华东师范大学学报（自然科学版） >

2018 , Vol. 2018 >Issue 4: 9 - 22,31

DOI: https://doi.org/10.3969/j.issn.1000-5641.2018.04.002

数学

具有非线性收获效应的捕食者-食饵系统的空间Turing斑图

  • 张道祥 ,
  • 孙光讯 ,
  • 胡伟 ,
  • 凯歌
展开
  • 1. 安徽师范大学 数学与统计学院, 安徽 芜湖 241002;
    2. 北京工业大学 机械工程与应用电子技术学院, 北京 100124
张道祥,男,博士,副教授,主要研究方向为微分方程理论及其应用.E-mail:zdxiang@ahnu.edu.cn

收稿日期: 2017-09-20

  网络出版日期: 2018-07-19

基金资助

国家自然科学基金青年项目（11302002）；安徽师范大学2017年研究生科研创新与实践项目（2017cxsj040）

收起

Spatial Turing pattern of a predator-prey system with nonlinear harvesting effect

  • ZHANG Dao-xiang ,
  • SUN Guang-xun ,
  • HU Wei ,
  • KAI Ge
Expand
  • 1. School of Mathematics and Statistics, Anhui Normal University, Wuhu Anhui 241002, China;
    2. College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China

Received date: 2017-09-20

  Online published: 2018-07-19

Fold

摘要

研究了一类带有非线性食饵收获效应的捕食者-食饵系统的Turing斑图的生成及选择问题.首先利用稳定性理论给出了由交叉扩散项引起的Turing不稳定条件和分支理论分析得到了系统Turing斑图的存在区域，然后运用多重尺度分析法推导了系统的振幅方程，给出了Turing斑图的选择结果.最后利用Matlab软件对系统Turing斑图的生成和选择结果进行了数值模拟.结果展示了系统有丰富的Turing斑图，如点状、条状以及二者共存.

关键词： 捕食者-食饵系统; Hopf分支; Turing斑图; 振幅方程

本文引用格式

张道祥 , 孙光讯 , 胡伟 , 凯歌 . 具有非线性收获效应的捕食者-食饵系统的空间Turing斑图[J]. 华东师范大学学报（自然科学版）, 2018 , 2018(4) : 9 -22,31 . DOI: 10.3969/j.issn.1000-5641.2018.04.002

Abstract

We study the formation and selection of Turing patterns for a class of predator-prey systems with nonlinear harvesting effect. Firstly, the conditions of Turing instability induced by cross-diffusion terms are given by stability theory, and the existence region of Turing patterns of the system are obtained by bifurcation theory. Secondly, the amplitude equations of the system are derived using the multi-scales analysis method, and the selection results of Turing patterns are given. Finally, Matlab is used to simulate the pattern formation and selection results of the system. The results show that the system has rich Turing patterns, such as spot, stripe, and coexistence of the two types.

Key words： predator-prey system; Hopf bifurcation; Turing patterns; amplitude equation

参考文献

[1] ZADORIN A S, RONDELEZ Y, GINES G, et al. Synthesis and materialization of a reaction-diffusion French flag pattern[J]. Nature Chemistry, 2017, 9(10):990-996.
[2] 李新政, 白占国, 李燕, 等. 双层非线性耦合反应扩散系统中复杂Turing斑图[J]. 物理学报, 2013, 62(22):45-51.
[3] ZHAO H Y, HUANG X X, ZHANG X B. Turing instability and pattern formation of neural networks with reaction-diffusion terms[J]. Nonlinear Dynamics, 2014, 76(1):115-124.
[4] SONG Y L, YANG R, SUN G Q. Pattern dynamics in a Gierer-Meinhardt model with a saturating term[J]. Applied Mathematical Modelling, 2017, 46:476-491.
[5] SUN G Q. Mathematical modeling of population dynamics with Allee effect[J]. Nonlinear Dynamics, 2016, 85(1):1-12.
[6] WANG X L, WANG W D, ZHANG G H. Vegetation pattern formation of a water-biomass model[J]. Commun Nonlinear Sci Numer Simulat, 2017, 42:571-584.
[7] 张道祥, 赵李鲜, 孙光讯, 等. 一类带负交叉扩散项二维系统的空间Turing斑图[J]. 吉林大学学报(理学版), 2017, 55(3):537-546.
[8] SUN G Q, WANG C H, WU Z Y. Pattern dynamics of aGierer-Meinhardt model with spatial effects. Nonlinear Dynamics,2017, 88(2):1385-1396.
[9] GUIN L N. Existence of spatial patterns in a predator-preymodel with self-and cross-diffusion. Applied Mathematics andComputation, 2014, 226:320-335.
[10] 张道祥, 赵李鲜, 胡伟.一类三种群食物链模型中交错扩散引起的Turing不稳定.山东大学学报(理学版), 2017, 52(1):88-97.
[11] SAMBATH M, BALACHANRAN K, SUVINTHRA M. Stability and Hopfbifurcation of a diffusive predator-prey model with hyperbolicmortality. Complexity, 2016, 21:34-43.
[12] XIAO M, CAO J D. Hopf bifurcation and non-hyperbolic equilibrium ina ratio-dependent predator-prey model with linear harvesting rate:Analysis and computation. Mathematical and Computer Modelling,2009, 50:360-379.
[13] ZHANG F R, LI Y. Stability and Hopf bifurcation of adelayed-diffusive predator-prey model with hyperbolic mortality andnonlinear prey harvesting. Nonlinear Dynamics, 2017, 88(2):1397-1412.
[14] LI Y. Dynamics of a delayed diffusive predator-prey modelwith hyperbolic mortality. Nonlinear Dynamics, 2016, 85(4):2425-2436.
[15] 欧阳颀. 非线性科学与斑图动力学导论. 北京:北京大学出版社, 2010.
[16] KUZNETSOV Y A. Elements of applied bifurcation theory. 2nd ed. New York:Springer, 1995.
Options
文章导航

/

〈 〉