientia, 2018, 38(5): 1014-1031.
[4]	Wu Bin, Gao Ying, Yan Lin, Yu Jun. Carleman Estimate for a 2×2 Strongly Coupled Partial Differential System with Nonsingular Coefficient Matrix of Principal Parts and Application to an Inverse Source Problem [J]. Acta mathematica scientia,Series A, 2018, 38(4): 779-799.
[5]	Zhang Chunguo, Liu Yubiao, Liu Weiwei. Boundary Optimal Control for the Timoshenko Beam [J]. Acta mathematica scientia,Series A, 2018, 38(3): 454-466.
[6]	Wang Junli, Zhang Xingwei, Liu Jian. Instability of Plane Couette-Poiseuille Flow of Compressible Navier-Stokes Equation [J]. Acta mathematica scientia,Series A, 2018, 38(2): 322-333.
[7]	Wang Wansheng, Zhong Peng, Zhao Xinyang. Long-Time Stability of Nonlinear Neutral Differential Equations with Variable Delay [J]. Acta mathematica scientia,Series A, 2018, 38(1): 96-109.
[8]	Li Haixia. Stability and Uniqueness of Positive Solutions for a Food-Chain Model [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1094-1104.
[9]	Meng Xiaoying. Analysis of a Stochastic Delayed Epidemic Model with a Non-Monotonic Incidence Rate [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1162-1175.
[10]	Wang Chun, Xu Tianzhou. Solution and Hyers-Ulam-Rassias Stability of a Mixed Type Quadratic-Additive Functional Equation with a Parameter in Quasi-Banach Spaces [J]. Acta mathematica scientia,Series A, 2017, 37(5): 846-859.
[11]	Chen Hongbin, Gan Siqing, Xu Da, Peng Yulong. A Formally Second-Order BDF Compact ADI Difference Scheme for the Two-Dimensional Fractional Evolution Equation [J]. Acta mathematica scientia,Series A, 2017, 37(5): 976-992.
[12]	Yang Jihua, Zhang Erli, Liu Mei. Dynamic Analysis and Chaos Control of a Finance System with Delayed Feedbacks [J]. Acta mathematica scientia,Series A, 2017, 37(4): 767-782.
[13]	Xiong Jun, Li Junmin, He Chao. Fuzzy Boundary Control Design for a Class of First-Order Hyperbolic PDEs [J]. Acta mathematica scientia,Series A, 2017, 37(3): 469-477.
[14]	Wei Aiju, Zhang Xinjian, Wang Junyi, Li Kezan. Dynamics Analysis of an Ebola Epidemic Model [J]. Acta mathematica scientia,Series A, 2017, 37(3): 577-592.
[15]	Wang Wei, Zhu Yucan. Stability of Fusion Frame Systems under Perturbation of the Local Frame Vectors [J]. Acta mathematica scientia,Series A, 2017, 37(2): 228-238.