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Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

  • * Corresponding author: Anna Marciniak-Czochra

    * Corresponding author: Anna Marciniak-Czochra 

This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Collaborative Research Center 1324 (SFB1324, project B6). IT has been supported in part by JSPS Kakenhi, Grant Numbers 16KT0128 and 19K03557

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  • The paper is devoted to analysis of far-from-equilibrium pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.

    Mathematics Subject Classification: Primary: 35K57, 35B36; Secondary: 35J25, 35K20.


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  • Figure 1.  Typical configurations of the zero sets of the kinetic functions

    Figure 2.  (A) Phase plane of $\gamma^{-1}U_{xx}+q \bar{u}(U)=0$. The blue trajectory $(U(x),U_x(x))$ connects the points $(u_0,0)$ and $(u_e,0)$ and is a solution of the boundary value problem satisfying $U_x(0)=U_x(1)=0$. (B) A monotone increasing stationary solution with jump at $\bar{u}$ and layer position at $\bar{x}$

    Figure 3.  The time-maps for the kinetic functions $f(u,v)=1.4v-u$, $g(u,v)=u-(v^3-6.3v^2+10v)$ and the jump $\bar{u}=3.1$. Here, it holds $Q \bar{u}(u_2)<0$ and umin=0.8957. We see that Tu1 is decreasing, whereas $T_{\bar{u}}^{2}$ is increasing, which leads to $\frac{\mathrm{d}}{\mathrm{d} u_{0}} T_{\bar{u}}(u_0)<0$. Furthermore, we observe that $\lim_{u_0\to {\bar{u}}}T_{\bar{u}}{}(u_0)=0$ and $\lim_{u_0\to u_{\rm{min}}}T_{\bar{u}}{}(u_0)=\infty$

    Figure 4.  Simulations of the generic model with hysteresis for different types of perturbations of a stationary solution. The plots show initial conditions (dotted lines) and the approached stationary solution (continuous lines) after a sufficiently large time $ t_{end} $

    Figure 5.  The layer position $ \bar{x}( \bar{u}) $ and the Interval $ I^0 $. (A) and (B) Plots for the kinetic functions eq. (60) and for diffusion coefficients $ 1/\gamma $ with $ \gamma=50 $ in A and $ \gamma=200 $ in B. (C) Plots for the kinetic functions eq. (61) and the diffusion coefficient $ 1/200 $

    Figure 6.  The phase planes of $\frac{1}{\gamma} U_{x x}+q_{H}(U)=0$ and $\frac{1}{\gamma} U_{x x}+q{T}(U)=0$ are overlapping. In blue we see a periodic solution with jump at u. We cannot determine the mode of a periodic solution in the phase plane. It corresponds to how often the trajectory has been traveled through. In red we see a irregular solution with three different jumps

    Figure 7.  An irregular solution $ \big(U(x), V(x)\big) $ with three jumps $ \bar{u}^{{1}}, \bar{u}^{{2}}, \bar{u}^{{3}} $, which is monotone increasing restricted to $ [0, x^1] $. We see that for continuity of $ U(x) $ we need to have $ u_{e}^1=u_0^2 $ and $ u_{e}^2=u_0^3 $ fulfilled. Furthermore, we see how the partition of the interval is determined: $ x^1=T(\bar{u}^{{1}}, u_0^1), x^2=x^1+T(\bar{u}^{{2}}, u_0^2) $ and $ 1=x^2+T(\bar{u}^{{3}}, u_0^3) $. The layer positions are given by $ \bar{x}^{{1}}=T_1(\bar{u}^{{1}}, u_0^1) $ and $ \bar{x}^{{3}}=x^2+T_1(\bar{u}^{{3}}, u_0^3) $, because $ U(x) $ is increasing on the corresponding subintervals and $ \bar{x}^{{2}}=x^1+T_2(\bar{u}^{{2}}, u_0^2)=x^2-T_1(\bar{u}^{{2}}, u_0^3) $

    Figure 8.  Simulations of model (1)-(3) for admissible kinetic functions, diffusion coefficient $ 1/\gamma=1/1000 $ and initial conditions of type (59) having four discontinuities. The $ u $-component is plotted in blue, whereas the $ v $-component is red. The initial condition $ \big(u(0, x), v(0, x)\big)=\big(u_0(x), v_0(x)\big) $ is indicated by dotted lines and the stationary solution $ \big(u(t_{end}, x), v(t_{end}, x)\big) $ is indicated by continuous bold lines. Here, $ t_{end} $ is a sufficiently large timepoint, such that the solution $ \big(u(t, x), v(t, x)\big) $ does not change in time anymore. A-C the kinetic functions are given by (60) D the kinetic functions are given by (62)

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