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Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media

  • * Corresponding author: Conghui Zhang

    * Corresponding author: Conghui Zhang

Currently, Mathematical Institute, Tohoku University, Sendai, 980-8578.
Dedicated to the memory of the late Professor Yuzo Hosono

This work is supported in part by JSPS Kakenhi, Grant Numbers 16KT0128 and 19K03557; CHZ is sponsored by the China Scholarship Council
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  • This paper is concerned with the existence and stability of steady states of a reaction-diffusion-ODE system arising from the theory of biological pattern formation. We are interested in spontaneous emergence of patterns from spatially heterogeneous environments, hence assume that all coefficients in the equations can depend on the spatial variable. We give some sufficient conditions on the coefficients which guarantee the existence of far-from-the-equilibrium patterns with jump discontinuity and then verify their stability in a weak sense. Our conditions cover the case where the number of equilibria of the kinetic system (i.e., without diffusion) changes from one to three in the spatial interval, which is not obtained by a small perturbation of constant coefficients. Moreover, we consider the asymptotic behavior of steady states as the diffusion coefficient tends to infinity. Some examples and numerical simulations are given to illustrate the theoretical results.

    Mathematics Subject Classification: Primary: 35B36, 35K57; Secondary: 35B35, 35J25.


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  • Figure 2.  Nullclines $ f(u,v) = 0 $ and $ g(u,v) = 0 $. The red curve represents $ f(u,v) = 0 $, while the blue curve represents $ g(u,v) = 0. $

    Figure 1.  The relationship between $ X_{i,\beta} $ and $ Y_{i,\beta} $. Cases (a) and (b) are exclusive each other; and cases (c) and (d) are exclusive each other. Theorem 3.2 treats cases (a) and (c); Theorem 3.1 (i) deals with case (a); Theorem 3.1 (ii) deals with case (c) and Theorem 3.3 treats cases (b) and (d)

    Figure 3.  Pattern formation in Example 1. The red curve represents receptor; the blue curve represents ligand; light blue is $ \mu_{2} $; green is $ m_{1}(x) $ and brown is $ m_{2}(x) $

    Figure 4.  Pattern formation in Example 2. The red curve represents receptor; the blue curve represents ligand; light blue is $ \mu_{2}(x) $ and brown is $ m_{2}(x) $

    Figure 5.  Pattern formation in Example 3. The red curve represents receptor; the blue curve represents ligand; light blue is $ \mu_{2}(x) $; green is $ m_{1}(x) $ and brown is $ m_{2}(x) $

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