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

This work was undertaken in the framework of German-Japanese University Partnership Program (HeKKSaGOn Alliance). The first two authors were supported by European Research Council Starting Grant No 210680 'Multiscale mathematical modelling of dynamics of structure formation in cell systems' and Emmy Noether Program of DFG. The second author was supported by 'Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences' and 'Baden-Württemberg Stipendium plus' scholarship of Baden-Württemberg Stiftung. The third author was supported in part by JSPS Grant-in-Aid for Scientific Research (A) #22244010 'Theory of Differential Equations Applied to Biological Pattern Formation|from Analysis to Synthesis' and #26610027 'Control of Patterns by Multi-component Reaction-Diffusion Systems of Degenerate Type'.
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  • 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 $(\varepsilon_0, A)$-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.

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


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  • Figure 2.1.  Illustration of the topology applied to problem (1.1) for scalar $u$. Model (1.1) exhibits steady states with jump discontinuity and global existence of classical solutions. $\tilde{u}$ represents a steady state while $u(t,x)$ represents a solution for some $t$

    Figure 3.1.  Plot of the nullclines of $f_r(u,v)=-(1+v)u+m_1(u^2/(1+ku^2))$ for $u\neq 0$ and $g_r(u,v)=-(\mu_3+u)v+m_2 (u^2/(1+ku^2))$.

    Figure 3.2.  Numerically obtained solution to model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=1$. We observe convergence towards a steady state with jump discontinuity. Left: Non-diffusive component $u$. Right: Diffusive component $v$.

    Figure 3.3.  Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01$ with varying diffusion coefficient: upper left: $D=5$, upper right: $D=1$, lower left: $D=0.5$, lower right: $D=0.1$. We observe emergence of more jump-type discontinuities for smaller diffusion coefficient. Initial conditions are $u_0(x)=1.725-0.1\cos(2 \pi x^2), v_0(x)=2.48615$

    Figure 3.4.  Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=5$. We observe emergence of jump-type discontinuities around local maxima of the initial conditions. Initial conditions are $u_0(x)=1.725-0.1x^4\cos(8 \pi x^2), v_0(x)=2.48615$

    Figure 5.1.  Nullclines of $f_r$ and $g_r$ and $v_{f_r,g_r}$ for parameters $m_1 = 1.44, m_2 = 2, \mu_3 = 4.2, k=0.1$

    Figure 5.2.  Illustration of the right-hand side of $-\partial^2 v/\partial x^2 = g_r(u,v)$ for different branches of the solution $u(v)$ of $u_t=f_r(u,v)=0$. The parameters for illustration are $D=1,m_1=1.44,m_2=2,\mu_3=4.2$. We can observe that all nontrivial homogeneous steady states are of type $u_-$

    Figure 5.3.  Illustration of the construction of weak steady states in the proof of Lemma 3.6

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