Stability of Localized Wave Fronts in Bistable Systems

Localized wave fronts are a fundamental feature of biological systems from cell biology to ecology. Here, we study a broad class of bistable models subject to self-activation, degradation, and spatially inhomogeneous activating agents. We determine the conditions under which wave-front localization is possible and analyze the stability thereof with respect to extrinsic perturbations and internal noise. It is found that stability is enhanced upon regulating a positional signal and, surprisingly, also for a low degree of binding cooperativity. We further show a contrasting impact of self-activation to the stability of these two sources of destabilization.

Figure 1

(a) Two types of gradients: exponential decay (dashed line) and a sigmoidal profile ensuing from regulating an exponentially decaying input (solid line). (b) The potential for different values of the front position $q$. The sliding ball analogy states that the front localizes where the two maximum values of the potential are equal. (c) Sketch of the bifurcation diagram and traveling wave solution of Eq. (1). Blue (dark gray) lines denote stable solutions whereas the dashed (red) line corresponds to the unstable branch. Wave fronts (black lines and shaded area) penetrating the bistable region slow down and eventually come to rest at a stable fixed point of the front dynamics. (d) Phase diagrams of possible parameter values allowing wave localization. The parameter range of wave localization increases with the Hill coefficient $n$.

Figure 2

Stability $\xi \sigma$, normalized to the steepness of the external profile, for (a) exponential and (b) sigmoidal [$m=5$] external profiles; the Hill coefficient for self-activation is $n=5$. Stability increases from blue to red (grayscale is from dark gray over light gray to medium gray): values of $\xi \sigma$ on lines of equal stability are indicated in the graph. While in both cases stability is optimized for weak self-activation $r$, they differ in the spatial position of the localized front as measured by the value of $M_0$. (c) For sigmoidal profiles, small Hill coefficients $n$ for self-activation are optimal for front stability. Parameters for plots (a)–(c) were $\xi =100$ and $k=0.2$. (d) To study if regulation of an exponential signal is biologically beneficial, we determined the optimal stabilities that can be achieved for a front localized at a specific position. For each $q_0$, there are parameters $r$, $k$, and $M_0$ such that the linear stability $\sigma$ is maximal. Parameters were $n=5$, $2\le r\le 6$, $0.1\le k\le 1$. The plot shows the corresponding optimal values for $\sigma$ for exponential (dashed line) and sigmoidal external profiles (solid line, $m$ indicated in the graph). Sigmoidal gradients are generally more stable and in addition allow stable localization of fronts a significant distance from the gradients source at $x=0$.