您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

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

• • 上一篇    下一篇

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

PDF (PC)

394

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
[1] 李海侠,李艳玲. 一类带B-D反应项的食物链模型正解的稳定性和唯一性[J]. 山东大学学报(理学版), 2013, 48(9):103-110. LI Haixia, LI Yanling. Stability and uniqueness of positive solutions for a food chain model with B-D functional response [J]. Journal of Shandong University(Natural Science), 2013, 48(9):103-110.
[2] 杨斌, 王静. 具有 Holling Ⅳ 型功能性反应的非自治三种群食物链模型的周期解[J]. 东北师大学报(自然科学版), 2012, 44(1):10-15. YANG Bin, WANG Jing. Periodic solution of a nonautonomous three-species food-chain model with the Holling Ⅳ functional response[J]. Journal of Northeast Normal University(Natural Science), 2012, 44(1):10-15.
[3] 王育全, 刘来福. 具有 Monod-Haldane功能反应的一类食物链模型的动力学行为[J]. 数学物理学报, 2007, 27(1):79-89. WANG Yuquan, LIU Laifu. On the dynamics of a food chain with Monod-Haldane functional response[J]. Acta Mathematics Scientia, 2007, 27(1):79-89.
[4] LV S, ZHAO M. The dynamic complexity of a three species food chain model[J]. Chaos, Solitons and Fractals, 2008, 37(5):1469-1480.
[5] MCCANN K, YODZIS P. Bifurcation structure of a three-species food-chain model[J]. Theoretical population biology, 1995, 48(2):93-125.
[6] 屈菲. 带扩散项的三级营养食物链模型斑图的存在性[D]. 兰州:西北师范大学硕士学位论文, 2014. QU Fei. The existence of stationary patterns for a tritrophic food chain model with diffusion[D]. Lanzhou: Masters Degree Thesis of Northwest Normal University, 2014.
[7] 伏升茂, 温紫娟, 崔尚斌. 三种群食物链交错扩散模型的整体解[J]. 数学学报, 2007, 50(1):75-89. FU Shengmao, WEN Zijuan, CUI Shangbin. On global solutions for the three-species food-chain model with cross-diffusion[J]. Acta Mathematica Sinica, 2007, 50(1):75-89.
[8] PARSHAD R D, KUMARI N, KASIMOV A R, et al. Turing patterns and long-time behavior in a three-species food-chain model[J]. Mathematical Biosciences, 2014, 254(8):83-102.
[9] TANG Xiaosong, SONG Yongli. Cross-diffusion induced spatiotemporal patterns in a predator—prey model with herd behavior[J]. Nonlinear Analysis: Real World Applications, 2015, 24:36-49.
[10] TURING A M. The chemical basis of morphogenesis[J]. Philosophical Transactions of the Royal of London. Series B, Biological Sciences, 1952, 237(64):37-72.
[11] LING Zhi, ZHANG Lai, LIN Zhigui. Turing pattern formation in a predator—prey system with cross diffusion[J]. Applied Mathematical Modelling, 2014, 38(21):5022-5032.
[12] PETER Y H P, WANG M X. Strategy and stationary pattern in a three-species predator-prey model[J]. Journal of Differential Equations, 2004, 200(2):245-273.
[13] LI Jianjun, GAO Wenjie. A strongly coupled predator-prey system with modified Holling-Tanner functional response[J]. Computer and Mathematics with Applications, 2010, 60(7):1908-1916.
[14] LOTFI E M, MAZIANE M, HATTAF K, et al. Partial differential equations of an epidemic model with spatial diffusion[J]. International Journal of Partial Differential Equations, 2014(2014):186437-186437.6
[15] 胡文勇, 邵元智. 局域浓度调控扩散系数的次氯酸-碘离子-丙二酸系统图灵斑图形成中的反常扩散[J]. 物理学报, 2014, 63(23):238202. HU Wenyong, SHAO Yuanzhi. Anomalous diffusion in the formation of Turing pattern for the chlorine-iodine-malonic-acid system with a local concentration depended diffusivity [J]. Journal of Physics, 2014, 63(23): 238202.
[16] CHIA-Ven Pao. Nonlinear Parabolic and Elliptic Equations[M] , New York: Plenum Press, 1992.
[17] YE Qixiao, LI Zhengyuan, WANG Mingxin, et al. Introduction to reaction-diffusion equations[M]. Beijing: Science Press, 1990.
[18] 马知恩, 周义仓. 种群生态学的数学建模与研究[M]. 安徽: 安徽教育出版社, 1996. MA Zhien, ZHOU Yicang. Mathematical modeling and research of population ecology[M]. Anhui: Anhui Education Press, 1996.
[19] Hale J K. Ordinary differential equations[M]. FL:Krieger, Malabar, 1980.
[1] 付娟,张睿,王彩军,张婧. 具有Beddington-DeAngelis功能反应项的捕食-食饵扩散模型的稳定性[J]. 山东大学学报（理学版）, 2016, 51(11): 115-122.
[2] 张丽娜,吴守妍. 修正的Leslie-Gower捕食者-食饵#br# 扩散模型解的整体性态[J]. 山东大学学报（理学版）, 2014, 49(1): 86-91.
[3] 李海侠1,2,李艳玲1. 一类带B-D反应项的食物链模型正解的稳定性和惟一性[J]. J4, 2013, 48(09): 103-110.
[4] 董春燕,伏升茂. 一类害虫流行病控制模型非常数正平衡解的存在性[J]. J4, 2011, 46(3): 80-88.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发