Chaos and pattern transformation of a spatiotemporal discrete predator-prey system

doi:10.3969/j.issn.1000-1565.2023.04.002

Abstract

Abstract: The complex spatiotemporal dynamics of a kind of spatiotemporal discrete Leslie-Gower type predator-prey system is discussed. The bifurcation theorem is used to analyze the conditions under which the system undergoes flip bifurcation, Neimark-Sacker bifurcation and Turing bifurcation near the immobility point. The bifurcation process of the system is demonstrated by the bifurcation diagram and the maximum Lyapunov exponent, the pattern transition on the route chaos triggered by the bifurcation is revealed by the changes of spatial amplitude and space-time development. The results show that the pattern indicate a similar period doubling cascade process on the route chaos caused by flip bifurcation, and the system goes through a change from frozen chaos to defect turbulence. The pattern is mainly shaped like annular and spiral waves on the route chaos caused by the Neimark-Sacker bifurcation, and the pattern formed by Turing instability on the periodic and quasi-periodic attractors displays the ordered spatial and temporal banded structure. The pattern formed on the chaotic attractor shows the spatial amplitude changes- DOI:10.3969/j.issn.1000-1565.2023.04.002一类时空离散捕食系统的混沌与斑图转变类维倩,郭丰路,黄头生(华北电力大学工程生态学与非线性科学研究中心,北京 102206)摘要:探讨了一类时空离散Leslie-Gower型捕食系统的复杂时空动力学行为.运用分岔定理分析了该系统在不动点附近发生倍周期分岔、Neimark-Sacker分岔和图灵分岔的条件.通过分岔图和最大李雅普诺夫指数展示了系统的分岔过程,借助空间振幅和时空发展的变化揭示了由分岔引发混沌路径上的斑图转变规律.结果表明:在倍周期分岔引发的混沌路径上,斑图呈现类似的周期加倍级联过程,系统经历了从冻结混沌到缺陷湍流的变化;在Neimark-Sacker分岔引发的混沌路径上,斑图以环状和螺旋波状为主,在周期和拟周期吸引子上经图灵失稳形成的斑图仍会呈现有序的时空带状结构,在混沌吸引子上形成的斑图虽然呈现无序的空间振幅变化,但在时空发展变化中会呈现某种整体有序局部混乱的湍流状态.关键词:时空离散;捕食系统;分岔;混沌路径;斑图自组织中图分类号:O175.13 文献标志码:A 文章编号:1000-1565(2023)04-0346-11Chaos and pattern transformation of a spatiotemporal discrete predator-prey systemLEI Weiqian,GUO Fenglu,HUANG Tousheng(Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing 102206, China)Abstract: The complex spatiotemporal dynamics of a kind of spatiotemporal discrete Leslie-Gower type predator-prey system is discussed. The bifurcation theorem is used to analyze the conditions under which the system undergoes flip bifurcation, Neimark-Sacker bifurcation and Turing bifurcation near the immobility point. The bifurcation process of the system is demonstrated by the bifurcation diagram and the maximum Lyapunov exponent, the pattern transition on the route chaos triggered by the bifurcation is revealed by the changes of spatial amplitude and space-time development. The results show that the pattern indicate a similar period doubling cascade process on the route chaos caused by flip bifurcation, and the system goes through a change from frozen chaos to defect turbulence. The pattern is mainly shaped like annular and spiral waves on the route chaos caused by the Neimark-Sacker bifurcation, and the pattern formed by Turing instability on the periodic and quasi-periodic attractors displays the ordered spatial and temporal banded structure. The pattern formed on the chaotic attractor shows the spatial amplitude changes- 收稿日期:2022-11-28 基金项目:国家自然科学基金青年基金资助项目(11802093);国家水体污染控制与治理科技重大专项(2017ZX07101002) 第一作者:类维倩(1997—),女,山东泰安人,华北电力大学在读硕士研究生,主要从事非线性动力学方向研究.E-mail: 120202209059@ncepu.edu.cn 通信作者:黄头生(1985—),男,江西吉水人,华北电力大学副教授,主要从事非线性动力学与斑图动力学方向研究.E-mail: tous_huang@ncepu.edu.cn第4期类维倩等:一类时空离散捕食系统的混沌与斑图转变of disorder, however, some turbulence state with overall order and local chaos appears in the changes of space-time development diagram.

Key words: space-time discrete, predator-prey system, bifurcation, route to chaos, self-organization of patterns

CLC Number:

O175.13

LEI Weiqian,GUO Fenglu,HUANG Tousheng. Chaos and pattern transformation of a spatiotemporal discrete predator-prey system[J]. Journal of Hebei University(Natural Science Edition), 2023, 43(4): 346-356.

