一类生物入侵模型的解析极限环解
Analytical Limit Cycle Solutions of Abiological Invasion Mode
摘要: 本文利用平方广义谐波函数摄动法研究了一类具有对流–反应–扩散类型的单种群生物入侵模型，求得了该系统极限环解的近似表达式，并得到了极限环幅值与控制参数之间的关系。利用此关系预测了系统在不同参数下的极限环振幅，通过将本方法计算所得结果与数值结果进行比较，验证了所得结果的有效性和可靠性。该结果表明平方广义谐波函数摄动法亦可推广至一类无限维动力系统Hopf分岔的解析定量分析，为无限维动力系统的研究相提供了新的思路和参考方法。
Abstract: In this paper, a single species biological invasion model with convection diffusion reaction type is studied by using the square generalized harmonic function perturbation method. The approximate expression of the limit cycle solution of the system is obtained, and the relationship between the limit cycle amplitudes and the control parameters is also obtained. Based on this relationship, the limit cycle amplitudes of the system under different parameters are predicted. By comparing the results calculated by this method with the numerical results, the effectiveness and reliability of the results are verified. The medium and long-term prediction of the population density development of a class of species and the effective control of Hopf bifurcation behavior are realized, which provides a new idea and reference method for the analytical and quantitative study of Hopf bifurcation of a class of infinite dimensional dynamic systems.
文章引用：李亚宁, 黄雨欣, 李震波. 一类生物入侵模型的解析极限环解[J]. 应用数学进展, 2024, 13(7): 3407-3416. https://doi.org/10.12677/aam.2024.137326

参考文献

[1] 赵斌. 生物数学的起源与形成[D]: [博士学位论文]. 西安: 西北大学, 2011.
[2] 周玉梅. 浅谈生物数学的发展及应用[J]. 课程教育研究, 2018(14): 181-182.
[3] 陆秋琴, 黄光球. 捕食-被食动力学优化算法[J]. 系统仿真学报, 2018, 30(10): 3975-3984.
[4] Spiegler, A., Kiebel, S.J., Atay, F.M. and Knösche, T.R. (2010) Bifurcation Analysis of Neural Mass Models: Impact of Extrinsic Inputs and Dendritic Time Constants. NeuroImage, 52, 1041-1058. [Google Scholar] [CrossRef] [PubMed]
[5] Basu, A., Petre, C. and Hasler, P.E. (2010) Dynamics and Bifurcations in a Silicon Neuron. IEEE Transactions on Biomedical Circuits and Systems, 4, 320-328. [Google Scholar] [CrossRef] [PubMed]
[6] Touboul, J., Wendling, F., Chauvel, P. and Faugeras, O. (2011) Neural Mass Activity, Bifurcations, and Epilepsy. Neural Computation, 23, 3232-3286. [Google Scholar] [CrossRef] [PubMed]
[7] 唐文艳, 焦建军. 具脉冲扩散效应的Gomportz种群动力学模型研究[J]. 湘潭大学自然科学学报, 2013, 35(2): 10-13.
[8] 任善静, 焦建军, 李利梅. 具脉冲与基因单点突变效应的种群动力学模型[J]. 信阳师范学院学报(自然科学版), 2016, 29(4): 494-496, 500.
[9] 焦建军, 曾熙轩, 李利梅. 毒素脉冲输入与种群脉冲出生切换阶段结构单种群动力学模型研究[J]. 数学杂志, 2018, 38(6):1066-1074.
[10] 焦建军, 李利梅. 污染环境下具瞬时与非瞬时脉冲出生单种群动力学模型研究[J]. 南京师大学报(自然科学版), 2018, 41(4): 1-6.
[11] Duan, D., Niu, B. and Wei, J. (2019) Local and Global Hopf Bifurcation in a Neutral Population Model with Age Structure. Mathematical Methods in the Applied Sciences, 42, 4747-4764. [Google Scholar] [CrossRef] 
[12] 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.
[13] Petrovskii, S. and Li, B. (2003) An Exactly Solvable Model of Population Dynamics with Density-Dependent Migrations and the Allee Effect. Mathematical Biosciences, 186, 79-91. [Google Scholar] [CrossRef] [PubMed]
[14] Sherratt, J.A. and Smith, M.J. (2008) Periodic Travelling Waves in Cyclic Populations: Field Studies and Reaction-Diffusion Models. Journal of the Royal Society Interface, 5, 483-505. [Google Scholar] [CrossRef] [PubMed]
[15] Sherratt, J.A. (2012) Numerical Continuation Methods for Studying Periodic Travelling Wave (Wavetrain) Solutions of Partial Differential Equations. Applied Mathematics and Computation, 218, 4684-4694. [Google Scholar] [CrossRef] 
[16] Wang, Q. and Huang, W. (2014) Limit Periodic Travelling Wave Solution of a Model for Biological Invasions. Applied Mathematics Letters, 34, 13-16. [Google Scholar] [CrossRef] 
[17] Liu, Y. and Li, J. (1990) Theory of Values of Singular Point in Complex Autonomous Differential System. Science China A, 33, 10-24.
[18] Haibo, C. and Yirong, L. (2004) Linear Recursion Formulas of Quantities of Singular Point and Applications. Applied Mathematics and Computation, 148, 163-171. [Google Scholar] [CrossRef] 
[19] Li, Z., Tang, J. and Cai, P. (2013) A Generalized Harmonic Function Perturbation Method for Determining Limit Cycles and Homoclinic Orbits of Helmholtz-Duffing Oscillator. Journal of Sound and Vibration, 332, 5508-5522. [Google Scholar] [CrossRef] 
[20] Li, Z., Tang, J. and Cai, P. (2015) Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-Van De Pol Oscillator. Qualitative Theory of Dynamical Systems, 15, 19-37. [Google Scholar] [CrossRef] 

为你推荐