Single Transition Layer in Mass-Conserving Reaction-Diffusion Systems with Bistable Nonlinearity

TL;DR

This work proves the existence and stability of stationary solutions with a single internal transition layer in mass-conserving reaction-diffusion systems with bistable nonlinearity, under general assumptions and Neumann boundary conditions. The authors combine matched asymptotics and the implicit-function theorem to construct solutions in which the $u$-component forms a jump at a layer $x^*(\varepsilon)$ and the $v$-component exhibits a smaller, $O(\varepsilon^2)$-scale transition, with the layer position and conserved mass linked by the conservation constraint. A key finding is that stability is governed by the derivative $J'(v^*)$ of the integral $J(v)=\int_{h^-(v)}^{h^+(v)} f(u,v)\,du$; specifically, stable layers occur when $J'(v^*)>0$, while instability is tied to $J'(v^*)<0$ and manifests as a separatrix between basins of attraction. The paper also provides numerical demonstrations of both stable and unstable regimes and discusses implications for cell polarity and wave-pinning phenomena, with remarks on extensions to higher dimensions and perturbed, non-conservative systems.

Abstract

Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of stationary solutions with a single internal transition layer in such reaction-diffusion systems under general assumptions by the singular perturbation theory. Moreover, we present a meaningful model for understanding the existence of an unstable transition layer solution; our numerical simulations show that the unstable solution is a separatrix of the dynamics of the model.

Figures (5)

• Figure 1: Schematic pictures of bistable nonlinearity and stationary solutions with a single internal transition layer at the leading order. (a) The graph of $f(u,v) = 0$ separates the $(u,v)$-plane into the upper-left and lower-right regions of $f > 0$ and $f<0$, respectively. Solid points on the graph indicate bistable equilibria of the ODE $u_t = f(u, v)$ in $u$ for $v = v^*$. (b) The red and blue solid lines show the profiles of $U^*(x)$ and $V^*(x)$, respectively. The dotted line shows the profile of $u^\varepsilon(x)$ around $x = x^*$, which jumps up from $h^-(v^*)$ to $h^+(v^*)$ at $x = x^*$ given by \ref{['a7']}.
• Figure 2: Stable stationary solutions with a single internal transition layer when the bistable nonlinearity is given by the cubic function (\ref{['cubic']}) with $\alpha = 0.2$. The value of the conserved mass $\xi$ is given by $\xi = 0.35$. Panels (a) and (b) show the graphs of $f(u,v) =0$ for $\beta = 0.5$ and $\beta = 0.3$, respectively. Panels (c) and (d) show the profiles of single transition layer solutions for $\beta = 0.5$ and $\beta = 0.3$, respectively, where the red and blue solid lines indicate the $u$- and $v$-components, respectively. The upper panels of (c) and (d) show that the $v$-components of the stationary solutions appear to be spatially homogeneous. However, the lower panels, enlarged views of the $v$-components show that they exhibit a single internal transition layer at $x= x^*$. The values of $v^*$ defined by $J(v) =0$ are given by $v^* = 0.0$ for (c) and $v^* \approx -0.164$ for (d), respectively, which are indicated by dotted lines. The layer positions are given by $x^* = 0.650$ for (c) and $x^* \approx 0.422$ for (d), respectively, which are indicated by broken lines. These values can be obtained by \ref{['a7']} when $\xi = 0.35$.
• Figure 3: Stable stationary solutions with a single internal transition layer when the bistable nonlinearity is given by the Hill type function \ref{['mori']}. The value of the conserved mass $\xi$ is given by $\xi = 2.3$. (a) The graph of $f(u,v) =0$ for $\kappa = 0.067$. (b) The graphs of $J (\underline{v})$ and $J (\overline{v})$, which are continuous functions in $\kappa$. (c) The profile of a single transition layer solution; the red and blue solid lines indicate the $u$- and $v$-components, respectively. The lower panel of (c), an enlarged view of the $v$-component shows that it exhibits a single internal transition layer at $x= x^*$. The value of $v^*$ defined by $J(v) = 0$ is given by $v^* \approx 1.802$. The layer position is given by $x^* \approx 0.660$, which can be obtained by \ref{['a7']} when $\xi = 2.3$.
• Figure 4: Unstable stationary solutions with a single internal transition layer when the bistable nonlinearity is given by the artificial function \ref{['multivalued']}. The value of the conserved mass $\xi$ is given by $\xi = 0.0$. Panels (a) and (b) show a curve defined by $f(u,v) =0$ and the graph of $J(v)$ defined by \ref{['a4']}, respectively. In (a), $\overline{v} = -\underline{v} \approx 1.072$. The solid points indicate the bistable equilibria of the ODE $u_t = f(u,v)$ in $u$ for $v= v^*_0$ and $v = v^*_\pm$, which are three zeros of $J(v)$ in (b). The values $v^*_0$, $v^*_+$ and $v^*_-$ are given by $v^*_0 = 0.0$, $v^*_+ \approx 0.55589$ and $v^*_- \approx -0.55589$, respectively. Panel (c) shows the profile of an unstable stationary solution with a single internal transition layer, where the red and blue solid lines indicate the $u$- and $v$-components, respectively. The lower panel of (c), an enlarged view of the $v$-component shows that it exhibits a single internal transition layer at $x = x^*$. The layer position is given by $x^* = 0.5$, which can be obtained by \ref{['a7']} when $\xi = 0.0$. Panel (d) shows the profile of the eigenfunction $\Phi^{\varepsilon} = (\varphi^\varepsilon, \psi^\varepsilon)$ associated with the critical eigenvalue $\lambda^{\varepsilon} \approx 6.2 \times 10^{-3}$ characterized by \ref{['cri']}, where the red and blue solid lines indicate $\varphi^\varepsilon$ and $\psi^\varepsilon$, respectively. The lower panel of (d), an enlarged view of the profile of $\Phi^{\varepsilon}$ is useful to understand that $\Phi^{\varepsilon}$ satisfies the constrained condition in \ref{['c2']}.
• Figure 5: Panel (a) shows that solutions starting from the unstable stationary solution in Figure \ref{['example3']}(c) with small perturbations converge to one or the other of stable transition layer solutions corresponding to $v^*_-$ and $v^*_+$ defined by $J(v) = 0$. Panels (b) and (c) show the profiles of these stable solutions when the bistable nonlinearity is given by \ref{['multivalued']}, where the red and blue solid lines indicate the $u$- and $v$-components, respectively. The lower panels of (b) and (c) are enlarged views of the profiles of the $v$-components. The value of the conserved mass $\xi$ is given by $\xi = 0.0$. The positions of the transition layers in (b) and (c) are $x^* \approx 0.164$ and $x^* \approx 0.836$, respectively, which are indicated by broken lines.

Theorems & Definitions (22)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• Proposition 2.1
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Corollary 2.1
• Remark 2.3