Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilibrium patterns with jump discontinuity. We derive conditions for stability of stationary solutions with jump discontinuity in a suitable topology which allows us to include the discontinuity points and leads to the definition of (ε0, A)-stability. Additionally, we provide conditions on stability of patterns in a quasi-stationary model reduction. The analysis is illustrated on the example of three-component model of receptor-ligand binding. The proposed model provides an example of a mechanism of de novo formation of far from the equilibrium patterns in reaction-diffusion-ODE models involving co-existence of DDI and hysteresis.

Original languageEnglish
Pages (from-to)757-800
Number of pages44
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number2
Publication statusPublished - 2017 Feb
Externally publishedYes


  • Discontinuous patterns
  • Hysteresis
  • Pattern formation
  • Quasi-steady state approximation
  • Reaction-diffusion-ODE systems
  • Turing instability

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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