Optimal Harvesting in a Periodic 3-Species Predator-Prey Model With Size Structure in Predators

LIU Rong; LIU Guirong

doi:10.21656/1000-0887.410285

Volume 42 Issue 5

May 2021

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LIU Rong, LIU Guirong. Optimal Harvesting in a Periodic 3-Species Predator-Prey Model With Size Structure in Predators[J]. Applied Mathematics and Mechanics, 2021, 42(5): 510-521. doi: 10.21656/1000-0887.410285

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Optimal Harvesting in a Periodic 3-Species Predator-Prey Model With Size Structure in Predators

doi: 10.21656/1000-0887.410285

LIU Rong¹,
LIU Guirong²

¹School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, P.R.China
²School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R.China

Funds: The National Natural Science Foundation of China（12001341；11971279）

Received Date: 2020-09-21
Rev Recd Date: 2021-03-12
Publish Date: 2021-05-01

Abstract

Abstract

The research on population dynamics and related control problems is not only of theoretical significance, but also closely related to biodiversity protection, pest control, and the development and utilization of renewable resources. The optimal harvesting problem was considered in a periodic 3-species predator-prey system with 1 predator and 2 competing preys, where the predator has size structure and was described with 1st-order partial differential equations. First, the existence of a unique non-negative solution of the controlled system was proven by means of the fixed-point reasoning, and the continuous dependence of the solution on the control variables was discussed. Then, the optimal harvesting conditions were given with the techniques of tangential-normal cones and the adjoint system. Finally, with Ekeland’s variational principle, the existence of the optimal harvesting strategy was derived. Here the objective functional represents the net economic benefit in the harvesting of 3 species. The results obtained would be beneficial for exploration of renewable resources.
- periodic environment,
- size structure,
- 3 species,
- predator-prey,
- optimal harvesting

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References(24)

References

[1]	王双明, 张明军, 樊馨蔓. 一类具时滞的周期logistic传染病模型空间动力学研究[J]. 应用数学和力学, 2018,39(2): 226-238.(WANG Shuangming, ZHANG Mingjun, FAN Xinman. Spatial dynamics of periodic reaction-diffusion epidemic models with delay and logistic growth[J]. Applied Mathematics and Mechanics,2018,39(2): 226-238.(in Chinese))
[2]	曹建智, 谭军, 王培光. 一类具有时滞的云杉蚜虫种群模型的Hopf分岔分析[J]. 应用数学和力学, 2019,40(3): 332-342.(CAO Jianzhi, TAN Jun, WANG Peiguang. Hopf bifurcation analysis of a model for spruce budworm populations with delays[J]. Applied Mathematics and Mechanics,2019,40(3): 332-342.(in Chinese))
[3]	HSU S B, RUAN S G, YANG T H. Analysis of three species Lotka-Volterra food web models with omnivory[J]. Journal of Mathematical Analysis and Applications,2015,426(2): 659-687.
[4]	SEN D, GHORAI S, BANERJEE M. Complex dynamics of a three species prey-predator model with intraguild predation[J]. Ecological Complexity,2018,34: 9-22.
[5]	LIU R, LIU G R. Dynamics of a stochastic three species prey-predator model with intraguild predation[J]. Journal of Applied Analysis and Computation,2020,10(1): 81-103.
[6]	LIU M, BAI C. Analysis of a stochastic tri-trophic food-chain model with harvesting[J]. Journal of Mathematical Biology,2016,73: 597-625.
[7]	MAGAL P, RUAN S G. Structured Population Models in Biology and Epidemiology [M]. Berlin: Springer, 2008.
[8]	ANI ?瘙塃 A S. Analysis and Control of Age-Dependent Population Dynamics [M]. Dordrecht: Kluwer Academic Publishers, 2000.
[9]	LUO Z X, HE Z R. Optimal control for age-dependent population hybrid system in a polluted environment[J]. Applied Mathematics and Computation,2014,228: 68-76.
[10]	LI L, FERREIRA C, AINSEBA B. Optimal control of an age-structured problem modelling mosquito plasticity[J]. Nonlinear Analysis: Real World Applications,2019,45: 157-169.
[11]	ANI A L, ANI A S. Note on some periodic optimal harvesting problems for age-structured population dynamics[J]. Applied Mathematics and Computation,2016,276: 21-30.
[12]	LU Y, LIU S. Threshold dynamics of a predator-prey model with age-structured prey[J]. Advances in Difference Equations,2018,2018(1): 164. DOI: 10.1186/s13662-018-1614-y.
[13]	FISTER K, LENHART S. Optimal harvesting in an age-structured predator-prey model[J]. Applied Mathematics and Optimization,2006,54: 1-15.
[14]	何泽荣, 刘荣, 刘丽丽. 捕食者具有年龄结构的种群系统的最优收获策略[J]. 数学进展, 2013,42(5): 691-700.(HE Zerong, LIU Rong, LIU Lili. Optimal harvesting for population system with age-dependent predator[J]. Advances in Mathematics,2013,42(5): 691-700.(in Chinese))
[15]	ANI A S, IANNELLI M, KIM M Y, et al. Optimal harvesting for periodic age-dependent population dynamics[J]. SIAM Journal on Applied Mathematics,1998,58: 1648-1666.
[16]	SAUER J R, SLADE N A. Size-based demography of vertebrates[J]. Annual Review of Ecology and Systematics,1987,18: 71-90.
[17]	CHU J X, DUCROT A, MAGAL P, et al. Hopf bifurcation in a size structured population dynamic model with random growth[J]. Journal of Differential Equations,2009,247(3): 956-1000.
[18]	YAN D, CAO Y, FU X. Asymptotic analysis of a size-structured population model with infinite states-at-birth[J]. Applicable Analysis,2019,98(5): 913-933.
[19]	LI Y, ZHANG Z, LV Y, et al. Optimal harvesting for a size-stage-structured population model[J]. Nonlinear Analysis: Real World Applications,2018,44: 616-630.
[20]	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, 2012,32〖WTHZ〗A〖WTB4〗(1): 90-102.(LIU Yan, HE Zerong. Optimal harvesting of a Size-structured predator-prey model[J]. Acta Mathematical Scientia,2012,32〖WTHZ〗A〖WTB4〗(1): 90-102.(in Chinese))
[21]	刘炎, 何泽荣. 一类基于尺度结构的种群系统的最优收获[J]. 系统科学与数学, 2015,35(4): 459-471.(LIU Yan, HE Zerong. Optimal harvesting for a population system with body size[J]. Journal of Systems Science and Complexity,2015,35(4): 459-471.(in Chinese))
[22]	曹雪靓. 污染环境下具有尺度结构的捕食种群模型解的存在唯一性[J]. 应用数学进展, 2018,7(7): 758-765.(CAO Xuejing. Existence and uniqueness of solution for predator-prey population model with size-structured in polluted environment[J]. Advances in Applied Mathematics,2018,7(7): 758-765.(in Chinese))
[23]	ZHANG F Q, LIU R, CHEN Y M. Optimal harvesting in a periodic food chain model with size structures in predators[J]. Applied Mathematics and Optimization,2017,75: 229-251.
[24]	BARBU V. Mathematical Methods in Optimization of Differential Systems [M]. Dordrecht: Kluwer Academic Publishers, 1994.