JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (1): 88-97.doi: 10.6040/j.issn.1671-9352.0.2016.286

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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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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

CLC Number: 

  • O175.21
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