Hopf Bifurcation Analysis of a Model for Spruce Budworm Populations With Delays
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摘要: 研究了一类具有时滞的云杉蚜虫种群阶段结构模型的动力学行为.首先,讨论了模型正平衡点的存在唯一性,并分析了该平衡点的局部稳定性和出现Hopf分岔的充分条件;其次,利用中心流形定理和正规形理论,讨论了分岔周期解的稳定性及方向;最后,通过数值模拟验证了相关结论的正确性.该文所得结论具有广泛的实际应用价值.Abstract: The dynamic behavior of a population model with stage structure for spruce budworms with time delay was investigated. Firstly, existence of a unique positive equilibrium of the model was discussed and sufficient conditions for local stability of the positive equilibrium and Hopf bifurcation occurrence were obtained. Next, the direction of the Hopf bifurcation and the stability of the periodic bifurcation solutions were analyzed with the normal form method combined with the center manifold theorem. Finally, some numerical simulations to verify the theoretical results were also conducted. The work provides an applicable reference for control of spruce budworms.
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Key words:
- spruce budworm population model /
- equilibrium /
- stability /
- Hopf bifurcation
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[1] FLEMING R A, SHOEMAKER C A. Evaluating models for spruce budworm-forest management: comparing output with regional field data[J]. Ecological Applications: a Publication of the Ecological Society of America,1992,2(4): 460-477. [2] MAGNUSSEN S, ALFARO R I, BOUDEWYN P. Survival-time analysis of white spruce during spruce budworm defoliation[J]. Silva Fenncia,2005,39(2): 177-189. [3] NIE Z Y, MACLEAN D A, TAYLOR A R. Forest overstory composition and seedling height influence defoliation of understory regeneration by spruce budworm[J]. Forest Ecology and Management,2018,409: 353-360. [4] ROYAMA T. Population dynamics of the spruce budworm choristoneura fumiferana[J]. Ecological Monographs,1984,54(4): 429-462. [5] MEIGS G W, KENNEDY R E, GRAY A N, et al. Spatiotemporal dynamics of recent mountain pine beetle and western spruce budworm outbreaks across the Pacific Northwest Region, USA[J]. Forest Ecology and Management,2015,339: 71-86. [6] ALFARO R I, BERG J, AXELSON J. Periodicity of western spruce budworm in Southern British Columbia, Canada[J]. Forest Ecology and Management,2014,315: 72-79. [7] NEALIS V G, TURNQUIST R, MORIN B, et al. Baculoviruses in populations of western spruce budworm[J]. Journal of Invertebrate Pathology,2015,127: 76-80. [8] 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)) [9] LUDWIG D, JONES D D, HOLLING C S. Qualitative analysis of insect outbreak systems: the spruce budworm forest[J]. Journal of Animal Ecology,1978,47: 315-332. [10] LUDWIG D, ARONSON D G, WEINBERGER H F. Spatial patterning of the spruce budworm[J]. Journal of Mathematical Biology,1979,〖STHZ〗 8(3): 217-258. [11] RASMUSSEN A, WYLLER J, VIK J O. Relaxation oscillations in spruce-budworm interactions[J]. Nonlinear Analysis: Real World Applications,2011,12(1): 304-319. [12] HASSELL D C, ALLWRIGHT D J, FOWLER A C. A mathematical analysis of Jone’s site model for spruce budworm infestation[J]. Journal of Mathematical Biology,1999,38(5): 377-421. [13] SINGH M, EASTON A, CUI G, et al. A numerical study of the spruce budworm reaction diffusion equation with hostile boundaries[J]. Natural Resource Modeling,2000,13(4): 535-549. [14] VAIDYA N K, WU J H. Modeling spruce budworm population revisited: impact of physiological structure on outbreak control[J]. Bulletin of Mathematical Biology,2008,70(3): 769-784. [15] XU X F, WEI J J. Bifurcation analysis of a spruce budworm model with diffusion and physiological structures[J]. Journal of Differential Equations,2017,262(10): 5206-5230. [16] 魏俊杰, 王洪滨, 蒋卫华. 时滞微分方程的分支理论及应用[M]. 北京: 科学出版社, 2012.(WEI Junjie, WANG Hongbin, JIANG Weihua. Bifurcation Theory and Application of Delay Differential Equation [M]. Beijing: Science Press, 2012.(in Chinese)) [17] WAN A Y, ZOU X F. Hopf bifurcation analysis for a model of genetic regulatory system with delay[J]. Journal of Mathematical Analysis and Applications,2009,356(2): 464-476.
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