一类捕食者-食饵共生互作离散耦合logistics模型的余维一分岔及Marotto混沌

数学物理学报 ›› 2026, Vol. 46 ›› Issue (5): 1932-1950.

一类捕食者-食饵共生互作离散耦合logistics模型的余维一分岔及Marotto混沌

郑凯玥¹, 刘娜², 余志恒¹^,^*()

¹ 西南交通大学数学学院成都 611756
² 成都工贸职业技术学院通识教育学院成都 611731

收稿日期:2025-07-16 修回日期:2025-10-25 出版日期:2026-10-26 发布日期:2026-09-07
通讯作者: 余志恒 E-mail:yuzhiheng9@163.com
基金资助:
国家自然科学基金(11701476);四川省自然科学基金(2023NSFSC0064)

Codimension-one Bifurcations and Marotto's Chaos in a Class of Discrete Coupled Predator-Prey Mutualistic Interaction Logistics Models

Kaiyue Zheng¹, Na Liu², Zhiheng Yu¹^,^*()

¹ School of Mathematics, Southwest Jiaotong University School, Chengdu 611756
² School of General Education, Chengdu Industry and Trade College, Chengdu 611731

Received:2025-07-16 Revised:2025-10-25 Online:2026-10-26 Published:2026-09-07
Contact: Zhiheng Yu E-mail:yuzhiheng9@163.com
Supported by:
NSFC(11701476);Sichuan Provincial Natural Science Foundation of China(2023NSFSC0064)

摘要/Abstract

摘要：

该文研究一类捕食者--食饵共生互作的离散时间logistics模型的分岔现象及Marotto混沌. 首先, 应用多项式完全判别系统理论求解对应的高阶半代数系统, 作者给出了各不动点的拓扑分类及其稳定性条件. 其次, 基于中心流形定理和分岔理论, 作者进一步得到了该系统在相应的非双曲情形下产生跨临界和翻转分岔的参数条件. 此外, 作者还证明了该系统具有回弹排斥子, 进而可产生Marotto意义下的混沌现象. 最后, 文中利用${\tt Matlab R2024a}$及${\tt Maple 2024}$数值模拟重构了该系统的分岔过程及混沌行为的演化轨迹, 并结合相应的Lyapunov指数进一步验证了前述理论结果的正确性.

关键词: 耦合logistics模型, 多项式完全判别系统, 分岔, Marotto 混沌, Lyapunov指数

Abstract:

In this paper, we investigate bifurcation phenomena and Marotto's chaos in a class of discrete-time predator-prey mutualistic interaction logistics models. Firstly, applying the theory of polynomial complete discrimination systems to solve the corresponding high-order semi-algebraic system, we determine and present the topological classifications of each fixed point alongside its stability conditions. Secondly, based on the center manifold theorem and bifurcation theory, we further establish the parameter conditions under which the system undergoes transcritical and flip bifurcations in the corresponding non-hyperbolic cases. Additionally, we prove that the system possess snap-back repellers, thereby exhibiting chaos in the sense of Marotto. Finally, utilizing ${\tt Matlab R2024a}$ and ${\tt Maple 2024}$ for numerical simulations, we reconstruct the system's bifurcation processes and the evolutionary trajectories of chaotic behavior. The corresponding Lyapunov exponents further validate the accuracy of the aforementioned theoretical results.

Key words: coupled logistics model, polynomial complete discrimination system, bifurcation, Marotto's chaos, Lyapunov exponents

中图分类号:

O178

郑凯玥, 刘娜, 余志恒. 一类捕食者-食饵共生互作离散耦合logistics模型的余维一分岔及Marotto混沌[J]. 数学物理学报, 2026, 46(5): 1932-1950.

Kaiyue Zheng, Na Liu, Zhiheng Yu. Codimension-one Bifurcations and Marotto's Chaos in a Class of Discrete Coupled Predator-Prey Mutualistic Interaction Logistics Models[J]. Acta mathematica scientia,Series A, 2026, 46(5): 1932-1950.

