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山东大学学报（理学版） ›› 2017, Vol. 52 ›› Issue (1): 88-97.doi: 10.6040/j.issn.1671-9352.0.2016.286

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一类三种群食物链模型中交错扩散引起的Turing不稳定

张道祥1,2, 赵李鲜1, 胡伟1   

  1. 1. 安徽师范大学数学计算机科学学院, 安徽 芜湖 241002;2. 赫尔辛基大学数学与统计学院, 芬兰 赫尔辛基 00014
  • 收稿日期:2016-06-22 出版日期:2017-01-20 发布日期:2017-01-16
  • 作者简介:张道祥(1979— ), 男, 博士, 副教授, 研究方向为微分方程理论及其应用.E-mail:18955302433@163.com
  • 基金资助:
    国家自然科学基金青年项目资助项目(11302002)

Turing instability induced by cross-diffusion in a three-species food chain model

ZHANG Dao-xiang1,2, ZHAO Li-xian1, HU Wei1   

  1. 1. School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241002, Anhui, China;
    2. Department of Mathematics and Statistics, University of Helsinki, Helsinki 00014, Finland
  • Received:2016-06-22 Online:2017-01-20 Published:2017-01-16

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摘要： 研究了一类三种群食物链模型的强耦合交错扩散系统。 首先通过构造Lyapunov函数证明唯一的正平衡点在ODE系统下是全局渐近稳定的, 当交错扩散系数均为零时, 唯一的正平衡点仍是全局渐近稳定的。 当引入交错扩散时, 正平衡点则变得不稳定。 利用Routh-Hurwitz准则和Descartes符号法则证明了大的交错扩散系数(k21或k32足够大时)可以导致平衡点由原来的稳定变得不稳定。 最后利用数学软件Matlab 对我们的结果进行数值模拟, 得到了不同类型的Turing斑图, 包括六边形、条状以及二者共存的斑图。

关键词: 食物链模型, Turing不稳定, Turing斑图, 交错扩散

Abstract: This paper considers a strong coupled cross-diffusion system about a three-species food chain model. We first prove that the unique positive equilibrium solution is globally linearly stable for the ODE system and remains globally linearly stable when the reaction-diffusion system without cross-diffusion by constructing Lyapunov functions. Then we use the Routh-Hurwitz criterion and Descartes' rule to illustrate that the unique positive equilibrium solution becomes linearly unstable only when the cross-diffusion plays a role in this reaction-diffusion system. Finally, numerical simulations are performed to test our theoretical results by means of Matlab. We can obtain different types of patterns including spotted, striped and mixture patterns.

Key words: cross-diffusion, turing instability, turing pattern, food chain model

中图分类号: 

  • O175.21
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