三种群捕食-食饵模型的分形特征与控制

摘要/Abstract

摘要：

种群数量变化规律是动物生态学、资源管理学的核心问题之一，通过研究种群数量的变化，可以有效掌握种群动态、生活习性，这对于合理利用资源、保护生态有着重要的意义.该文从分形理论的角度讨论了三种群的捕食-食饵模型.研究了三维捕食-食饵模型的分形行为，通过研究Julia分形集的性质，探讨了使该模型趋于稳定的条件，并引入反馈控制项，实现了模型由不稳定向稳定的转换.此外，分析了单一种群数量变化对另外两个种群数量及生态系统的影响.最后，构造不同参数的非线性耦合项，将响应系统转变为目标系统，实现不同系统之间的同步.仿真证实了这些方法的有效性.

关键词: 捕食-食饵模型, 稳定, Julia集, 反馈控制, 同步

Abstract:

The law of population quantity change is one of the key problems in Animal Ecology and Resource Management. By studying the change of population quantity, we can effectively grasp the population dynamics and living habits, which is of great significance for the rational utilization of resources and the protection of ecology. In this paper, we discuss the three-species predator-prey model from the point of fractal theory. We construct the Julia set of 3D predatorprey model. By studying the property of Julia sets, we discuss the conditions for the model to be stable, and take feedback control terms to realize the transformation from instability to stability. In addition, the effects of single population changes on the other two populations and ecosystems were analyzed. Finally, the nonlinear coupling terms with different parameters are constructed, the response system is transformed into the target system, and the synchronization between different systems is realized. Simulation results show the effectiveness of the method.

Key words: Predator-prey model, Stability, Julia set, Feedback control, Synchronization

中图分类号:

邵长旭,刘树堂. 三种群捕食-食饵模型的分形特征与控制[J]. 数学物理学报, 2019, 39(4): 951-962.

Changxu Shao,Shutang Liu. Fractal Feature and Control of Three-Species Predator-Prey Model[J]. Acta mathematica scientia,Series A, 2019, 39(4): 951-962.

图/表 6

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参考文献 20

1	Lotka A J . Contribution to the theory of periodic reaction. The Journal of Physical Chemistry, 1910, 14: 271- 274 doi: 10.1021/j150111a004
2	Goel N S , Maitra S C , Montroll E W . On the volterra and other nonlinear models of interacting populations. Reviews of Modern Physics, 1971, 43: 231- 276 doi: 10.1103/RevModPhys.43.231
3	Lotka A J . Analytical note on certain rhythmic relations in organic systems. Proc Natl Acad Sci USA, 1920, 6 (7): 410- 415 doi: 10.1073/pnas.6.7.410
4	Lotka A J . Elements of Physical Biology. New York: Dover Publications, 1956
5	Volterra V . Variazione e fluttuazioni de numero d'individui in specie animali conviventi. Mem Acad Lincei, 1926, 2: 31- 113
6	Volterra V . Variations and fluctuations of the number of individuals in animal species living together. J Cons Int Explor Mer, 1928, 3 (1): 351- 400
7	Egerton F N . Modeling nature:episodes in the history of population ecology, by Sharon E. Kingsland. American Historical Review, 1985, 92 (2): 183- 200
8	Gilpin M E . Do hares eat lynx?. The American Naturalist, 1973, 107: 727- 730 doi: 10.1086/282870
9	Jost C , Devulder G , Vucetich J A , et al. The wolves of Isle Royale display scale-invariant satiation and ratio-dependent predation on moose. Journal of Animal Ecology, 2005, 74: 809- 816 doi: 10.1111/j.1365-2656.2005.00977.x
10	Yuan Lou , Wei Mingni , Shoji Yotsutani . On a limiting system in the Lotka-Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems-Series A, 2004, 10: 435- 458
11	Liu Shaoping , Liao Xiaoxin . Permanence and persistence of time varying Lotka-Volterra systems. Acta Mathematica Scientia, 2006, 26B (1): 49- 58
12	傅金波, 陈兰荪. 基于生态环境和反馈控制的多种群竞争系统的正周期解. 数学物理学报, 2017, 37A (3): 553- 561 doi: 10.3969/j.issn.1003-3998.2017.03.015
	Fu Jinbo , Chen Lansun . Positive periodic solution of multiple species comptition system with ecological environment and feedback controls. Acta Mathematica Scientia, 2017, 37A (3): 553- 561 doi: 10.3969/j.issn.1003-3998.2017.03.015
13	Liu Xianning , Chen Lansun . Complex dynamics of Holling type Ⅱ Lotka-Volterra predator-prey system with impulsive perturbations on the predator. Chaos, Solitons & Fractals, 2003, 16 (2): 311- 320
14	徐瑞, 陈兰荪. 具有时滞和基于比率的三种群捕食系统的持久性与全局渐近稳定性. 系统科学与数学, 2001, 4: 204- 212 doi: 10.3969/j.issn.1000-0577.2001.02.010
	Xu Rui , Chen Lansun . Persistence and global stability for three-species ratio-dependent predator-prey system with time delays. Acta Mathematica Scientia, 2001, 4: 204- 212 doi: 10.3969/j.issn.1000-0577.2001.02.010
15	Lu Zhonghua , Chen Lansun . Analysis of the periodic Lotka-Volterra three-species mixed model. Pure and Applied Mathematics, 1995, 2: 81- 85
16	Fu Shengmao , Gao Haiyan , Cui Shangbin . Uniform boundedness and stability of solutions to the threespecies Lotka-Volterra competition model with self and cross-diffusion. Chinese Ann Math Ser A, 2006, 27 (3): 345- 356
17	Levin M . A Julia set model of field-directed morphogenesis:developmental biology and artificial life. Computer Applications in the Biosciences, 1994, 10: 85- 105
18	Sun Yuanyuan , Xu Rudan , Chen Lina , et al. Image compression and encryption scheme using fractal dictionary and Julia set. IET Image Processing, 2015, 9 (3): 173- 183 doi: 10.1049/iet-ipr.2014.0224
19	Shudo A , Ishii Y , Lkeda K S . Julia set describes quantum tunnelling in the presence of chaos. Journal of Physics A:Mathematical and Genera, 2002, 35: 225- 231 doi: 10.1088/0305-4470/35/17/101
20	Sun Weihua , Zhang Yongping , Zhang Xin . Fractal analysis and control in the predator-prey model. International Journal of Computer Mathematics, 2017, 94 (4): 737- 746 doi: 10.1080/00207160.2015.1130825

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