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参考文献 25

[1]	Rogers T L, Johnson B J, Munch S B. Chaos is not rare in natural ecosystems. Nat Ecol Evol, 2022, 6(8): 1105-1111 doi: 10.1038/s41559-022-01787-y pmid: 35760889
[2]	Sibly R M, Barker D, Denham M C, et al. On the regulation of populations of mammals, birds, fish, and insects. Science, 2005, 309(5734): 607-610 pmid: 16040705
[3]	Zhang M, Wang X, Tang S. Integrating dynamic models and neural networks to discover the mechanism of meteorological factors on Aedes population. PLoS Comput Biol, 2024, 20(9): Art 1012499
[4]	Andrén H, Liberg O. Numerical response of predator to prey: Dynamic interactions and population cycles in Eurasian lynx and roe deer. Ecol Monogr, 2024, 94(1): Art 1594
[5]	Hale K R S, Valdovinos F S. Ecological theory of mutualism: Robust patterns of stability and thresholds in two-species population models. Ecol Evol, 2021, 11(24): 17651-17671 doi: 10.1002/ece3.8453 pmid: 35003630
[6]	Spaak J W, Schreiber S J. Building modern coexistence theory from the ground up: the role of community assembly. Ecol Lett, 2023, 26(11): 1840-1861 doi: 10.1111/ele.14302 pmid: 37747362
[7]	Alves M T, Hilker F M. Hunting cooperation and Allee effects in predators. J Theor Biol, 2017, 419: 13-22 doi: S0022-5193(17)30057-7 pmid: 28174111
[8]	Hastings A, Powell T. Chaos in a three-species food chain. Ecology, 1991, 72(3): 896-903 doi: 10.2307/1940591
[9]	Perkins D M, Hatton I A, Gauzens B, et al. Consistent predator-prey biomass scaling in complex food webs. Nat Commun, 2022, 13(1): Art 4990 doi: 10.1038/s41467-022-32578-5 pmid: 36008387
[10]	Dong Y, Wu D, Shen C, et al. Influence of fear effect and predator-taxis sensitivity on dynamical behavior of a predator-prey model. Z Angew Math Phys, 2022, 73(1): Art 25 doi: 10.1007/s00033-021-01659-8
[11]	Mougi A. Predator interference and complexity-stability in food webs. Sci Rep-UK, 2022, 12(1): Art 2464
[12]	Valencia D E, Génin A, Rojas S, et al. The complexity of a "simple'' predator-prey system: non-trophic positive interactions generate unsuspected dynamics and dependencies. Web Ecol, 2025, 25(1): 103-120 doi: 10.5194/we-25-103-2025
[13]	López-Ruiz R, Fournier-Prunaret D. Complex behaviour in a discrete coupled logistic model for the symbiotic interaction of two species. Math Biosci Eng, 2004, 1(2): 307-324 doi: 10.3934/mbe.2004.1.307
[14]	López-Ruiz R, Fournier-Prunaret D. Indirect Allee effect, bistability and chaotic oscillations in a predator-prey discrete model of logistic type. Chaos Soliton Fract, 2005, 24(1): 85-101 doi: 10.1016/j.chaos.2004.07.018
[15]	López-Ruiz R, Fournier-Prunaret D. Periodic and chaotic events in a discrete model of logistic type for the competitive interaction of two species. Chaos Soliton Fract, 2009, 41(1): 334-347 doi: 10.1016/j.chaos.2008.01.015
[16]	Yang L, Hou X, Xia B. A complete algorithm for automated discovering of a class of inequality-type theorems. Sci China Ser F, 2001, 44(1): 33-49
[17]	Yang L, Xia B. Andreas D, Andreas S, Thomas S. Real solution classifications of parametric semi-algebraic systems.// Andreas D, Andreas S, Thomas S. Algorithmic Algebra and Logic, Proceedings of the A3L 2005. Norderstedt: Herstellung und Verlag, 2005, 281-289
[18]	张锦炎, 冯贝叶. 常微分方程几何理论与分支问题. 北京: 北京大学出版社, 2000
	Zhang J Y, Feng B Y. Geometric Theory of Ordinary Differential Equations and Bifurcation Problems. Beijing: Peking University Press, 2000
[19]	Kuznetsov Y A. Elements of Applied Bifurcation Theory. Fourth Edition. New York: Springer-Verlag, 2023
[20]	Carr J. Application of Centre Manifold Theory. New York: Springer-Verlag, 1981
[21]	Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Second Edition. New York: Springer-Verlag, 2003
[22]	Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983
[23]	Marotto F R. Snap-back repellers imply chaos in $\mathbb{R}^n$. J Math Anal Appl, 1978, 63(1): 199-223 doi: 10.1016/0022-247X(78)90115-4
[24]	Marotto F R. On redefining a snap-back repeller. Chaos Soliton Fract, 2005, 25(1): 25-28 doi: 10.1016/j.chaos.2004.10.003
[25]	Salman S M, Elsadany A A. On the bifurcation of Marotto's map and its application in image encryption. J Comput Appl Math, 2018, 328: 177-196 doi: 10.1016/j.cam.2017.07.010