References

[1] RIETKERK M, VAN DE KOPPEL J. Regular pattern formation in real ecosystems[J]. Trends Ecol Evol, 2008, 23(3): 169-175. DOI: 10.1016/j.tree.2007.10.013.
[2] BASCOMPTE J, SOLÉR V. Rethinking complexity: modelling spatiotemporal dynamics in ecology[J]. Trends Ecol Evol, 1995, 10(9): 361-366. DOI: 10.1016/S0169-5347(00)89134-X.
[3] PARROTT L. Measuring ecological complexity[J]. Ecol Indic, 2010, 10(6): 1069-1076. DOI: 10.1016/j.ecolind.2010.03.014.
[4] TURING A. The chemical basis of morphogenesis[J]. Bull Math Biol, 1990, 52(1/2): 153-197. DOI: 10.1016/s0092-8240(05)80008-4.
[5] GUIN L N, MANDAL P K. Effect of prey refuge on spatiotemporal dynamics of the reaction–diffusion system[J]. Comput Math Appl, 2014, 68(10): 1325-1340. DOI: 10.1016/j.camwa.2014.08.025.
[6] PUNITHAN D, KIM D K, MCKAY R. Spatio-temporal dynamics and quantification of daisyworld in two-dimensional coupled map lattices[J]. Ecol Complex, 2012, 12: 43-57. DOI: 10.1016/j.ecocom.2012.09.004.
[7] SHAO Y Y, MENG Y, XU X Y. Turing instability and spatiotemporal pattern formation induced by nonlinear reaction cross-diffusion in a predator–prey system with allee effect[J]. Mathematics, 2022, 10(9): 1500. DOI: 10.3390/math10091500.
[8] ZHU M, LI J, LIAN X Z. Pattern dynamics of cross diffusion predator–prey system with strong allee effect and hunting cooperation[J]. Mathematics, 2022, 10(17): 3171. DOI: 10.3390/math10173171.
[9] GHORAI S, PORIA S. Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity[J]. Chaos Solitons Fractals, 2016, 91: 421-429. DOI: 10.1016/j.chaos.2016.07.003.
[10] 黄头生. 基于耦合映像格子的生态学时空复杂性研究[D].北京: 华北电力大学, 2016.
[11] SHANG Z C, QIAO Y H. Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey[J]. Nonlinear Anal Real World Appl, 2022, 64: 103453. DOI: 10.1016/j.nonrwa.2021.103453.
[12] LESLIE P H. Some further notes on the use of matrices in population mathematics[J]. Biometrika, 1948, 35(3/4): 213-245. DOI: 10.1093/biomet/35.3-4.213.
[13] GUCKENHEIMER J, HOLMES P, SLEMROD M. Nonlinear oscillations dynamical systems, and bifurcations of vector fields[J]. J Appl Mech, 1984, 51(4): 947. DOI: 10.1115/1.3167759.
[14] ZHANG H Y, MA S N, HUANG T S, et al. Complex dynamics on the routes to chaos in a discrete predator-prey system with crowley-martin type functional response[J]. Discrete Dyn Nat Soc, 2018, 2018: 1-18. DOI: 10.1155/2018/2386954.
[15] 吕宁.双脉冲阶段结构的种群系统动力学特性[J].山东大学学报(理学版), 2021, 56(12): 100-110. DOI: 10.6040/j.issn.1671-9352.0.2021.179.
[16] WEN G L. Criterion to identify Hopf bifurcations in maps of arbitrary dimension[J]. Phys Rev E, 2005, 72(2): 026201. DOI: 10.1103/physreve.72.026201.
[17] HAN Y T, HAN B, ZHANG L, et al. Turing instability and wave patterns for a symmetric discrete competitive Lotka-Volterra system[J]. WSEAS Trans Math, 2011, 10(5): 181-189.
[18] LU Y, LIU S. Threshold dynamics of a predator-prey model with age-structured prey [J]. Advances in Difference Equations, 2018(1): 164.
[19] KANEKO K. Pattern dynamics in spatiotemporal chaos[J]. Phys D Nonlinear Phenom, 1989, 34(1/2): 1-41. DOI: 10.1016/0167-2789(89)90227-3.
[20] ZHANG Y Q, HE Y, WANG X Y. Spatiotemporal chaos in mixed linear–nonlinear two-dimensional coupled logistic map lattice[J]. Phys A Stat Mech Appl, 2018, 490: 148-160. DOI: 10.1016/j.physa.2017.07.019.
[21] ZHANG Y Q, WANG X Y. Spatiotemporal chaos in mixed linear–nonlinear coupled logistic map lattice[J]. Phys A Stat Mech Appl, 2014, 402: 104-118. DOI: 10.1016/j.physa.2014.01.051.
[22] NAGANO S, MAEDA Y. Phase transitions in predator-prey systems[J]. Phys Rev E Stat Nonlin Soft Matter Phys, 2012, 85(1): 011915. DOI: 10.1103/PhysRevE.85.011915.
[23] LIU J N, QI Q, LIU B, et al. Pest control switching models with instantaneous and non-instantaneous impulsive effects[J]. Math Comput Simul, 2023, 205: 926-938. DOI: 10.1016/j.matcom.2022.10.027. (