Applications of Mathematics

, Volume 64, Issue 1, pp 61–73 | Cite as

Dynamics and patterns of an activator-inhibitor model with cubic polynomial source

  • Yanqiu LiEmail author
  • Juncheng Jiang


The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns.


activator-inhibitor model cubic polynomial source Turing pattern global stability weakly linear coupling 

MSC 2010

35B32 35B35 35B40 92C15 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2019

Authors and Affiliations

  1. 1.Nanjing University of TechnologyNanjing, JiangsuChina

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