Oto-Escaff-Cisternas模型的定性分析
The Qualitative Analysis of Oto-Escaff-Cisternas Model
摘要: 考虑Oto-Escaff-Cisternas植物–食草动物模型的动力学性质。首先，讨论了模型解的正性及有界性，分析对应平衡点处雅可比矩阵的特征值，给出了模型平衡点的拓扑类型；其次，应用分支分析的理论及解析方法，讨论了该模型的鞍结点分支和Hopf分支，得到了正平衡点的稳定性和Hopf分支的存在条件；再次，通过计算第一李雅普洛夫系数，证明了该系统会发生亚临界Hopf分支，并且在正平衡点附近产生了唯一一个不稳定的极限环。最后，对所获得的理论结果给出了数值模拟。
Abstract: In this paper, the dynamics properties of the Oto-Escaff-Cisternas plant-herbivore model are con-sidered. Firstly, the positiveness and boundedness of the model solution are discussed, the eigen-values of the Jacobian matrix at the corresponding equilibrium point are analyzed, and the topolog-ical type of the model equilibrium point is given. Secondly, via the analytical method of bifurcation analysis, the saddle node bifurcation and Hopf bifurcation of the model are discussed, and the sta-bility of the positive equilibrium point and the existence conditions of the Hopf bifurcation are ob-tained. Furthermore, by calculating the first Lyapunov coefficient, it is proved that the system will have a subcritical Hopf bifurcation and produce the only unstable limit ring near the positive equi-librium point. Finally, some numerical simulations were carried out to illustrate our theoretical re-sults.
文章引用：胡亚斌, 高红亮. Oto-Escaff-Cisternas模型的定性分析[J]. 应用数学进展, 2023, 12(11): 4862-4873. https://doi.org/10.12677/AAM.2023.1211479

参考文献

[1] Stavi, I., Zaady, E., Gusarov, A., et al. (2021) Dead Shrub Patches as Ecosystem Engineers in Degraded Drylands. Journal of Geographical Sciences, 31, 1187-1204. [Google Scholar] [CrossRef] 
[2] Cheng, Y., Zhan, H., Yang, W., et al. (2021) An Ecohydrological Perspective of Reconstructed Vegetation in the Semi-Arid Region in Drought Seasons. Agricultural Water Management, 243, 106488. [Google Scholar] [CrossRef] 
[3] Zhang, C.H. and Li, Z.Z. (2014) Dynamics in a Diffusive Plant-Herbivore Model with Toxin-Determined Functional Response. Computers & Mathematics with Applications, 67, 1439-1449. [Google Scholar] [CrossRef] 
[4] Elsayed, E.M. and Din, Q. (2019) Period-Doubling and Neimark-Sacker Bifurcations of Plant-Herbivore Models. Advances in Difference Equations, 2019, Article No. 271. [Google Scholar] [CrossRef] 
[5] Din, Q., Shabbir, M.S., Khan, M.A., et al. (2019) Bifurcation Analysis and Chaos Control for a Plant-Herbivore Model with Weak Predator Functional Response. Journal of Biologi-cal Dynamics, 13, 481-501. [Google Scholar] [CrossRef] [PubMed]
[6] Din, Q., Elsadany, A.A. and Khalil, H. (2017) Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model. Discrete Dynamics in Na-ture and Society, 2017, Article ID 6312964. [Google Scholar] [CrossRef] 
[7] Khan, M.S., Samreen, M., Ozair, M., et al. (2022) On the Qualitative Study of a Two-Trophic Plant-Herbivore Model. Journal of Mathematical Biology, 85, Article No. 34. [Google Scholar] [CrossRef] [PubMed]
[8] Klausmeier, C.A. (1999) Regular and Irregular Patterns in Semi-arid Vegetation. Science, 284, 1826-1828. [Google Scholar] [CrossRef] [PubMed]
[9] Sherratt, A.J. (2005) An Analysis of Vegetation Stripe For-mation in Semi-Arid Landscapes. Journal of Mathematical Biology, 51, 183-197. [Google Scholar] [CrossRef] [PubMed]
[10] Sherratt, A.J. (2011) Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semi-Arid Environments II: Patterns with the Largest Possible Propagation Speeds. Proceed-ings: Mathematical, Physical and Engineering Sciences, 467, 3272-3294. [Google Scholar] [CrossRef] 
[11] Fernandez-Oto, C., Escaff, D. and Cisternas, J. (2019) Spiral Vegeta-tion Patterns in High-Altitude Wetlands. Ecological Complexity, 37, 38-46. [Google Scholar] [CrossRef] 
[12] Wiggins, S., Wiggins, S. and Golubitsky, M. (2003) Introduc-tion to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York.
[13] Pirayesh, B., Pazirandeh, A. and Akbari, M. (2016) Local Bifurcation Analysis in Nuclear Reactor Dynamics by Sotomayor’s Theorem. Annals of Nucle-ar Energy, 94, 716-731. [Google Scholar] [CrossRef] 

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