具有Size结构的捕食种群系统的最优收获策略

Abstract

Abstract:

This work is concerned with an optimal harvesting problem for a predator-prey model, in which the prey population is described by a first order partial differential equation (PDE) in a density function and the predator by an ordinary di?erential equation in total size. The existence and uniqueness of solutions to the state system and the dual system is proven via fixed point theorem. Necessary optimality conditions of first order are established by use of tangent-normal cones and dual system technique. The existence of a unique optimal control
pair is derived by means of Ekeland’s variational principle. The resulting conclusion extends some existing results involving age-dependent populations.

Key words: Optimal control, Predator-prey model, Size-structure, Tangent-normal cones, Ekeland’s variational principle

CLC Number:

92B05

LIU Yan, HE Ze-Rong. Optimal Harvesting of a Size-structured Predator-prey Model[J].Acta mathematica scientia,Series A, 2012, 32(1): 90-102.

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References

[1] Webb G F. Theory of Nonlinear Age-dependent Population Dynamics. New York: Marcel Dekker, 1985

[2] Iannelli M. Mathematical Theory of Age-structured Population Dynamics. Pisa: Giardini Editori, 1994

[3] Cushing J M. An Introduction to Structured Population Dynamics. Philadelphia: SIAM, 1998

[4] Ani$\c{t}$a S. Analysis and Control of Age-dependent Population Dynamics. Dordrecht: Kluwer, 2000

[5] Murphy L F. Maximum sustainable yield of a nonlinear population model with continuous age structure.
Math Biosci, 1991, 104: 259--270

[6] Barbu V, Iannelli M. Optimal control of population dynamics. J Optim Theory Appl, 1999, 102: 1--14

[7] Barbu V. Mathematical Methods in Optimization of Differential Systems. Dordrecht: Kluwer, 1994

[8] Barbu V, Iannelli M, Martcheva M. On the controllability of the Lotka-McKendrick model of population
dynamics. J Math Anal Appl, 2001, 253: 142--165

[9] Fister K R, Lenhart S. Optimal control of a competitive system with age-structure. J Math Anal Appl, 2004, 291: 526--537

[10] Fister K R, Lenhart S. Optimal harvesting in an age-structured predator-prey model. Appl Math Optim, 2006, 54: 1--15

[11] 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. 数学物理学报, 2005, 25(6): 905--912

[12] 何泽荣, 刘炎. 一类基于时滞和年龄分布的种群控制问题. 系统科学与数学, 2010, 30(1): 53--59

[13] 何泽荣. 具有年龄结构和约束的群落系统的最优收获. 数学物理学报, 2010, 30A(2): 477--486

[14] He Z R. Optimal birth control of age-dependent competitive species. J Math Anal Appl, 2004, 296: 286--301

[15] He Z R, Hong S H, Zhang C G. Double control problems of age-distributed population dynamics. Nonlinear Anal RWA, 2009, 10: 3112--3121

[16] Zhao C, Zhao P, Wang M S. Optimal harvesting for nonlinear age-dependent population dynamics. Math Comput Model, 2006, 43: 310--319

[17] Luo Z X. Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model.
Appl Math Comput, 2007, 186: 1742--1752

[18] Metz J A J, Diekmann O. The Dynamics of Physiologically Structured Populations. Berlin: Springer, 1986

[19] Tucker S, Zimmerman S. A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J Appl Math, 1988, 48: 549--591

[20] Cushing J M. A competition model for size-structured species. SIAM J Appl Math, 1989, 49: 838--858

[21] Calsina A, Salda\~{n}a J. A model of physiologically structured population dynamics with a nonlinear
individual growth rate. J Math Biol, 1995, 33: 335--364

[22] Kato N. A general model of size-dependent population dynamics with nonlinear growth rate. J Math Anal Appl, 2004, 297: 234--256

[23] Farkas J Z. Stability conditions for a non-linear size-structured model. Nonlinear Anal RWA, 2005, 6: 962--969

[24] Farkas J Z, Hagen T. Stability and regularity results for a size-structured population model. J Math Anal Appl, 2007, 328: 119--136

[25] 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

[26] Ackleh A S, Deng K, Wang X B. Competitive exclusion and coexistence for a quasilinear size-structured population model. Math Biosci, 2004, 192: 177--192

[27] 刘克英, 刘伟安. 大小结构种群模型的平衡解稳定性. 数学物理学报, 2010, 30A(2): 417--424

[28] Hadeler K P, M\"{u}ller J. Optimal harvesting and optimal vaccination. Math Biosci, 2007, 206: 249--272

[29] Hritonenko N, et al. Maximum principle for a size-structured model of forest and carbon sequestration management. Applied Mathematics Letters, 2008, 21: 1090--1094

[30] Kato N. Optimal harvesting for nonlinear size structured population dynamics. J Math Anal Appl, 2008, 342: 1388--1398

Optimal Harvesting of a Size-structured Predator-prey Model

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