Hiroshima Mathematical Journal

Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis

Ying Li, Anna Marciniak-Czochra, Izumi Takagi, and Boying Wu

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This paper is devoted to the existence and (in)stability of nonconstant steady-states in a system of a semilinear parabolic equation coupled to an ODE, which is a simplified version of a receptor-ligand model of pattern formation. In the neighborhood of a constant steady-state, we construct spatially heterogeneous steady-states by applying the bifurcation theory. We also study the structure of the spectrum of the linearized operator and show that bifurcating steady-states are unstable against high wave number disturbances. In addition, we consider the global behavior of the bifurcating branches of nonconstant steady-states. These are quite different from classical reaction-diffusion systems where all species diffuse.

Article information

Hiroshima Math. J., Volume 47, Number 2 (2017), 217-247.

Received: 5 July 2016
Revised: 12 January 2017
First available in Project Euclid: 7 July 2017

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Zentralblatt MATH identifier

Primary: 35B36: Pattern formation 35K57: Reaction-diffusion equations 35B35: Stability

reaction-diffusion-ODE system pattern formation bifurcation analysis steady-states global behavior of solution branches instability


Li, Ying; Marciniak-Czochra, Anna; Takagi, Izumi; Wu, Boying. Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis. Hiroshima Math. J. 47 (2017), no. 2, 217--247. doi:10.32917/hmj/1499392826. https://projecteuclid.org/euclid.hmj/1499392826

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